Fuzzy mathematics: Approximation theory

George A. Anastassiou Fuzzy Mathematics: Approximation Theory

Studies in Fuzziness and Soft Computing, Volume 251 Editor-in-Chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw Poland E-mail: [email protected] Further volumes of this series can be found on our homepage: springer.com Vol. 235. Kofi Kissi Dompere Fuzzy Rationality, 2009 ISBN 978-3-540-88082-0 Vol. 236. Kofi Kissi Dompere Epistemic Foundations of Fuzziness, 2009 ISBN 978-3-540-88084-4 Vol. 237. Kofi Kissi Dompere Fuzziness and Approximate Reasoning, 2009 ISBN 978-3-540-88086-8 Vol. 238. Atanu Sengupta, Tapan Kumar Pal Fuzzy Preference Ordering of Interval Numbers in Decision Problems, 2009 ISBN 978-3-540-89914-3 Vol. 239. Baoding Liu Theory and Practice of Uncertain Programming, 2009 ISBN 978-3-540-89483-4 Vol. 240. Asli Celikyilmaz, I. Burhan Türksen Modeling Uncertainty with Fuzzy Logic, 2009 ISBN 978-3-540-89923-5 Vol. 241. Jacek Kluska Analytical Methods in Fuzzy Modeling and Control, 2009 ISBN 978-3-540-89926-6 Vol. 242. Yaochu Jin, Lipo Wang Fuzzy Systems in Bioinformatics and Computational Biology, 2009 ISBN 978-3-540-89967-9 Vol. 243. Rudolf Seising (Ed.) Views on Fuzzy Sets and Systems from Different Perspectives, 2009 ISBN 978-3-540-93801-9 Vol. 243. Rudolf Seising (Ed.) Views on Fuzzy Sets and Systems from Different Perspectives, 2009 ISBN 978-3-540-93801-9

Vol. 244. Xiaodong Liu and Witold Pedrycz Axiomatic Fuzzy Set Theory and Its Applications, 2009 ISBN 978-3-642-00401-8 Vol. 245. Xuzhu Wang, Da Ruan, Etienne E. Kerre Mathematics of Fuzziness – Basic Issues, 2009 ISBN 978-3-540-78310-7 Vol. 246. Piedad Brox, Iluminada Castillo, Santiago Sánchez Solano Fuzzy Logic-Based Algorithms for Video De-Interlacing, 2010 ISBN 978-3-642-10694-1 Vol. 247. Michael Glykas Fuzzy Cognitive Maps, 2010 ISBN 978-3-642-03219-6 Vol. 248. Bing-Yuan Cao Optimal Models and Methods with Fuzzy Quantities, 2010 ISBN 978-3-642-10710-8 Vol. 249. Bernadette Bouchon-Meunier, Luis Magdalena, Manuel Ojeda-Aciego, José-Luis Verdegay, Ronald R. Yager (Eds.) Foundations of Reasoning under Uncertainty, 2010 ISBN 978-3-642-10726-9 Vol. 250. Xiaoxia Huang Portfolio Analysis, 2010 ISBN 978-3-642-11213-3 Vol. 251. George A. Anastassiou Fuzzy Mathematics: Approximation Theory, 2010 ISBN 978-3-642-11219-5

George A. Anastassiou

Fuzzy Mathematics: Approximation Theory

ABC

Author George A. Anastassiou,Ph.D Professor of Mathematics The University of Memphis Department of Mathematical Sciences TN 38152 Memphis USA E-mail: [email protected]

ISBN 978-3-642-11219-5

e-ISBN 978-3-642-11220-1

DOI 10.1007/978-3-642-11220-1 Studies in Fuzziness and Soft Computing

ISSN 1434-9922

Library of Congress Control Number: 2009941640 c 2010 Springer-Verlag Berlin Heidelberg  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typeset & Cover Design: Scientific Publishing Services Pvt. Ltd., Chennai, India. Printed in acid-free paper 987654321 springer.com

Dedicated to my daughters Angela and Peggy.

"No assertion is ever known with certainty ... but that does not stop us making assertions." Carneades, 214-129 BCE

"Any useful logic must concern itself with Ideas with a fringe of vagueness and a Truth that is a matter of degree." Norbert Wiener

"The facts were always fuzzy or vague or inexact ... Science treated the gray or fuzzy facts as if they were the black-white facts of math. Yet no one had put forth a single fact about the world that was 100% true or 100% false." Bart Kosko, Fuzzy Thinking, 1994, Preface.

Preface

This monograph is the …rst in Fuzzy Approximation Theory. It contains mostly the author’s research work on fuzziness of the last ten years and relies a lot on [10]-[32] and it is a natural outgrowth of them. It belongs to the broader area of Fuzzy Mathematics. Chapters are self-contained and several advanced courses can be taught out of this book. We provide lots of applications but always within the framework of Fuzzy Mathematics. In each chapter is given background and motivations. A complete list of references is provided at the end. The topics covered are very diverse. In Chapter 1 we give an extensive basic background on Fuzziness and Fuzzy Real Analysis, as well a complete description of the book. In the following Chapters 2,3 we cover in deep Fuzzy Di¤erentiation and Integration Theory, e.g. we present Fuzzy Taylor Formulae. It follows Chapter 4 on Fuzzy Ostrowski Inequalities. Then in Chapters 5, 6 we present results on classical algebraic and trigonometric polynomial Fuzzy Approximation. In Chapters 7-11 we develop completely the theory of convergence with rates of Fuzzy Positive linear operators to Fuzzy Unit operator, the so called Fuzzy Korovkin Theory. We include there the related topic of Fuzzy Global Smoothness, see Chapter 9. In Chapters 12-14 we deal with Fuzzy Wavelet type operators and their convergence with rates to Fuzzy Unit operator. In Chapters 15-16 we discuss similarly as above the Fuzzy Neural Network Operators. In Chapter 17 we deal with Fuzzy Random Korovkin type approximation theory. In Chapter 18 we deal with Fuzzy Random Neural Network approximations.

viii

Preface

In Chapters 19, 20 we present Fuzzy Korovkin type approximations in the Sense of Summability. Finally in Chapter 21 we estimate in the fuzzy sense di¤erences of Fuzzy Wavelet type operators. The monograph’s approach is Quantitative and almost all main results are given through Fuzzy inequalities, involving fuzzy moduli of continuity, that is fuzzy Jackson type inequalities. Thus all fuzzy convergences are given with rates and the proofs are constructive. The exposed theory is destined and expected to …nd applications to all aspects of Fuzziness from theoretical to practical in almost all sciences, technology and industry; in our real world we mostly perform fuzzy approximations. On the other hand our theory has its own theoretical merit and interest within the framework of Pure Mathematics. So this monograph is suitable for researchers, graduate students and seminars of theoretical and applied mathematics, computer science, statistics, engineering, etc., also suitable for all science libraries. Fuzzy set theory and applications has experienced a rapid development since its discovery by L. Zadeh in 1965, see [103], its growth and applications now cover almost all kinds of mathematics and applied sciences with great applications to real life. We mention here only a few: …nance and stock market, weather prediction, nuclear science, robotics, biomedicine, handwriting analysis, space exploration and satellites, radars, electronics, rheology, agriculture, elevators, ecology, geography and philosophy. For a much lenghtier list of applications of Fuzzy sets and Fuzzy logic see Chapter 1, Section 1.1. I would like to thank Professor Sorin Gal, University of Oradea, Romania, for introducing me into Fuzzy Sets. The …nal preparation of book took place during 2009 in Memphis, Tennessee, USA. I would like to thank my family for their dedication and love to me, which was the strongest support during the writing of the monograph. I am also indebted and thankful to my graduate student Razvan Mezei for the typing preparation of the manuscript in a short time. November 1, 2009 George A. Anastassiou Department of Mathematical Sciences The University of Memphis, TN, USA

Contents

1 INTRODUCTION 1.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Chapters description . . . . . . . . . . . . . . . . . . . . . .

1 1 4 10

2 ABOUT H-FUZZY DIFFERENTIATION 2.1 Introduction . . . . . . . . . . . . . . . . . . 2.2 Background . . . . . . . . . . . . . . . . . . 2.3 Basic Results . . . . . . . . . . . . . . . . . 2.4 Main Results . . . . . . . . . . . . . . . . .

. . . .

15 15 16 17 30

3 ON FUZZY TAYLOR FORMULAE 3.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Addendum . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 58

4 FUZZY OSTROWSKI 4.1 Introduction . . . . . 4.2 Background . . . . . 4.3 Results . . . . . . . .

65 65 66 67

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

INEQUALITIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 A FUZZY TRIGONOMETRIC APPROXIMATION THEOREM OF WEIERSTRASS-TYPE 75 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 76

x

Contents

5.3 5.4

Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . General Remarks . . . . . . . . . . . . . . . . . . . . . . . .

6 ON BEST APPROXIMATION AND JACKSON-TYPE ESTIMATES BY GENERALIZED FUZZY POLYNOMIALS 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Best approximation, trigonometric case . . . . . . . . . . . 6.3 Best approximation, algebraic case . . . . . . . . . . . . . . 6.4 Application to fuzzy interpolation . . . . . . . . . . . . . . .

77 80

83 83 84 88 96

7 BASIC FUZZY KOROVKIN THEORY 99 7.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 7.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . 102 8 FUZZY TRIGONOMETRIC KOROVKIN THEORY 105 8.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 8.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . 111 9 FUZZY GLOBAL SMOOTHNESS PRESERVATION 115 9.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 9.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . 121 10 FUZZY KOROVKIN 10.1 Background . . . . 10.2 Related Results . . 10.3 Main Results . . .

THEORY . . . . . . . . . . . . . . . . . . . . .

AND INEQUALITIES 125 . . . . . . . . . . . . . . . . 126 . . . . . . . . . . . . . . . . 132 . . . . . . . . . . . . . . . . 136

11 HIGHER ORDER FUZZY KOROVKIN THEORY ING INEQUALITIES 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Univariate Results . . . . . . . . . . . . . . . . . . . . . 11.3 Multidimensional Results . . . . . . . . . . . . . . . . . 11.4 Lp -estimates, p 1 . . . . . . . . . . . . . . . . . . . . . 11.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . .

US159 . . 159 . . 167 . . 175 . . 185 . . 188

12 FUZZY WAVELET LIKE OPERATORS 191 12.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 12.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 13 ESTIMATES TO DISTANCES BETWEEN FUZZY WAVELET LIKE OPERATORS 209 13.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 13.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

Contents

xi

14 FUZZY APPROXIMATION BY FUZZY CONVOLUTION OPERATORS 237 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 14.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 14.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 15 DEGREE OF APPROXIMATION OF FUZZY NEURAL NETWORK OPERATORS, UNIVARIATE CASE 263 15.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 15.2 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . 265 15.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . 271 16 HIGHER DEGREE OF FUZZY APPROXIMATION BY FUZZY WAVELET TYPE AND NEURAL NETWORK OPERATORS 289 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 16.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 17 FUZZY RANDOM KOROVKIN EQUALITIES 17.1 Introduction . . . . . . . . . . . . 17.2 Auxilliary Material . . . . . . . . 17.3 Main Results . . . . . . . . . . . 17.4 Application . . . . . . . . . . . .

THEOREMS AND IN309 . . . . . . . . . . . . . . . 309 . . . . . . . . . . . . . . . 315 . . . . . . . . . . . . . . . 318 . . . . . . . . . . . . . . . 343

18 FUZZY-RANDOM NEURAL NETWORK APPROXIMATION OPERATORS, UNIVARIATE CASE 347 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 18.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . 350 18.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 19 A-SUMMABILITY AND FUZZY KOROVKIN APPROXIMATION 373 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 19.2 A Fuzzy Korovkin Type Theorem . . . . . . . . . . . . . . . 374 19.3 Rate of Convergence . . . . . . . . . . . . . . . . . . . . . . 381 20 A-SUMMABILITY AND FUZZY TRIGONOMETRIC KOROVKIN APPROXIMATION 385 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 20.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 20.3 A Fuzzy Korovkin Type Theorem . . . . . . . . . . . . . . . 387 20.4 Rate of Convergence . . . . . . . . . . . . . . . . . . . . . . 394

xii

Contents

21 UNIFORM REAL AND FUZZY ESTIMATES FOR DISTANCES BETWEEN WAVELET TYPE OPERATORS AT REAL AND FUZZY ENVIRONMENT 399 21.1 Estimates for Distances of Real Wavelet type Operators . . 400 21.2 Estimates for Distances of Fuzzy Wavelet type Operators . 421 References

431

List of Symbols

439

Index

441

1 INTRODUCTION

1.1 Basics The concept of fuzziness was …rst discovered and introduced in the seminal article written by Lot… A. Zadeh in 1965, see [103]. So in our description next we follow [103]. Frequently classes of objects encountered in the real natural world do not have exactly de…ned criteria of membership. For example, the class of animals clearly includes lions, tigers, horses, birds, …sh, etc. as its members and obviously excludes objects such as trees, gases, cars, stones, houses, metals, etc. However there are objects such as star…sh, bacteria, etc. that have an ambiguous status in comparison to the class of animals. Similar ambiguity arises when we compare the number 20 to the class of real numbers much greater than zero. Clearly, "the class of real numbers much greater than zero", or "the class of beautiful women", or "the class of tall men" or "the class of smart students" are not de…ned precisely, thus they do not constitute sets of objects in the usual mathematical sense where each element of a set is 100% there. However such imprecisely considered "classes" of objects exist frequently and play an important role in every aspect of our lives,they show up a lot especially in engineering, computer science, pattern recognition, industry, etc. So the concept under consideration is the fuzzy set, which is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership function which assigns to each object a grade of membership varying from zero to one. The notions of inclusion, union, intersection, complementation,

G.A. Anastassiou: Fuzzy Mathematics: Approximation The., STUDFUZZ 251, pp. 1–14. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

2

1. INTRODUCTION

relation, convexity, etc., are extended to fuzzy sets, see [103]. So the above framework of consideration gives us a natural way of dealing with such imprecise phenomena, when classes of objects lack precise criteria of membership for their elements. Clearly the concept of fuzziness is distinctively di¤erent than the concepts of deterministic and random phenomena. So random variables cannot describe the fuzzy sets. Thus a fuzzy set A related to a space of objects X has a membership function fA (x) which assigns to each element x 2 X a number in the interval [0; 1] that represents the "grade of membership" of x in A: Hence, the closer this value to 1, the higher the grade of membership of x in A: If A is an ordinary set, then fA (x) = 1 or 0 according to if x 2 A or x 2 = A: In this last case fA = A the characteristic function on A: As another example let X = R and A be the fuzzy set of real numbers much larger than 0. Then one can de…ne fA (0) = fA (1) = 0; fA (10) = 0:02; fA (1000) = 0:96; fA (10000) = 1: Again, the membership function though it resembles to a probability measure has nothing to do with it. They have totally di¤erent properties and are based on totally di¤erent concepts. So most clearly the notion of fuzziness is totally non-stochastic. For all the details of the above see again [103]. A fuzzy set is empty if and only if its membership function is identically equal to zero on X: The fuzzy sets A and B are equal, A = B; if and only if their membership functions are identically equal on X: The complement of a fuzzy set A is a fuzzy set denoted by A0 and its membership function is de…ned by fA0 = 1 fA : Also A is contained in B; A B; if and only if fA fB ; where A; B are fuzzy sets. The union A [ B is a fuzzy set with membership function fA[B (x) = max ffA (x); fB (x)g ; x 2 X; abbreviated as fA[B = fA _ fB : The intersection A \ B is a fuzzy set with membership function fA\B (x) = min ffA (x); fB (x)g ; x 2 X; in short written as fA\B = fA ^ fB : We say the fuzzy sets A and B are disjoint i¤ they are fuzzy-empty. For the rest of de…nitions and all properties of the algebra of the fuzzy sets see [103], where were …rst given and described. L. A. Zadeh has done an extensive body of work on Fuzziness and he shaped this subject. From all the above we derive that the concept of "membership function" plays a pivotal role in the study of fuzzy sets, in fact it is the most important component describing and de…ning a fuzzy set. Therefore in the Fuzzy Mathematics usually we deal with membership functions from the sets of real or complex numbers to [0; 1]: These are

1.1 Basics

3

the real or complex fuzzy numbers. For the description of the real fuzzy numbers see Section 1.2. Based on the above in formal mathematical calculations we identify fuzzy sets with membership functions. The extensive works on the fuzzy subject by D. Dubois and H. Prade are well known and play an important role there. To Fuzzy set theory corresponds the Fuzzy logic which also was founded by Zadeh [103]. The Fuzzy logic is a generalization of classical and Boolean logic. It relaxes the Aristotelian law of the excluded middle. Fuzzy logic via the concept of "membership function" is a multivalued logic as opposed to the Aristotle’s bi-valued logic: true or false / Yes or No only. Earlier than Zadeh was Jan Lukasiewicz to propose a 3-valued logic, with the third value "Possible". Typical applications of Fuzzy logic include to control small devices, to control large systems, to learning computer programs and neural networks, to pattern recognition in speech and handwriting, etc. It has been established that fuzzy mathematics calculations are much simpler, e¢ cient and faster than numerical and other approximate calculations. In our natural world’s computations there is not complete precision, so fuzzy mathematics have and will have a major e¤ect to control complicated machines from robots to rockets which need quick computations due to quick changes. So Fuzzy logic is reasoning with the use of Fuzzy sets, in particular with the use of membership functions. Fuzziness is a measure of completeness and has no relation to uncertainty. Probability is a measure of certainty and increases to 1 as more information is provided. So they are two completely di¤erent concepts. Words themselves may be fuzzy values. Saying "airplane" or "boat" we cover all kinds of airplanes or boats. Assigning adjectives we reduce the size of fuzzy sets, e.g. "big airplane" or "small boat". So the concept of Fuzziness appears everywhere in real life including the natural languages we speak. In engineering, triangular and trapezoidal fuzzy membership functions are examples of su¢ ciently accurate and with simple implementation such functions. Let a b c d denote some characteristic numbers, we de…ne

"Triangular" : "Trapezoidal"

:

f (x; a; b; c) = max min f (x; a; b; c; d) = max min

x b

a c x ; ;0 ; a c b x a d x ; 1; ;0 : b a d c

Multi-valued fuzzy logic mathematics belong to the wider umbrella of non-classical mathematics. They include fuzzy subgroupoids and fuzzy subgroups, fuzzy …elds, fuzzy Galois theory, fuzzy topology, fuzzy geometry,

4

1. INTRODUCTION

fuzzy relations, fuzzy graphs, possibility theory, non-additive measures, fuzzy measure theory and fuzzy integrals. This monograph is the …rst one in Fuzzy Approximation Theory. The fuzzy logic and fuzzy mathematics development so far has been through three important stages: (i) the direct fuzzi…cation of most mathematical concepts during 1965-80, (ii) the development of all possible ways in fuzzy generalization process during 1980-90, (iii) the axiomatic foundation and standardization of fuzziness during 1990-2000. The applications of fuzziness are uncountable and varied. Additionally to consumer applications in Japanese electronics, auto industry in Germany, and home appliances; Fuzzy logic is applied to …nance, stock market, biomedicine, ecology, philosophy, agriculture, geography, rheology, satellite remote control, nuclear science, weather prediction, elevators, robotics and rocket science, to mention a few of them. In general fuzziness is applied to engineering and control theory very widely. Finally, we would like to emphasize that the fuzzy-neural network methodologies …nd lots of applications to learning strategies of radars, sonar signal processing and classi…cation, speech recognition, classi…cation of remotely sensed images, handwritten character recognition, semiconductor manufacturing processes, smart home appliances, hybrid fuzzy/expert systems for medical diagnosis, etc. The fuzzy literature is vast and huge and goes to all possible directions. Thus we have decided the list of references of this book to contain only articles and books that are used here. This monograph contains mainly the research work of the author in Fuzzy Approximation Theory that was conducted during the last ten years. The chapters of the book are self-contained and each chapter can be read independently from the rest.

1.2 Background We need the following De…nition 1.1 (see [53]). Let : R ! [0; 1] with the following properties. (i) is normal, i.e., 9x0 2 R; (x0 ) = 1. (ii)

( x + (1 )y) minf (x); (y)g, 8x; y 2 R, 8 2 [0; 1] ( is called a convex fuzzy subset).

(iii)

is upper semicontinuous on R, i.e., 8x0 2 R and 8" > 0, 9 neighborhood V (x0 ): (x) (x0 ) + ", 8x 2 V (x0 ).

1.2 Background

5

(iv) The set supp( ) is compact in R (where supp( ) := fx 2 R; (x) > 0g). We call a fuzzy real number. Denote the set of all with RF . E.g., Xfx0 g 2 RF , for any x0 2 R, where Xfx0 g is the characteristic function at x0 . For 0 < r 1 and 2 RF de…ne [ ]r := fx 2 R: (x) rg and [ ]0 := fx 2 R : (x) > 0g: Then it is well known that for each r 2 [0; 1], [ ]r is a closed and bounded interval of R ([67]). For u; v 2 RF and 2 R, we de…ne uniquely the sum u v and the product u by [u

v]r = [u]r + [v]r ;

u]r = [u]r ;

[

8r 2 [0; 1];

where [u]r + [v]r means the usual addition of two intervals (as subsets of R) and [u]r means the usual product between a scalar and a subset of R (see, e.g., [53]). Notice 1 u = u and it holds u v = v u, u=u . (r) (r) [u]r1 . Actually [u]r = [u ; u+ ], where If 0 r1 r2 1 then [u]r2 (r) (r) (r) (r) (r) (r) u u+ , u ; u+ 2 R, 8r 2 [0; 1]. For > 0 one has u = ( u) , respectively. De…ne D : RF RF ! R+ by D(u; v)

:

= sup maxfju r2[0;1]

=

(r)

v

(r)

(r)

v+ jg

r

r

j; ju+

(r)

sup Hausdor¤ distance ([u] ; [v] ) ; r2[0;1]

(r)

(r)

where [v]r = [v ; v+ ]; u; v 2 RF . We have that D is a metric on RF . Then (RF ; D) is a complete metric space, see [50],[53], with the properties D(u w; v w) = D(u; v); 8u; v; w 2 RF ; D(k u; k v) = jkjD(u; v); 8u; v 2 RF ; 8k 2 R; D(u v; w e) D(u; w) + D(v; e); 8u; v; w; e 2 RF : Let f; g : R ! RF be fuzzy number valued functions. The distance between f; g is de…ned by D (f; g) := sup D(f (x); g(x)): x2R

On RF we de…ne a partial order by “ ”(or " "): u; v 2 RF , u (r) (r) (r) (r) v and u+ v+ , 8r 2 [0; 1]. u v) i¤ u We mention

v (or

6

1. INTRODUCTION

Lemma 1.2 ([31] and Lemma 5.3). For any a; b 2 R : a b u 2 RF we have D(a

u; b

u)

ja

0 and any

bj D(u; o~);

where o~ 2 RF is de…ned by o~ := Xf0g . Lemma 1.3 ([31] and Lemma 5.6). (i) If we denote o~ := Xf0g , then o~ 2 RF is the neutral element with respect to , i.e., u o~ = o~ u = u, 8u 2 RF . (ii) With respect to o~, none of u 2 RF , u 6= o~ has opposite in RF . (iii) Let a; b 2 R : a b 0, and any u 2 RF , we have (a + b) u = a u b u. For general a; b 2 R, the above property is false. (iv) For any

2 R and any u; v 2 RF , we have

(v) For any ;

2 R and u 2 RF , we have

(u v) =

u

u) = (

)

(

v. u.

If we denote kukF := D(u; o~), 8u 2 RF , then k kF has the properties of a usual norm on RF , i.e., kukF ku vkF

=

0 i¤ u = o~; k ukF = j j kukF ; kukF + kvkF ; kukF kvkF D(u; v):

Notice that (RF ; ; ) is not a linear space P over R, and consequently (RF ; k kF ) is not a normed space. Here denotes the fuzzy summation. De…nition 1.4. Let a1 ; a2 ; b1 ; b2 2 R such that a1 Then we de…ne [a1 ; b1 ] + [a2 ; b2 ] = [a1 + a2 ; b1 + b2 ]: Let a; b 2 R such that a

b1 and a2

b2 . (1.1)

b and k 2 R, then we de…ne, if k 0; if k < 0;

k[a; b] = [ka; kb]; k[a; b] = [kb; ka]:

(1.2)

Here we use Lemma 1.5. Let f : [a; b] ! RF be fuzzy continuous and let g : [a; b] ! R+ be continuous. Then f (x) g(x) is fuzzy continuous function 8x 2 [a; b]. Proof. The same as of Lemma 2 ([14]), using Lemma 2 of [11].

1.2 Background

7

Let f; g : U ! RF ; U (M; d) metric space, be fuzzy real number valued functions. The distance between f; g is also de…ned by D (f; g) := sup D(f (x); g(x)): x2U

A subset K 8u; v 2 K: We need

RF is called fuzzy bounded, i¤ D(u; v)

M; M > 0;

De…nition 1.6. (see [53]). Let x; y 2 RF . If there exists a z 2 RF such that x = y z, then we call z the H-di¤ erence of x and y, denoted by z := x y. De…nition 1.7 ([53]). Let T := [x0 ; x0 + ] R, with > 0. A function f : T ! RF is H-di¤ erentiable at x 2 T if there exists an f 0 (x) 2 RF such that the limits (with respect to metric D) lim

h!0+

f (x + h) h

f (x)

;

lim

f (x)

h!0+

f (x h

h)

exist and are equal to f 0 (x). We assume that the H di¤erences f (x + h) f (x); f (x) f (x h) 2 RF in a neighborhood of x:We call f 0 the derivative or H-derivative of f at x. If f is H-di¤erentiable at any x 2 T , we call f di¤ erentiable or H-di¤ erentiable and it has H-derivative over T the function f 0 . The last de…nition was given …rst by M. Puri and D. Ralescu [93]. E x a m p l e 1.8. Let f : R+ ! RF be such that for any ; 0 it holds f ( x + y) =

f (x)

f (y);

8x; y 2 R+ :

Then the H-derivative f 0 (x) = f (1), 8x 2 R+ . Proof. By f (x + h) = f (x) f (h), that is the H-di¤erence f (x + h)

f (x) = f (h) 2 RF :

Thus

f (x + h) f (x) = f (1); h > 0: h Similarly, f (x) = f (x h) f (h), for h > 0 small, that is the H-di¤erence f (x) f (x h) = f (h) 2 RF . Hence f (x)

f (x h

h)

= f (1):

But lim D(f (1); f (1)) = 0:

h!0+

Clearly for f 0 (0) we take the right-hand side H-derivative.

8

1. INTRODUCTION

We need also a particular case of the Fuzzy Henstock integral ( (x) = 2 ) introduced in [53], De…nition 2.1, there. That is, De…nition 1.9 ([66], p. 644). Let f : [a; b] ! RF . We say that f is FuzzyRiemann integrable to I 2 RF if for any " > 0, there exists > 0 such that for any division P = f[u; v]; g of [a; b] with the norms (P ) < , we have ! X D (v u) f ( ); I < "; P

where

P

denotes the fuzzy summation. We choose to write Z b f (x)dx: I := (F R) a

We also call an f as above (F R)-integrable. We need Theorem 1.10 ([67]). If f; g : [c; d] ! RF are (F R)-integrable fuzzy functions, and , are real numbers, then Z d Z d (F R) ( f (x) g(x))dx = (F R) f (x)dx c

(F R)

Z

c

d

g(x)dx:

c

Corollary 1.11 (Corollary 13.2 of [66]). If f 2 C ([a; b]; RF ) then f is (FR) integrable on [a; b]: We use the following fundamental theorem of Fuzzy Calculus: Corollary 1.12 ([11]). If f : [a; b] ! RF has a fuzzy continuous Hderivative f 0 on [a; b], then f 0 (x) is (F R)-integrable over [a; b] and Z s f (s) = f (t) (F R) f 0 (x)dx; for any s t; s; t 2 [a; b]: t

N o t e . In Corollary 1.12 when s < t the formula is invalid! since fuzzy real numbers correspond to closed intervals etc. We mention Lemma 1.13 ([11]). If f; g : [a; b] R ! RF are fuzzy continuous functions, then the function F : [a; b] ! R+ de…ned by F (x) := D(f (x); g(x)) is continuous on [a; b], and ! Z Z b Z b b D (F R) f (x)dx; (F R) g(x)dx D(f (x); g(x))dx: a

a

a

1.2 Background

9

Lemma 1.14 ([11]). Let f : [a; b] ! RF fuzzy continuous (with respect to metric D), then D(f (x); o~) M , 8x 2 [a; b], M > 0, that is f is fuzzy bounded. Equivalently we get f (x) M M , 8x 2 [a; b]. Lemma 1.15 ([11]). Let f : [a; b] R ! RF be fuzzy continuous. Then Z x (F R) f (t)dt is a fuzzy continuous function in x 2 [a; b]: a

Lemma 1.16 ([11]). Let f : [a; b] R ! RF fuzzy continuous, r 2 N. Then the following integrals Z sr 1 Z s r 2 Z sr 1 f (sr )dsr dsr 1 ; (F R) f (sr )dsr ; (F R) a a a Z s Z s1 Z s r 2 Z sr 1 : : : ; (F R) f (sr )dsr dsr 1 ds1 ; a

a

a

a

are fuzzy continuous functions in sr sr 1 sr 2 s b.

1 ; sr 2 ; : : : ; s,

respectively. Here a

Lemma 1.17 ([17]). Let f : [a; b] ! RF have an existing H-fuzzy derivative f 0 at c 2 [a; b]. Then f is fuzzy continuous at c. We need Theorem 1.18 ([67]). Let f : [a; b] ! RF be fuzzy continuous. Then Rb (F R) a f (x)dx exists and belongs to RF , furthermore it holds "

(F R)

Clearly f

Z

#r

b

f (x)dx

a

(r)

=

"Z

b

(r)

(f )

(x)dx;

Z

b

#

(r) (f )+ (x)dx

a

a

;

8r 2 [0; 1]: (1.3)

: [a; b] ! R are continuous functions.

We also need Theorem 1.19 ([71]). Let f : [a; b] Let t 2 [a; b], 0 r 1. (Clearly (r)

[f (t)]r = (f (t)) (r)

Then (f (t))

(r)

; (f (t))+

R ! RF be H-fuzzy di¤ erentiable.

R; denoted by [f ]r = [f

(r)

(r)

; f+ ]): (1.4)

are di¤ erentiable and (r) 0

[f 0r = ((f (t)) The last can be used to …nd f 0 .

(r)

) ; (f (t))+ )0 :

(1.5)

10

1. INTRODUCTION

Here C n ([a; b]; RF ), n 1 denotes the space of n-times fuzzy continuously H-di¤erentiable functions from [a; b] R into RF . By Theorem 5.2 of [71] (here Theorem 1.19), for f 2 C n ([a; b]; RF ) we obtain (r) (i)

[f (i) (t)]r = ((f (t))

(r)

) ; ((f (t))+ )(i) ;

(1.6)

for i = 0; 1; 2; : : : ; n and in particular we have (r)

(f (i) )

= (f

(r) (i)

) ;

8r 2 [0; 1]:

(1.7)

1.3 Chapters description Here we describe our monograph’s chapters. In Chapter 2 the concept of H-fuzzy di¤erentiation is discussed thoroughly in the univariate and multivariate cases. Basic H-derivatives are calculated and then important theorems are presented on the topic, such as, the H-mean value theorem, the univariate and multivariate H-chain rules, and the interchange of the order of H-fuzzy di¤erentiation. Finally is given a multivariate H-fuzzy Taylor formula. In Chapter 3 we produce Fuzzy Taylor formulae with integral remainder in the univariate and multivariate cases, analogs of the real setting. In Chapter 4 we present optimal upper bounds for the deviation of a fuzzy continuous function from its fuzzy average over [a; b] R, error is measured in the D-fuzzy metric. The established fuzzy Ostrowski type inequalities are sharp, in fact attained by simple fuzzy real number valued functions. These inequalities are given for fuzzy Hölder and fuzzy di¤erentiable functions and these facts are re‡ected in their right-hand sides. In Chapter 5 we establish that any 2 -periodic fuzzy continuous function from R to the fuzzy number space RF , can be uniformly approximated by some fuzzy trigonometric polynomials. About Chapter 6: In [31] was proved that any 2 -periodic continuous fuzzy-number-valued function can be uniformly approximated by sequences of generalized fuzzy trigonometric polynomials, but without giving any estimate for the approximation error. In this chapter, connected to the best approximation problem we present Jackson-type estimates. For the algebraic case we also give a Jackson-type estimate, using the Szabados-type polynomials. Finally, as an application we study the convergence of fuzzy Lagrange interpolation polynomials.

1.3 Chapters description

11

In Chapter 7 we present the basic fuzzy Korovkin theorem via a fuzzy Shisha–Mond inequality given here. This determines the degree of convergence with rates of a sequence of fuzzy positive linear operators to the fuzzy unit operator. The surprising fact is that only the real case Korovkin assumptions are enough for the validity of the fuzzy Korovkin theorem, along with a natural realization condition ful…lled by the sequence of fuzzy positive linear operators. The last condition is ful…lled by almost all operators de…ned via fuzzy summation or fuzzy integration. In Chapter 8 we present the fuzzy Korovkin trigonometric theorem via a fuzzy Shisha–Mond trigonometric inequality presented here too. This determines the degree of approximation with rates of a sequence of fuzzy positive linear operators to the fuzzy unit operator. The astonishing fact is that only the real case trigonometric assumptions are enough for the validity of the fuzzy trigonometric Korovkin theorem, along with a very natural realization condition ful…lled by the sequence of fuzzy positive linear operators. The latter condition is satis…ed by almost all operators de…ned via fuzzy summation or fuzzy integration. In Chapter 9 we present the property of global smoothness preservation for fuzzy linear operators acting on spaces of fuzzy continuous functions. Basically we transfer the property of real global smoothness preservation into the fuzzy setting, via some natural realization condition ful…lled by almost all example-fuzzy linear operators. The derived inequalities involve fuzzy moduli of continuity and we give examples. In Chapter 10 we study the fuzzy positive linear operators acting on fuzzy continuous functions. We prove the fuzzy Riesz representation theorem, the fuzzy Shisha–Mond type inequalities and fuzzy Korovkin type theorems regarding the fuzzy convergence of fuzzy positive linear operators to the fuzzy unit in various cases. Special attention is paid to the study of fuzzy weak convergence of …nite positive measures to the unit Dirac measure. All convergences are with rates and are given via fuzzy inequalities involving the fuzzy modulus of continuity of the engaged fuzzy valued function. The assumptions for the Korovkin theorems are minimal and of natural realization, ful…lled by almost all example – fuzzy positive linear operators. The surprising fact is that the real Korovkin test functions assumptions carry over here in the fuzzy setting and they are the only enough to impose the conclusions of fuzzy Korovkin theorems. We give a lot of examples and applications to our theory, namely: to fuzzy Bernstein operators, to fuzzy Shepard operators, to fuzzy Szasz–Mirakjan and fuzzy Baskakov-type operators and to fuzzy convolution type operators. We work in general, basically over real normed vector space domains that are compact and convex or just convex. On the way to prove the main theorems we establish a lot of other interesting and important side results.

12

1. INTRODUCTION

In Chapter 11 is studied with rates the fuzzy uniform and Lp , p 1, convergence of a sequence of fuzzy positive linear operators to the fuzzy unit operator acting on spaces of fuzzy di¤erentiable functions. This is done quantitatively via fuzzy Korovkin type inequalities involving the fuzzy modulus of continuity of a fuzzy derivative of the engaged function. From there we deduce general fuzzy Korovkin type theorems with high rate of convergence. The surprising fact is that basic real positive linear operator simple assumptions enforce here the fuzzy convergences. At the end we give applications. The results are univariate and multivariate. The assumptions are minimal and natural ful…lled by almost all example— fuzzy positive linear operators. About Chapter 12: The basic wavelet type operators Ak , Bk , Ck , Dk , k 2 Z were studied extensively in the real case, e.g., see [9]. Here they are extended to the fuzzy setting and are de…ned similarly via a real valued scaling function. Their pointwise and uniform convergence with rates to the fuzzy unit operator I is presented. The produced Jackson type inequalities involve the fuzzy …rst modulus of continuity and usually are proved to be sharp, in fact attained. Furthermore all fuzzy wavelet like operators Ak , Bk , Ck , Dk preserve monotonicity in the fuzzy sense. Here we do not assume any kind of orthogonality condition on the scaling function ', and the operators act on fuzzy valued continuous functions over R. About Chapter 13: The basic fuzzy wavelet type operators Ak , Bk , Ck , Dk , k 2 Z were …rst introduced in [14], see also Chapter 12, where they were studied among others for their pointwise/uniform convergence with rates to the fuzzy unit operator I. Here we continue this study by estimating the fuzzy distances between these operators. We give the pointwise convergence with rates of these distances to zero. The related approximation is of higher order since we involve these higher order fuzzy derivatives of the engaged fuzzy continuous function f . The derived Jackson type inequalities involve the fuzzy (…rst) modulus of continuity. Some comparison inequalities are also given so we get better upper bounds to the distances we study. The de…ning of these operators scaling function ' is of compact support in [ a; a], a > 0 and is not assumed to be orthogonal. About Chapter 14: Here we study four sequences of naturally arising fuzzy integral operators of convolution type that are integral analogs of known fuzzy wavelet type operators, de…ned via a scaling function. Their fuzzy convergence with rates to the fuzzy unit operator is established through fuzzy inequalities involving the fuzzy modulus of continuity. Also their high order fuzzy approximation is given similarly by involving the fuzzy modulus of continuity of the N th order (N 1) H-fuzzy derivative of the engaged fuzzy number valued function. The fuzzy global smoothness preservation property of these operators is presented too.

1.3 Chapters description

13

In Chapter 15 we study the rate of convergence to the unit operator of very speci…c well described univariate Fuzzy neural network operators of Cardaliaguet–Euvrard and “Squashing” types. These Fuzzy operators arise in a very natural and common way among Fuzzy neural networks. These rates are given through Jackson type inequalities involving the Fuzzy modulus of continuity of the engaged Fuzzy valued function or its derivative in the Fuzzy sense. Also several interesting results in Fuzzy real analysis are presented to be used in the proofs of the main results. In Chapter 16 are studied in terms of fuzzy high approximation to the unit several basic sequences of fuzzy wavelet type operators and fuzzy neural network operators. These operators are fuzzy analogs of earlier studied real ones. The produced results generalize earlier real ones into the fuzzy setting. Here the high order fuzzy pointwise convergence with rates to the fuzzy unit operator is established through fuzzy inequalities involving the fuzzy modulus of continuity of the N th order (N 1) H-fuzzy derivative of the engaged fuzzy number valued function. At the end we present a related Lp result for fuzzy neural network operators. In Chapter 17 we study the fuzzy random positive linear operators acting on fuzzy random continuous functions. We establish a series of fuzzy random Shisha–Mond type inequalities of Lq -type 1 q < 1 and related fuzzy random Korovkin type theorems, regarding the fuzzy random q-mean convergence of fuzzy random positive linear operators to the fuzzy random unit operator for various cases. All convergences are with rates and are given using the above fuzzy random inequalities involving the fuzzy random modulus of continuity of the engaged fuzzy random function. The assumptions for the Korovkin theorems are minimal and of natural realization, ful…lled by almost all example fuzzy random positive linear operators. The astonishing fact is that the real Korovkin test functions assumptions are enough for the conclusions of the fuzzy random Korovkin theory. We give at the end applications. In Chapter 18 we study the rate of pointwise convergence in the qmean to the Fuzzy-Random unit operator of very precise univariate FuzzyRandom neural network operators of Cardaliaguet–Euvrard and “Squashing”types. These Fuzzy-Random operators arise in a natural and common way among Fuzzy-Random neural networks. These rates are given through Probabilistic-Jackson type inequalities involving the Fuzzy-Random modulus of continuity of the engaged Fuzzy-Random function or its Fuzzy derivatives. Also several interesting results in Fuzzy-Random Analysis are given of independent merit, which are used then in the proofs of the main results of the chapter.

14

1. INTRODUCTION

The aim of Chapter 19 is to present a fuzzy Korovkin-type approximation theorem by using a matrix summability method. We also study the rates of convergence of fuzzy positive linear operators. The aim of Chapter 20 is to present a fuzzy trigonometric Korovkintype approximation theorem by using a matrix summability method. We also study the rates of convergence of fuzzy positive linear operators in trigonometric environment. Finally about Chapter 21: The basic fuzzy wavelet type operators Ak ; Bk ; Ck ; Dk ; k 2 Z were studied in [3], [5], for their pointwise and uniform convergence with rates to the fuzzy unit operator. Also they were studied in [6], in terms of estimating their fuzzy di¤erences and giving their pointwise convergence with rates to zero. For prior related and similar study of convergence to the unit of real analogs of these wavelet type operators see [1], section II. Here in Section 1 we present the complete study of …nding uniform estimates for the distances between the real Wavelet type operators Ak ; Bk ; Ck ; Dk ; k 2 Z: Their di¤erences converge to zero with rates. This is done via elegant tight Jackson type inequalities involving the modulus of continuity of the higher order derivative of the engaged real function. Based on these real analysis results in Section 2 we establish the corresponding fuzzy results regarding uniform estimates for the fuzzy differences between the fuzzy wavelet type operators. These fuzzy di¤erences converge to zero with rates given via fuzzy Jackson type tight inequalities. The last inequalities involve the fuzzy modulus of continuity of the higher order fuzzy derivative of the engaged fuzzy function. The de…ning all these operators real scaling function is not assumed to be orthogonal and is of compact support.

2 ABOUT H-FUZZY DIFFERENTIATION

The concept of H-fuzzy di¤erentiation is discussed thoroughly in the univariate and multivariate cases. Basic H-derivatives are calculated and then important theorems are presented on the topic, such as, the H-mean value theorem, the univariate and multivariate H-chain rules, and the interchange of the order of H-fuzzy di¤erentiation. Finally is given a multivariate Hfuzzy Taylor formula. This treatment relies in [10].

2.1 Introduction Fuzziness was …rst introduced in the celebrated paper [103]. For the notion of H-fuzzy derivative see [93] and [53]. First we give some background from Fuzziness, motivation and justi…cation, necessary for the results to follow. In Propositions 2.5, 2.7, 2.8, 2.10 we calculate basic H-fuzzy derivatives. In Lemmas 1.13 and 1.14 we give results on fuzzy continuity, and in Propositions 2.13 and 2.14 we give basic properties of H-fuzzy di¤erentiation. Then come the main results. Theorem 2.15 is on H-Fuzzy Mean Value Theorem, Lemmas 2.16, 2.17 and 2.20 are auxiliary on fuzzy convergence and fuzzy continuity, Theorem 2.18 is on univariate H-fuzzy chain rule, and Theorem 2.19 is on multivariate H-fuzzy chain rule. We conclude with Theorem 2.21 on the interchange of the order of Hfuzzy di¤erentiation, and the development of a multivariate H-fuzzy Taylor formula with integral remainder, see Theorem 2.22 and Corollary 2.23. G.A. Anastassiou: Fuzzy Mathematics: Approximation The., STUDFUZZ 251, pp. 15–50. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

16

2. About H-Fuzzy Di¤erentiation

2.2 Background We need the Fuzzy Taylor formula Theorem 2.1 ([11], see also Theorem 15.14). Let T := [x0 ; x0 + ] R, with > 0. We assume that f (i) : T ! RF are H-di¤ erentiable for all i = 0; 1; : : : ; n 1, for any x 2 T . (I.e., there exist in RF the H-di¤ erences f (i) (x + h) f (i) (x), f (i) (x) f (i) (x h), i = 0; 1; : : : ; n 1 for all small h : 0 < h < . Furthermore there exist f (i+1) (x) 2 RF such that the limits in D-distance exist and f (i+1) (x) = lim+ h!0

f (i) (x + h) h

f (i) (x)

= lim+

f (i) (x)

f (i) (x h

h!0

h)

;

for all i = 0; 1; : : : ; n 1.) Also we assume that f (n) , is fuzzy continuous on T . Then for s a; s; a 2 T we obtain f (s)

f 0 (a)

= f (a)

f (n

(s 1)

a)

f 00 (a)

(s a)n 1 (n 1)!

(a)

(s

a)2 2!

Rn (a; s);

where Rn (a; s) := (F R)

Z

s

a

Z

s1

a

Z

sn

1

f (n) (sn )dsn dsn

1

: : : ds1 :

a

Here Rn (a; s) is fuzzy continuous on T as a function of s. N o t e . This formula is invalid when s < a, as it is totally based on Corollary 1.12. Next C[0; 1] stands for the class of all real-valued bounded functions f on [0; 1] such that f is left continuous for any x 2 (0; 1] and f has a right limit for any x 2 [0; 1), especially f is right continuous at 0. With the norm kf k = sup jf (x)j, C[0; 1] is a Banach space [50]. x2[0;1]

We mention Theorem 2.2 (Wu and Ma [50]). For u 2 RF , denote j : j(u) := (r) (u ; u+ ), where u = u (r) := u ; 0 r 1. Then j(RF ) is a closed convex cone with vertex 0 in C[0; 1] C[0; 1] (here C[0; 1] C[0; 1] is a Banach space with the norm de…ned by k(f; g)k := max(kf k; kgk)), and j : RF ! C[0; 1] C[0; 1] satis…es (1) for all u; v 2 RF , s

0, t

0, j(su + tv) = sj(u) + tj(v),

(2) D(u; v) = kj(u) j(v)k, i.e., j embeds RF into C[0; 1] metrically and isomorphically.

C[0; 1] iso-

2.3 Basic Results

17

We …nally mention the important connections of the H-fuzzy derivative to the Fréchet derivative. Lemma 2.3 (Wu and Ma [51]). If f : [a; b] R ! RF satis…es the condition H : for any x 2 [a; b], there exists > 0 such that the Hdi¤ erences of f (x + h) f (x), f (x) f (x h) exist for all 0 < h < , then the H-di¤ erentiability of f (x) implies the di¤ erentiability of (j f )(x) and (j f )0 (x) 2 j(RF ), where the di¤ erentiability of (j f )(x) on C[0; 1] C[0; 1] is in the Fréchet’s sense. Lemma 2.4 (Wu and Ma [51]). If (j f )(x) is Fréchet di¤ erentiable and (j f )0 (x) 2 j(RF ), then f (x) is H-di¤ erentiable, and f 0 (x) = j 1 ((j f )0 (x)). Here f : [a; b] ! RF , j : RF ! (C[0; 1])2 , and (j f ) : [a; b] ! (C[0; 1])2 .

2.3 Basic Results We present Proposition 2.5. Let F (t) := tn Then (the H-derivative)

u, t

F 0 (t) = ntn

0, n 2 N, and u 2 RF be …xed.

1

u:

(2.1)

In particular when n = 1 then F 0 (t) = u. Proof. We need to establish that F 0 (t) = F+0 (t) = F 0 (t); where F+0 (t) := lim+

(t + h)n

u h

h!0

and

u

;

(t h)n u ; h h!0 the limits are taken with respect to the D-metric. First we take care of the case t > 0, n 2. Here h is a small positive quantity approaching zero. By Lemma 4.1 (iii) of [31] we notice that ! n X n n k k n n (t + h) u=t u t h u; k F 0 (t) := lim+

tn

tn

u

k=1

where tn ;

n X n n t k

k=1

k k

h > 0:

18

2. About H-Fuzzy Di¤erentiation

That is the H-di¤erence (t + h)

n

u

t

n

u=

n X n n t k

k k

h

k=1

!

u

exists, and (t + h)n

tn

u h

u

n X n n t k

=

k k 1

h

k=1

!

u:

Then we observe that (t + h)n

lim+ D

h!0

= lim+ D h!0

u tn u ; ntn 1 h ! n X n n k k 1 t h k

u u; nt

n 1

u

k=1

!

(by Lemma 2.2 of [31]) ! n X n n k k 1 t h ntn 1 D(u; o~) lim k h!0+ k=1 ! n X n n k k 1 = lim t h D(u; o~) = 0D(u; o~) = 0: k h!0+ k=2

That is F+0 (t) = ntn

1

u;

t > 0;

n

2:

Furthermore we notice that ((t

F 0 (t) = lim+

h) + h)n

u h

h!0

We set Thus

:= t

(t

h)n

u

:

h, which for su¢ ciently small h > 0 is positive, i.e., F 0 (t) = lim

( + h)n

u h

h!0+

n

u

:

Again we have n

( + h)

u=

n

u

n X n k

n k k

k=1

where n

;

n X n k

k=1

n k k

h > 0:

h

!

u;

> 0.

2.3 Basic Results

19

That is the H-di¤erence n

( + h)

n

u

n X n k

u=

n k k

h

k=1

!

u

exists, and ( + h)n

n

u h

u

n X n k

=

n k k 1

h

k=1

!

u:

Then we observe that lim D

tn

u

h!0+

= lim+ D h!0

(t h)n h n X n k

u

; ntn

n k k 1

h

k=1

n X n k

lim+

h!0

1

n k k 1

ntn

h

u

!

u; nt 1

n 1

u

!

D(u; o~)

k=1

n X n (t + k

n 1

= lim+ n(t

h)

h!0

h)n

k k 1

h

ntn

1

D(u; o~)

k=2

= 0D(u; o~) = 0: Hence F 0 (t) = ntn

1

u, t > 0, n

F 0 (t) = ntn

1

2. That is, u;

t > 0;

n

2:

Next we treat separately the case of n = 1, t > 0 for the sake of clarity. Here lim D

(t + h)

h!0+

u h

t

u

;u

= =

u ;u h lim+ D(u; u) = 0: lim D

h

h!0+ h!0

I.e., F+0 (t) = u, t > 0, n = 1. And we see that lim D

t

u

h!0+

= lim+ D h!0

= lim+ D h!0

= lim D h!0+

(t h) u ;u h ((t h) + h) u (t h) u ;u h ( + h) u u ;u h h u ; u = lim D(u; u) = 0; h h!0+

20

2. About H-Fuzzy Di¤erentiation

where := t h > 0, for su¢ ciently small h > 0. I.e., F 0 (t) = u, t > 0, n = 1. That is F 0 (t) = u; t > 0; n = 1: At last we do the case of t = 0. Here we need to …nd hn

F+0 (0) = lim+

u

= lim+ hn

h

h!0

1

u:

h!0

For n = 1, we see that lim D(hn

1

u; u) = lim D(u; u) = 0:

h!0+

h!0+

Thus F 0 (0) = F+0 (0) = u; For n

for n = 1:

2 we see that lim D(hn

1

u; o~) = D(~ o; o~) = 0:

h!0+

Therefore F 0 (0) = F+0 (0) = o~;

for n

2:

That is F 0 (t) = ntn

1

u

is true for t = 0.

R e m a r k 2.6. Let ai , i = 1; : : :, be a sequence of real numbers all of 1 P the same sign such that ai < +1. Then i=1

n X i=1

i

!

u=

n X

(ai

u);

u 2 RF ;

i=1

by Lemma 4.1 (iii) of [31]. Since ! n n X X D ai u; ( i=1

i

u)

i=1

!

8n 2 N;

= 0;

one obtains n X

lim D

n!+1

That is

i=1

1 X i=1

ai

!

ai

!

u=

u;

n X

i=1

1 X

i=1

(ai

(ai

u)

!

u) 2 RF :

= 0:

2.3 Basic Results

21

Next we give Proposition 2.7. Let F (x) = xp u, x 0, u 2 RF , and p > 0 not an integer. Then F 0 (x) = pxp 1 u; p > 0; x > 0; (2.2) and F 0 (o) = o~;

for p > 1:

(2.3)

Proof. When p > 0 and 1 x 1 from [98], p. 232 we obtain the Binomial series, which converges absolutely (1 + x)p = 1 + px +

p(p 1) 2 x + 2!

+

p(p

1)

(p n!

n + 1)

In the last we plug in instead of x, hx for h; x > 0 and h 1 hx 1 is automatically ful…lled and x + h > 0. That is 1+

h x

p

=

xn +

:

x. Clearly

h p(p 1) h2 + + x 2! x2 p(p 1) (p n + 1) hn + + n! xn 1+p

:

And p(p 1) 2 p 2 h x + 2! p(p 1) (p n + 1) n p n + h x + : n! By x+h > x we have (x+h)p > xp > 0 and (x+h)p xp > 0. Consequently it holds p(p 1) 2 p 2 := phxp 1 + h x + 2! p(p 1) (p n + 1) n p n + h x + > 0: n! Therefore (x + h)p u xp u = u exists in RF . (x + h)p

= xp + phxp

1

+

Hence lim+ D

(x + h)p

u h

h!0

= lim+ D h!0

xp

u; pxp

h

u 1

; pxp u

1

u lim

h!0+

p(p 1) p 2 hx + 2! p(p 1) (p n + 1) n 1 p h x + n! = 0D(u; o~) = 0: = lim pxp h!0+

1

h

pxp

1

D(u; o~)

+

n

+

pxp

1

D(u; o~)

22

2. About H-Fuzzy Di¤erentiation

That is, F+0 (x) = (xp

u)0+ = pxp

1

u;

p > 0;

x > 0:

Next we evaluate in D-metric F 0 (t)

(x h)p u h h!0+ ((x h) + h)p u (x = lim+ h h!0 p ( + h)p u u ; = lim+ h h!0 =

xp

lim

u

h)p

u

where := x h > 0, for h > 0 small enough. In fact we choose h such that 2h < x, that is, h < x h = . I.e., 0 < h < . Next we apply the Binomial series for h . Thus ( + h)p

p

=

+ ph p(p

+ Clearly Hence

p(p 1) 2 p h 2! 1) (p n + 1) n h n! p 1

and ( + h)p >

+h >

:=

p 1

ph +

p(p

p

2

+

+

p n

+

: p

> 0, by p > 0. And ( + h)p

p(p 1) 2 p 2 h + 2! 1) (p n + 1) n h n!

+

p n

+

> 0:

Therefore ( + h)p

p

u

u=

u

exists in RF :

Furthermore we have ( + h)p

lim+ D

h!0

= lim+ D

h

= lim D

p

h!0

h!0+

+

p(p

= D(pxp I.e., F 0 (x) = (xp

1) 1

u

u; pxp p 1

(p n! u; pxp

u)0 = pxp

F 0 (x) = (xp

p

u h

1

1

; pxp

u

u

p(p 1) h 2! n + 1) n 1 h

+

1

1

p 2

+

p n

u; pxp

u) = 0:

u, p > 0, x > 0. That is

u)0 = pxp

1

u;

p > 0;

x > 0:

1

u

> 0.

2.3 Basic Results

23

Finally at x = 0 we get F+00 (0) = lim

h!0+

(o + h)p h

u

= lim hp

1

h!0+

u:

Hence lim D(hp

1

h!0+

I.e., F 0 (0) = (xp

u)0

x=0

u; o~) = D(~ o; o~) = 0;

p > 1:

= o~, p > 1.

It follows Proposition 2.8. Let u 2 RF be …xed. Then u)0 = ex

(ex

u;

any x 2 R:

(2.4)

Proof. We have x3 x2 + + 2! 3!

ex = 1 + x +

+

xn + n!

;

1 < x < +1:

Then ex+h = 1 + (x + h) +

(x + h)2 (x + h)3 + + 2! 3!

+

(x + h)n + n!

;

h > 0:

Consequently we get ex+h

ex

= h+

+

2xh + h2 + 2! 0P n n n x B k=1 k B +@ n!

3x2 h + 3xh2 + h3 3! 1 k k h C C+ =: : A

Here x 2 R and x + h > x. Since ex is increasing then ex+h > ex > 0 and ex+h ex > 0. I.e., > 0. Therefore the next H-di¤erence and quotient makes sense in RF , ex+h

u ex h

u

= =

h

u

1+ 0P n

2x + h 2!

B k=1 +B @

n k

xn n!

+

3x2 + 3xh + h2 3! 1

k k 1

h

C C+ A

+

u =: K

u; K > 0:

24

2. About H-Fuzzy Di¤erentiation

Thus u; ex

lim D(K

h!0+

u)

1+x+

= D(u; o~) = jex

We prove that (ex u)0+ = ex Next we evaluate

u.

ex

u

u)0 = lim

(ex By setting

ex h

u)0 = lim+

h

e

+h

e

u

;

and e

+h

> e

2 h + h2 + 2! 0P n n n B k=1 k +B @ n!

= h+

+

+h

u e h

u

= =

=:

h

ex :

: +h

e

2

3 k k

h

h + 3 h 2 + h3 3! 1 C C+ A

=:

:

u

1+ 0P n

2 +h 2!

B k=1 +B @ K

u

> 0, and e

Clearly 0 < < +1. The next make sense in RF e

xn + n!

x 2 R; u 2 RF :

u e h

h!0

Again we have + h > Furthermore it holds +h

+

h we get (ex

e

x2 + 2!

ex jD(u; o~) = 0:

h!0+

:= x

ex jD(u; o~)

lim jK

h!0+

n k

+

n k k

h

n!

u;

K > 0:

2

+ 3 h + h2 3! 9 1 1 > > = C C+ u A > > ; 3

Thus lim D(K

h!0+

= 1+x+

u; ex

u)

x2 + 2

+

lim jK

h!0+ n

x + n!

ex jD(u; o~) ex D(u; o~) = 0:

+

> 0.

2.3 Basic Results

25

We have established (ex

u)0 = ex

u;

and …nally proved (2.4). N o t e . Clearly (ex Next we need

u)(`) = ex

u, ` 2 N, u 2 RF is …xed, x 2 R.

Bernstein’s Theorem 2.9 (see [37], p. 418). Assume that f 2 C 1 on an open interval of the form (a ; b), where > 0, and suppose that f and all its derivatives are non-negative in the half-open interval [a; b). Then, for every x0 in [a; b), we have 1 X f (n) (x0 ) f (x) = (x n! n=0

x0 )n ;

if x0

x < b:

We present Proposition 2.10. Let u 2 RF be …xed, and f 2 C 1 ( "; r), " > 0, r > 0 and assume that f , f 0 , f 00 ; : : : 0 on [0; r), with f (0) = 0. Then (f (x)

u)0 = f 0 (x)

u;

for 0

x < r:

(2.5)

Clearly (f (x)

u)(`) = f (`) (x)

u;

for 0

x < r;

` 2 N:

(2.6)

E.g., f (x) = sin hx. Proof. By Bernstein’s Theorem we have f (x) = and

1 X f (n) (0) n x ; n! n=0

1 X f (n) (0) (x + h)n ; f (x + h) = n! n=0

x 2 [0; r)

and h > 0 such that x + h 2 [0; r). Since f is non-decreasing we have f (x + h) f (x) 0, and f (x + h) f (x) 0. Consequently we see that f (x + h)

f (x)

=

1 X f (n) (0) ((x + h)n n! n=0

1 X f (n) (0) = n! n=0

xn )

n X n n x k

h

k=1

Thus f (x + h) h

f (x)

1 X f (n) (0) = n! n=0

n X n n x k

k=1

!

0:

!

0:

k k

k k 1

h

26

2. About H-Fuzzy Di¤erentiation

Therefore the next makes sense in RF f (x + h)

u h

f (x)

u

1 X f (n) (0) = n! n=0

n X n n x k

k k 1

h

!

)

k=1

u:

Then 1 X f (n) (0) n! n=0

lim D

h!0+

n X n n x k

k=1 n X

1 X f (n) (0) lim+ n! h!0 n=0

1 X f (n) (0) (nxn = n! n=1

= jf 0 (x)

k=1 1

)

k k 1

h

n n x k

k k 1

h

!!

u; f 0 (x)

!

f 0 (x) D(u; o~)

)

f 0 (x) D(u; o~)

f 0 (x)jD(u; o~) = 0:

That is, (f (x) Call

:= x

u)0+ = f 0 (x)

u;

0

x < r:

h, x > 0, x > h as h ! 0+ . Clearly f (x) =

> 0. Here

1 X f (n) (0) ( + h)n ; n! n=0

and f (x Also f (x), f (x f (x)

h) f (x

h) = f ( ) =

0 and f (x) h)

=

1 X f (n) (0) n! n=0

f (x

n

:

h). Thus

1 X f (n) (0) (( + h)n n! n=0

1 X f (n) (0) = n! n=0

n X n k

n

) !

0:

!

0:

n k k

h

k=1

Furthermore f (x)

f (x h

h)

1 X f (n) (0) = n! n=0

n X n k

k=1

u

!

n k k 1

h

2.3 Basic Results

27

Consequently lim D

f (x)

u

h!0+

f (x h

h)

1 X f (n) (0) n! n=0

= lim D h!0+

1 X f (n) (0) lim n! h!0+ n=0

D(u; o~)

=

u

; f 0 (x)

n X n k

n k k 1

h

k=1

n X n k

n k k 1

h

k=1

1 X f (n) (0) (nxn n! n=1

= jf 0 (x)

u

1

)

!

!

0

u; f (x)

u

!

f 0 (x) :

f 0 (x) D(u; o~)

f 0 (x)jD(u; o~) = 0:

I.e., (f (x)

u)0 = f 0 (x)

u;

0 < x < r:

We have established (2.5). N o t e . One can do other examples of calculation of H-derivatives of basic fuzzy functions, working as above with power series over appropriate intervals. We mention Lemma 2.11. Let f; g : (a; b) R ! RF be fuzzy continuous functions. Assume that the H-di¤ erence function f g exists on (a; b). Then f g is a fuzzy continuous function on (a; b). Proof. Let xn ; x 2 (a; b) such that xn ! x, as n ! +1. We observe that D(f (xn )

g(xn ); f (x) g(x)) = D(f (xn ) g(xn ) g(xn ); g(xn ) f (x) g(x)) = D(f (xn ); g(xn ) f (x) g(x)) = D(f (xn ) g(x); g(xn ) f (x) g(x) g(x)) = D(f (xn ) g(x); g(xn ) f (x)) D(f (xn ); f (x)) + D(g(xn ); g(x)) ! 0:

Lemma 2.12. Let U be an open subset of R2 and let f; g : U ! RF be fuzzy continuous (jointly) in (x; y) 2 U . Then D(f (x; y); g(x; y)) is continuous (jointly) in (x; y). Proof. It is similar to [66], p. 644, Lemma 13.2 (ii). It goes as follows: Let U 3 zn := (xn ; yn ) ! z := (x; y), as n ! +1. We have D(f (zn ); g(zn ))

D(f (zn ); f (z)) + D(f (z); g(z)) + D(g(z); g(zn ));

28

2. About H-Fuzzy Di¤erentiation

and D(f (z); g(z))

D(f (z); f (zn )) + D(f (zn ); g(zn )) + D(g(zn ); g(z)):

Passing to the limit as n ! +1, from the continuity of f and g we obtain lim D(f (zn ); g(zn )) = D(f (z); g(z)):

n!+1

We give Proposition 2.13. Let I be an open interval of R and let f; g : I ! RF be fuzzy di¤erentiable functions with H-derivatives f 0 , g 0 . Then (f g)0 exists and (f g)0 = f 0 g 0 : (2.7) Proof. Let h ! 0+ , then by assumption := f (x + h) Hence f (x + h) =

f (x);

:= g(x + h)

f (x), g(x + h) = (f

g)(x + h) =

(f

g)(x + h)

g(x) 2 RF :

g(x). Thus (f

g)(x);

i.e., (f

g)(x) =

:

Therefore D

(f

g)(x + h) h

= D D

h

h

(f

g)(x)

; f 0 (x)

g 0 (x)

g 0 (x)

; f 0 (x) + D

h

; f 0 (x)

h

; g 0 (x)

as h ! 0+ :

! 0;

Next we set := f (x)

f (x

h);

Clearly ; 2 RF . Then f (x) = (f

:= g(x) f (x

g)(x) = (

)

g(x

h):

h), g(x) = (f

g)(x

g(x h);

i.e., (f

g)(x)

(f

g)(x

h) =

:

h). Hence

2.3 Basic Results

29

Therefore D

(f

g)(x)

= D

h

D

h

(f h

g)(x

; f 0 (x)

h)

; f 0 (x)

g 0 (x)

g 0 (x)

; f 0 (x) + D

h

; g 0 (x)

! 0;

as h ! 0+ :

That is, proving the claim. The counterpart of the above follows. Proposition 2.14. Let I be an open interval of R and let f : ! RF be H-fuzzy di¤erentiable, c 2 R. Then f )0 exists and (c

(c

f )0 = c

f 0 (x):

(2.8)

Proof. We see (c

f )(x + h) (c f )(x) ; c f 0 (x) h c f (x + h) c f (x) = D ; c f 0 (x) h

D

Here

:= f (x + h)

f (x) 2 RF , so that f (x + h) = c

I.e., c

f (x + h)

c

( )

Next let

f (x + h) = c f (x) = c

c

:= f (x)

f (x

f (x). Then

f (x):

a. Therefore

a ; c f 0 (x) h a 0 = jcjD ; f (x) ! 0; h = D

=: ( ):

c

as h ! 0+ :

h) 2 RF , so that f (x) =

c

f (x) = c

c

f (x)

c

f (x

f (x

h). Hence

h);

i.e., c

f (x

h) = c

:

Therefore D

(c

f )(x)

= D = D

c c h

(c f )(x h) ; c f 0 (x) h f (x) c f (x h) ; c f 0 (x) h ;c

f 0 (x)

= jcjD

h

; f 0 (x)

! 0; as h ! 0+ :

30

2. About H-Fuzzy Di¤erentiation

That is establishing the claim. N o t e . Linearity is true in H-fuzzy di¤erentiation, that is ( when ;

g)0 =

f

f0

g0 ;

2 R and f; g are H-fuzzy di¤erentiable.

2.4 Main Results We present the “Fuzzy Mean Value Theorem". Theorem 2.15. Let f : [a; b] ! RF be a fuzzy di¤ erentiable function on [a; b] with H-fuzzy derivative f 0 which is assumed to be fuzzy continuous. Then D(f (d); f (c)) (d c) sup D(f 0 (t); o~); (2.9) t2[c;d]

for any c; d 2 [a; b] with d

c.

Proof. By Corollary A of [11] it holds that f (c) = f (a)

(F R)

Z

c

f 0 (t)dt;

a

and f (d) = f (a)

(F R)

Z

d

f 0 (t)dt:

a

Then D(f (d); f (c))

= D f (a)

(F R)

Z

d 0

f (t)dt; f (a)

(F R)

a

= D (F R)

Z

d

f 0 (t)dt; (F R)

a

= D (F R)

Z

= D (F R)

c 0

f (t)dt d

o~ = o~

o~ = o~ for k 2 R. And (d

c) = o~

Z

c

(F R) !

f 0 (t)dt; o~

c

Clearly k

c

f 0 (t)dt

a

a

Z

Z

d

1 dt = (F R)

Z

c

Z

!

Z

c 0

f (t)dt

a

d 0

f (t)dt; (F R)

c

Z

c 0

f (t)dt

a

=: ( ):

d

(~ o

1)dt = (F R)

Z

c

!

d

o~ dt:

!

2.4 Main Results

31

Hence ( ) = D (F R)

Z

d 0

f (t)dt; (F R)

c

Z

!

d

o~ dt

c

(by Lemma 1, [11]) Z

d

D(f 0 (t); o~)dt

c) sup D(f 0 (t); o~) < +1;

(d

t2[c;d]

c

by Lemma 2 of [11]. We need Lemma 2.16. Let un ; vn ; u; v 2 RF , n 2 N. Let un ! u, vn ! v, as n ! +1. Then D(un ; vn ) ! D(u; v), as n ! +1 (i.e., D(u; v) is continuous in (u; v)). In particular D(un ; v) ! D(u; v), as n ! +1. We write lim D(un ; vn ) = D

n!+1

lim un ; lim vn

n!+1

n!+1

= D(u; v):

Lemma 2.17. Let un ; u 2 RF ; cn ; c 2 R+ , such that un ! u and cn ! c, as n ! +1. Then in D-metric un

cn ! u

c;

as n ! +1;

i.e., lim (un

n!+1

cn ) =

lim un

lim cn

n!+1

n!+1

=u

c:

Proof. We notice that D(un cn ; u c) D(un (by Lemma 2.2, [31])

cn ; un jcn

c) + D(un

c; u

c)

cjD(un ; o~) + cD(un ; u)

(by Lemma 2.16) ! 0D(u; o~) + c0 = 0: That is lim D(un

n!+1

cn ; u

c) = 0:

We present the “Univariate Fuzzy Chain Rule". Theorem 2.18. Let I be a closed interval in R. Here g : I ! := g(I) R is di¤ erentiable, and f : ! RF is H-fuzzy di¤ erentiable. Assume that g is strictly increasing. Then (f g)0 (x) exists and (f

g)0 (x) = f 0 (g(x))

Proof. Call u := g(x). Let

g 0 (x);

x > 0, such that

8x 2 I: x ! 0+ .

(2.10)

32

2. About H-Fuzzy Di¤erentiation

i) Let u := g(x + x) g(x). Then u > 0, and as x ! 0+ we get u ! 0+ by continuity of g. See that g(x + x) = u + u. We observe that lim D

f (g(x +

x!0+

=

lim D

x!0+

g(x +

=

lim D

x!0+

g(x + = D(f 0 (u)

x)) f (g(x)) 0 ; f (g(x)) g 0 (x) x f (g(x + x)) f (g(x)) g(x + x) g(x) x) g(x) ; f 0 (g(x)) g 0 (x) x f (u + u) f (u) u x) g(x) ; f 0 (g(x)) g 0 (x) x g 0 (x); f 0 (g(x)) g 0 (x)) = 0;

by Lemmas 2.16 and 2.17. I.e., (f

g)0+ = f 0 (g(x))

g 0 (x):

ii) Let u := g(x) g(x x). Then u > 0, and as u ! 0+ by continuity of g. Notice that g(x x) = u that

x ! 0+ we get u. We observe

f (g(x x)) 0 ; f (g(x)) g 0 (x) x f (g(x)) f (g(x x)) = lim + D g(x) g(x x) x!0 g(x) g(x x ; f 0 (g(x)) g 0 (x) x f (u) f (u u) = lim D u x!0+ g(x) g(x x) ; f 0 (g(x)) g 0 (x) x = D(f 0 (u) g 0 (x); f 0 (g(x)) g 0 (x)) = 0;

lim D

f (g(x))

x!0+

by Lemmas 2.16 and 2.17. I.e., (f

g)0 = f 0 (g(x))

g 0 (x):

At the endpoints of I we take one-sided derivatives. Next follows the multivariate fuzzy chain rule. Theorem 2.19. Let i : [a; b] R ! i ([a; b]) := Ii R, i = 1; : : : ; n, n 2 N, are strictly increasing and di¤ erentiable functions. Denote xi :=

2.4 Main Results

33

xi (t) := i (t), t 2 [a; b], i = 1; : : : ; n. Consider U an open subset of Rn such that ni=1 Ii U . Consider f : U ! RF a fuzzy continuous function. Assume that fxi : U ! RF , i = 1; : : : ; n, the H-fuzzy partial derivatives of f , exist and are fuzzy continuous. Call z := z(t) := f (x1 ; : : : ; xn ). Then dz dt exists and n dz X dz dxi = ; 8t 2 [a; b] (2.11) dt dxi dt i=1 dz where dz dt , dxi , i = 1; : : : ; n are the H-fuzzy derivatives of f with respect to t, xi , respectively.

Proof. Let …rst t 2 (a; b). Let a general (x1 ; x2 ; : : : ; xn ) 2 U be …xed and let xi > 0, i = 1; : : : ; n, be small. I) Call 1

:= f (x1 + x1 ; x2 + x2 ; : : : ; xn + xn ) f (x1 ; x2 + x2 ; : : : ; xn + xn ) 2 RF :

That is f (x1 + x1 ; x2 + x2 ; : : : ; xn + xn ) =

1

f (x1 ; x2 + x2 ; : : : ; xn + xn ):

Call 2

:= f (x1 ; x2 + x2 ; : : : ; xn + xn ) f (x1 ; x2 ; x3 + x3 ; : : : ; xn +

xn ) 2 RF :

That is f (x1 ; x2 +

x2 ; : : : ; xn +

xn ) =

f (x1 ; x2 ; x3 +

2

x3 ; : : : ; xn +

xn ):

Call 3

:= f (x1 ; x2 ; x3 + x3 ; : : : ; xn + xn ) f (x1 ; x2 ; x3 ; x4 + x4 ; : : : ; xn +

xn ) 2 RF :

That is f (x1 ; x2 ; x3 + x3 ; : : : ; xn + xn ) =

3

f (x1 ; x2 ; x3 ; x4 + x4 ; : : : ; xn + xn ):

Etc. Call an := f (x1 ; x2 ; : : : ; xn

1 ; xn

+

xn )

f (x1 ; x2 ; : : : ; xn ) 2 RF :

That is f (x1 ; x2 ; : : : ; xn

1 ; xn

+

xn ) =

n

f (x1 ; x2 ; : : : ; xn ):

34

2. About H-Fuzzy Di¤erentiation

I.e., it holds RF 2 f (x1 +

x1 ; x2 +

x2 ; : : : ; xn +

xn )

f (x1 ; x2 ; : : : ; xn ) =

n X

i:

i=1

Since the partial derivatives fxi exist, the above H-di¤erences 1; : : : ; n exist in RF for small xi > 0. In particular we de…ne xi :=

i (t

+

t)

i (t);

t > 0;

i,

i =

i = 1; : : : n

(i.e., i (t

+

t) = xi +

xi ;

xi :=

i (t)):

Since i , i = 1; : : : ; n are strictly increasing we have that t ! 0+ , then xi ! 0+ by continuity of i . We observe that f(

lim D

1 (t

+

t); : : : ;

n (t

+

n (t))

x0i (t)

fxi (x1 ; : : : ; xn )

= lim + D

f (x1 +

x1 ; : : : ; xn +

t!0

lim + D

P

n i=1

i

t

f (x1 +

;

f (x1 ; x2 +

n X

fxi (x1 ; : : : ; xn )

x1 ; x2 +

x2 ; : : : ; xn + t

f (x1 ; x2 +

f (x1 ; x2 ; x3 +

+ lim D

!

x0i (t)

i=1

t!0

x2 ; : : : ; xn + t xn )

f (x1 ; x2 ; x3 +

f (x1 ; x2 ; x3 ; x4 +

xn )

; fx1 (x1 ; : : : ; xn )

x2 ; : : : ; xn + t

x3 ; : : : ; xn + t

t!0+

;

x0i (t)

t!0

+ lim + D

f (x1 ; : : : ; xn )

!

fxi (x1 ; : : : ; xn )

= lim + D

xn ) t

t!0

i=1

;

!

i=1

n X

f ( i (t); : : : ;

t

t!0+

n X

t))

xi > 0. So as

xn )

xn )

; fx2 (x1 ; : : : ; xn )

x3 ; : : : ; xn + t

x4 ; : : : ; xn + t

!

x01 (t)

xn )

!

x02 (t)

xn )

; fx3 (x1 ; : : : ; xn )

!

x03 (t)

2.4 Main Results

+

+ lim D

f (x1 ; x2 ; : : : ; xn

1 ; xn

+

xn )

!

f (x1 +

x1 ; x2 +

t!0

x2 ; : : : ; xn + x1

+ lim D

x2 ; : : : ; xn + xn ) x1 ! xn ) x1 ; fx1 (x1 ; : : : ; xn ) t

f (x1 ; x2 +

t!0+

f (x1 ; x2 ; x3 +

x2 ; : : : ; xn + x2 ! x3 ; : : : ; xn + xn ) x2

+

f (x1 ; x2 ; : : : ; xn

t!0

xn ; fxn (x1 ; : : : ; xn ) t

=

(by Corollary A, [11]) Rx + (F R) x11 lim + D t!0

x1

!

1 ; xn

x01 (t)

x2 ; fx2 (x1 ; : : : ; xn ) t

f (x1 ; x2 ; x3 +

+ lim + D

!

xn )

x3 ; : : : ; xn + x3 t!0 ! f (x1 ; x2 ; x3 ; x4 + x4 ; : : : ; xn + xn ) x3

+ lim + D

;

x0n (t)

fxn (x1 ; : : : ; xn )

f (x1 ; x2 +

f (x1 ; x2 ; : : : ; xn )

t

t!0+

= lim + D

35

+ xn

x02 (t)

xn ) ! x3 ; fx3 (x1 ; : : : ; xn ) x03 (t) t ! xn ) f (x1 ; : : : ; xn )

x0n (t)

fx1 (t; x2 + x1

!

x2 ; : : : ; xn +

xn )dt

!

! x1 0 ; fx1 (x1 ; : : : ; xn ) x1 (t) t ! R x + x2 (F R) x22 fx2 (x1 ; t; x3 + x3 ; : : : ; xn + xn )dt + lim + D x2 t!0 ! x2 ; fx2 (x1 ; : : : ; xn ) x02 (t) t ! R x + x3 (F R) x33 fx3 (x1 ; x2 ; t; x4 + x4 ; : : : ; xn + xn )dt + lim + D x3 t!0

36

2. About H-Fuzzy Di¤erentiation

x3 ; fx3 (x1 ; : : : ; xn ) t (F R) +

+ lim + D t!0

xn t

1

!

x03 (t) R xn

1+

xn

xn

1

fxn 1 (x1 ; : : : ; xn

1

x0n

; fxn 1 (x1 ; : : : ; xn )

!

xn

+

xn )dt

1

!

1 (t)

x0n (t); fxn (x1 ; : : : ; xn )

+ D(fxn (x1 ; : : : ; xn )

2 ; t; xn

x0n (t))

(by Lemmas 2.16 and 2.17)

=

x01 (t)

1 D (F R) x1

lim +

t!0

x1 + x02 (t)

t!0+

xn )dt;

lim +

t!0

xn )dt;

1 (t) lim

=

i=1

x0i (t)

x2 ; : : : ; xn +

xn )dt;

!

t!0+

lim +

t!0

t; xi+1 +

x2

1 xn

x3

Z

x2 + x2

fx2 (x1 ; t; x3 +

x2

Z

fx3 (x1 ; x2 ; t; x4 +

x3

!

fx3 (x1 ; : : : ; xn )

1

Z

xn

xn

2 ; t; xn

xi+1 ; : : : ; xn +

!

x3 + x3

D (F R)

1 D (F R) xi

x3 ;

fx2 (x1 ; : : : ; xn )

1 D (F R) x3

fxn 1 (x1 ; x2 ; : : : ; xn n X1

fx1 (t; x2 +

x1

1 D (F R) x2

lim

: : : ; xn + + x0n

x1 + x1

fx1 (x1 ; : : : ; xn )

: : : ; xn + + x03 (t)

Z

Z

+

1+

x4 ;

+

xn

1

1

xn )dt; xn

1

!

fxn 1 (x1 ; : : : ; xn )

xi + xi

fxi (x1 ; x2 ; : : : ; xi

1;

xi

xn )dt; (F R)

Z

xi + xi

xi

fxi (x1 ; : : : ; xn )dt

!

2.4 Main Results

(by Lemma 1 of [11]) nX1

x0i (t)

lim +

t!0

i=1

x2 ; : : : ; xi n X1

1 ; t; xi+1

1 xi

x0i (t) lim + t!0

i=1

xi+1 +

+

sup

xi+1 ; : : : ; xn +

(as

xn ); fxi (x1 ; : : : ; xn ))dt

!

xn ); fxi (x1 ; : : : ; xn ))) 2 [xi ; xi +

i

1;

i ; xi+1

1;

;

xi

xi ]) +

xi+1 ;

xn ); fxi (x1 ; : : : ; xn )

t ! 0+ , then all =

!

(D(fxi (x1 ; x2 ; : : : ; xi

t!0+

: : : ; xn +

D(fxi (x1 ;

2[xi ;xi + xi ]

x0i (t) lim D fxi (x1 ; x2 ; : : : ; xi

i=1

xi + xi

xi

xi+1 ; : : : ; xn +

(by Lemma 1 of [11]) = (for some n X1

Z

1 xi

37

n X1

xi ! 0+ and thus

i

! xi , for all i = 1; : : : ; n)

x0i (t)D fxi (x1 ; : : : ; xn ); fxi (x1 ; : : : ; xn )

i=1

=

n X1

x0i (t) 0 = 0;

i=1

by continuity of fxi , i = 1; : : : ; n dz dt

= +

1. I.e., we have proved that n X dz dx i i=1

dxi : dt

II) Call 1

:= f (x1 ; x2 ; : : : ; xn )

f (x1 ; x2 ; : : : ; xn

1 ; xn

xn ) 2 RF :

That is f (x1 ; x2 ; : : : ; xn ) =

f (x1 ; x2 ; : : : ; xn

1

1 ; xn

xn ):

Call 2

:=

f (x1 ; x2 ; : : : ; xn 1 ; xn f (x1 ; x2 ; : : : xn 2 ; xn

xn ) 1

xn

1 ; xn

xn ) 2 RF :

That is f (x1 ; x2 ; : : : ; xn

1 ; xn

xn )

=

2

xn

f (x1 ; x2 ; : : : ; xn xn ):

2 ; xn 1

xn

1;

38

2. About H-Fuzzy Di¤erentiation

Call 3

:=

f (x1 ; x2 ; : : : ; xn 2 ; xn 1 f (x1 ; x2 ; : : : ; xn 3 ; xn 2 RF :

xn 1 ; xn xn 2 ; xn

2

xn ) xn

1

1 ; xn

xn )

That is f (x1 ; x2 ; : : : ; xn 2 ; xn 1 = 3 f (x1 ; x2 ; : : : ; xn

xn 1 ; xn xn ) xn 2 ; xn 3 ; xn 2

xn

1

1 ; xn

xn ):

Etc. Call n

:= f (x1 ; x2

x2 ; : : : ; xn

xn ) f (x1

x1 ; x2

x2 ; : : : ; xn

xn ) 2 RF :

That is f (x1 ; x2

x2 ; : : : ; xn

xn ) =

f (x1

n

x1 ; x2

x2 ; : : : ; xn

xn ):

I.e., it holds RF 3 f (x1 ; x2 ; : : : ; xn )

f (x1

x1 ; x2

x2 ; : : : ; xn

xn ) =

n X

i:

i=1

Since the partial derivatives fxi exist, the above H-di¤erences i , i = 1; : : : ; n exist in RF for small xi > 0. In particular we de…ne xi := t), t > 0, i = 1; : : : ; n (i.e., i (t t) = xi xi , i (t) i (t xi := i (t)). Since i , i = 1; : : : ; n are strictly increasing we have that xi > 0. So as t ! 0+ , then xi ! 0+ by continuity of i . We observe that f(

lim D

1 (t); : : : ;

n (t))

f(

n X

fxi (x1 ; : : : ; xn )

lim D

i=1

n (t

t))

!

f (x1 ; : : : ; xn )

f (x1

fxi (x1 ; : : : ; xn )

x1 ; : : : ; xn t

t!0+ n X

t); : : : ;

x0i (t)

i=1

=

1 (t

t

t!0+

!

x0i (t)

xn )

;

;

2.4 Main Results

=

lim D

t!0+

P

n i=1

i

t

;

X

n i=1

fxi (x1 ; x2 ; : : : ; xn )

f (x1 ; x2 ; :::; xn )

lim + D

f (x1 ; x2 ; : : : ; xn

2 ; xn 1

xn

00

1 ; xn

f (x1 ; x2 ; : : : ; xn 2 ; xn 1 BB f (x1 ; x2 ; : : : ; xn 3 ; xn B lim D B @@ t!0+ ; fxn 2 (x1 ; : : : ; xn )

=

f (x1 ; x2

lim D

t!0+

;

xn ) xn )

t

f (x1

xn )

t!0+

f (x1 ; x2 ; : : : ; xn 1 ; xn t

D

1 ; xn

x0n (t)) + lim

fxn (x1 ; :::; xn ) D

!

x0i (t)

f (x1 ; x2 ; :::; xn xn

t!0

39

!

x0n 2 (t)

x2 ; : : : ; xn t

x1 ; x2

+

1 (t)

1

xn ) 1

xn

xn ) C C A

1 ; xn

+ lim + t!0

xn )

x2 ; : : : ; xn t

f (x1 ; x2 ; :::; xn )

xn 1 ; xn xn 2 ; xn t

2

x0n

; fxn 1 (x1 ; : : : ; xn )

!

xn )

; fx1 (x1 ; : : : ; xn )

f (x1 ; x2 ; :::; xn xn

1 ; xn

!

x01 (t)

xn

xn ; fxn (x1 ; :::; xn ) x0n (t) + lim + t t!0 f (x1 ; x2 ; :::; xn 1 ; xn xn ) f (x1 ; x2 ; :::; xn 2 ; xn 1 D xn 1 xn 1 ; fxn 1 (x1 ; :::; xn ) x0n 1 (t) + lim + t t!0 00 f (x1 ; x2 ; : : : ; xn 2 ; xn 1 xn 1 ; xn xn ) BB f (x1 ; x2 ; : : : ; xn 3 ; xn 2 xn 2 ; xn 1 xn 1 ; xn B DB @@ xn 2

xn

1 ; xn

1

xn ) C C A

xn

+

40

2. About H-Fuzzy Di¤erentiation

xn 2 ; fxn 2 (x1 ; :::; xn ) x0n 2 (t) + lim t t!0+ f (x1 ; x2 x2 ; :::; xn xn ) f (x1 x1 ; x2 x1 x1 ; fx1 (x1 ; :::; xn ) x01 (t) t

D

x2 ; :::; xn

xn

(by Corollary A, [11]) =

x0n (t); fxn (x1 ; : : : ; xn ) x0n (t) R xn 1 (F R) xn 1 xn 1 fxn 1 (x1 ; x2 ; : : : ; xn

D fxn (x1 ; : : : ; xn ) + lim + D t!0

xn t (F R) D xn t

D

1

; fxn 1 (x1 ; : : : ; xn )

R xn xn

2

2 2

xn

2

R x1 x1

1 (t)

x1

x0n

fx1 (t; x2

x1 ; fx1 (x1 ; : : : ; xn ) t (by Lemmas 2.16, 2.17) 0 = xn

1

3 ; t; xn 1

+

xn

t!0+

xn )dt

!

!

x01 (t) 1 (t)

lim

t!0+

1 ; xn

+ lim

x2 ; : : : ; xn x1

!

t!0

xn 2 !

2 (t)

xn )dt

+ lim +

fxn 2 (x1 ; x2 ; : : : ; xn

; fxn 2 (x1 ; : : : ; xn )

(F R)

x0n

xn !

2 ; t; xn

1 xn

D (F R) 1

xn )dt

!

2.4 Main Results Z

xn 1

fxn xn 1

xn 1

xn

fxn

+ x0n

1

1

1

t!0+

t; xn

1

+

+ x01 (t) lim

2

1 ; xn

xn

2

(x1 ; x2 ; : : : ; xn

xn 2

fxn

2

x2 ; : : : ; xn

1 D (F R) xi

xi+1 ; : : : ; xn

Z

1

xi

2

3;

!

(x1 ; : : : ; xn )

xn )dt;

!

fx1 (x1 ; : : : ; xn )

x1

xi

fxi x1 ; x2 ; : : : ; xi xi

xi

xn !

1 ; xn

1;

xn ) dt;

fxi (x1 ; : : : ; xn )dt xi

xi

x0i (t) lim

t!0+

i=1

xi+1

xi+1 ; : : : ; xn

t!0+

: : : ; xn

1

xn

(for some

i

(by Lemma 1 of [11]) =

xn

1

1 xi

x0i (t) lim

i=1

i ; xi+1

=

n X1

=

i=1

xi

D(fxi (x1 ; x2 ; : : : ; xi xi

xi

!

D(fxi (x1 ; x2 ; : : : ; xi

xn ); fxi (x1 ; : : : ; xn ))

2 [xi

xi ; xi ])

n X1

!

t!0+

xn

1 ; xn

xi ! 0+ and thus

i

1;

1;

xn ); fxi (x1 ; : : : ; xn )

! xi , for all i = 1; : : : ; n)

x0i (t)D fxi (x1 ; : : : ; xn ); fxi (x1 ; : : : ; xn ) x0i (t) 0 = 0;

; xi+1

xi

x0i (t) lim D fxi (x1 ; x2 ; : : : ; xi 1

1 ; t;

xn ); fxi (x1 ; : : : ; xn ))dt

xi ;xi ]

i=1

n X1

Z

1 ; xn

xi+1 ; : : : ; xn

t ! 0+ , then all

1 xi

1 ; xn

sup 2[xi

i=1

(as

fxn xn 2

1 x1

(by Lemma 1 of [11]) nX1

n X1

xn 2

x1

t!0+

t; xi+1 Z (F R)

Z

xn )dt;

fx1 (t; x2

x0i (t) lim

i=1

xn )dt;

x1

x1

n X1

D (F R)

t!0+

Z

2 ; t; xn

(x1 ; : : : ; xn ) 1 xn

xn

(x1 ; x2 ; : : : ; xn !

2 (t) lim

D (F R)

=

41

xi+1 ;

42

2. About H-Fuzzy Di¤erentiation

by continuity of fxi , i = 1; : : : ; n dz dt

1. I.e., we have proved that

=

n X dz dx i i=1

dxi : dt

dz dz , respectively. Clearly When t = a, or b, then dz dt equals dt + , or dt here dxi dxi dxi dxi = ; and = : dt t=a dt + t=a dt t=b dt t=b

Etc., the same proof as before. The theorem basically is proved. A further explanation follows. If t ! 0+ , then all xi ! 0+ ; i = 1; : : : ; n: We notice that R x + x1 (F R) x11 fx1 (t; x2 + x2 ; : : : ; xn + lim + D x1 t!0 ! R x1 + x1 (F R) x1 fx1 (t; x2 ; : : : ; xn )dt

xn )dt

;

x1

= (i.e.

x1

x1 + x1

1 D (fx1 (t; x2 + x2 ; : : : ; xn + xn ); x1 x1 t!0 fx1 (t; x2 ; : : : ; xn )) dt (the integrand just above is continuous in t) 1 lim D (fx1 (t ; x2 + x2 ; : : : ; xn + xn ); x1 t!0+ fx1 (t ; x2 ; : : : ; xn )) x1 (t in [x1 ; x1 + x1 ]) lim D (fx1 (t ; x2 + x2 ; : : : ; xn + xn ); fx1 (t ; x2 ; : : : ; xn )) lim +

=

Z

t!0+

! 0; and t ! x1 ) = D (fx1 (x1 ; x2 ; : : : ; xn ); fx1 (x1 ; x2 ; : : : ; xn )) = 0; by continuity of fx1 ; etc.

We further see that f (x1 +

x1 ; x2 ; : : : ; xn ) f (x1 ; x2 ; : : : ; xn ) =

Z

x1 + x1

fx1 (t; x2 ; : : : ; xn ) dt:

x1

That is f (x1 +

x1 ; x2 ; : : : ; xn ) x1

f (x1 ; x2 ; : : : ; xn )

=

R x1 + x1

x1

fx1 (t; x2 ; : : : ; xn ) dt x1

:

2.4 Main Results

43

Also it holds f (x1 +

lim + D

x1 ; x2 ; : : : ; xn ) x1

t!0

f (x1 ; x2 ; : : : ; xn )

; fx1 (x1 ; x2 ; : : : ; xn )

= 0: Therefore (F R)

lim D

t!0+

R x1 +

x1

x1

fx1 (x1 ; x2 ; : : : ; xn )) Rx + (F R) x11 lim D

fx1 (t; x2 +

R x1 +

x1

x1

x1

fx1 (t; x2 +

t!0+

=

fx1 (t; x2 ; : : : ; xn )dt

x1 Rx + (F R) x11

+ lim D

xn )dt

x2 ; : : : ; xn +

xn )dt

x1

t!0+

(F R)

x2 ; : : : ; xn +

x1

x1 !

fx1 (t; x2 ; : : : ; xn )dt x1

;

;

!

; fx1 (x1 ; x2 ; : : : ; xn )

0:

Therefore we get R (F R)

x1 + x1

x1

fx1 (t;x2 + x2 ;:::;xn + xn )dt x1

D

as

!

fx1 (x1 ; x2 ; : : : ; xn ); etc.

t!0+

The proof of the theorem now is clear and completed. We need Lemma 2.20. Let f be a fuzzy continuous function from the open set (r) U Rn , n 2 N, into RF . Then f are continuous functions from U into R, for all r 2 [0; 1]. Proof. Let xm ; x 2 U , m 2 N, be such that xm ! x as m ! +1. Then by continuity of f we get D(f (xm ); f (x)) ! 0, as m ! +1. Hence we have (r)

(r)

D(f (xm ); f (x)) = sup maxfj(f (xm ))

(f (x))

r2[0;1] (r)

j(f (xm ))+ (r)

(r)

j;

(r)

(f (x))+ jg ! 0: (r)

(r)

Therefore j(f (xm )) (f (x)) j ! 0 and j(f (xm ))+ (f (x))+ j ! 0, as (r) (r) m ! +1, for all r 2 [0; 1]. Consequently (f (xm )) ! (f (x)) , proving (r) that f 2 C(U; R), for all 0 r 1. We present the interchange of the order of H-fuzzy di¤erentiation.

44

2. About H-Fuzzy Di¤erentiation

Theorem 2.21. Let U be an open subset of Rn , n 2 N, and f : U ! RF be a fuzzy continuous function. Assume that all H-fuzzy partial derivatives of f up to order m 2 N exist and are fuzzy continuous. Let x := (x1 ; : : : ; xn ) 2 U . Then the H-fuzzy mixed partial derivative of order k, Dx`1 ;:::;x`k f (x) is unchanged when the indices `1 ; : : : ; `k are permuted. Each `i is a positive integer n. Here some or all of `i ’s can be equal. Also k = 2; : : : ; m and there are nk partials of order k. Proof. We only need to demonstrate the proof for the case n = k = 2. The rest is true by induction on k, and similarly true for n > 2. So here 2 2 @2f @2f , @y@x exist and are fuzzy z = f (x; y) : U R2 ! RF and @@xf2 , @@yf2 , @x@y continuous functions from U into RF . We make use of Theorem 5.2 from [71] repeatedly. Here we have (r)

[f (x; y)]r = (f (x; y))

By that theorem and the above assumptions r

@ f (x; y) @x for all 0 r exist and

r

0

r

(r) @ @x (f (x; y))

1: exist and

@ (r) @ (r) (f (x; y)) ; (f (x; y))+ ; @x @x

1 and all (x; y) 2 U . Furthermore, the same way

@2 f (x; y) @y@x for all 0

=

(r)

; (f (x; y))+ ;

r

=

(r) @2 @y@x (f (x; y))

@2 @2 (r) (r) (f (x; y)) ; (f (x; y))+ ; @y@x @y@x

1 and all (x; y) 2 U . Similarly we obtain

@2 f (x; y) @x@y

r

=

@2 @2 (r) (r) (f (x; y)) ; (f (x; y))+ ; @x@y @x@y

for all 0 r 1 and all (x; y) 2 U . Clearly it also holds that @2 f (x; y) @x2

r

@2 f (x; y) @y 2

r

and

for all 0

r

=

2 @2 (r) @ (r) (f (x; y)) ; (f (x; y))+ ; @x2 @x2

=

2 @2 (r) @ (r) (f (x; y)) ; 2 (f (x; y))+ ; 2 @y @y

1 and all (x; y) 2 U . By Lemma 2.20 we …nd that

2 @2 @2 @2 (r) @ (r) (r) (r) ; (f (x; y)) ; (f (x; y)) ; (f (x; y)) (f (x; y)) @x2 @y 2 @x@y @y@x

2.4 Main Results

45

are all continuous for any r 2 [0; 1]. But by basic real analysis, Theorem 6-20, p. 121 of [37] we have @2 @2 (r) (r) (f (x; y)) = (f (x; y)) ; @x@y @y@x for any r 2 [0; 1]. Thus we get @2 f (x; y) @x@y

r

=

@2 f (x; y) @y@x

r

;

for all 0 r 1. That is the H-fuzzy partial derivatives are equal, @ 2 f (x;y) @2 f (x; y) = for all (x; y) 2 U . @x@y @y@x Finally it follows a multivariate Fuzzy Taylor’s formula. Theorem 2.22. Let U be an open convex subset of Rn , n 2 N and f : U ! RF be a fuzzy continuous function. Assume that all H-fuzzy partial derivatives of f up to order m 2 N exist and are fuzzy continuous. Let z := (z1 ; : : : ; zn ), x0 := (x01 ; : : : ; x0n ) 2 U such that zi x0i , i = 1; : : : ; n. Let 0 t 1, we de…ne xi := x0i + t(zi x0i ), i = 1; 2; : : : ; n and gz (t) := f (x0 + t(z x0 )). (Clearly x0 + t(z x0 ) 2 U .) Then for N = 1; : : : ; m we obtain 2 !N 3 n X @ (zi x0i ) gz(N ) (t) = 4 f 5 (x1 ; x2 ; : : : ; xn ): (2.12) @xi i=1 Furthermore it holds the following fuzzy multivariate Taylor formula f (z) = f (x0 )

m 1 (N ) X gz (0) N! N =1

Rm (0; 1);

(2.13)

where Rm (0; 1) := (F R)

Z

1

0

Z

s1

0

Z

sm

1

gz(m) (sm )dsm

0

!

dsm

1

!

!

ds1 :

(2.14)

N o t e (Explaining formula (2.12)). When N = n = 2 we have (zi i = 1; 2) gz (t) = f (x01 + t(z1

x01 ); x02 + t(z2

x02 ));

0

t

1:

We apply Theorems 2.19 and 2.21 repeatedly, etc. Thus we have gz0 (t) = (z1

x01 )

@f (x1 ; x2 ) @x1

(z2

x02 )

@f (x1 ; x2 ): @x2

x0i ,

46

2. About H-Fuzzy Di¤erentiation

Furthermore it holds gz00 (t)

=

(z1

@2f (x1 ; x2 ) @x21

x01 )2

@ 2 f (x1 ; x2 ) @x1 @x2

(z2

2(z1

x02 )2

x01 ) (z2

x02 )

@2f (x1 ; x2 ): @x22

(2.15)

x01 )2 (z2

x02 ) (2.16)

When n = 2 and N = 3 we get gz000 (t)

=

(z1

x01 )3

@3f (x1 ; x2 ) @x31

@ 3 f (x1 ; x2 ) @x21 @x2 (z2

x02 )3

3(z1

x02 )2

x01 )(z2

@ 3 f (x1 ; x2 ) @x1 @x22

@3f (x1 ; x2 ): @x32

When n = 3 and N = 2 we obtain (zi gz00 (t)

3(z1

x0i , i = 1; 2; 3)

@2f @2f (x1 ; x2 ; x3 ) (z2 x02 )2 (x1 ; x2 ; x3 ) 2 @x1 @x22 @2f (z3 x03 )2 (x1 ; x2 ; x3 ) 2(z1 x01 )(z2 x02 ) @x23 @ 2 f (x1 ; x2 ; x3 ) 2(z2 x02 )(z3 x03 ) (2.17) @x1 @x2 @ 2 f (x1 ; x2 ; x3 ) @2f 2(z3 x03 )(z1 x01 ) (x1 ; x2 ; x3 ): @x2 @x3 @x3 @x1

=

x01 )2

(z1

Etc. Proof of Theorem 2.22. Let z := (z1 ; : : : ; zn ), x0 := (x01 ; : : : ; x0n ) 2 U , n 2 N, such that zi > x0i , i = 1; 2; : : : ; n. We de…ne xi := Thus

dxi dt

= zi

i (t)

:= x0i + t(zi

x0i );

0

t

1;

i = 1; 2; : : : ; n:

x0i > 0. Consider

Z := gz (t) := f (x0 + t(z

x0 ))

= f (x01 + t(z1 = f ( 1 (t); : : : ;

x01 ); : : : ; x0n + t(zn n (t)):

Since by assumptions f : U ! RF is fuzzy continuous, also fxi exist and are fuzzy continuous, by Theorem 2.19 (2.11) we get dZ(x1 ; : : : ; xn ) dt

n X @Z(x1 ; : : : ; xn ) = @xi i=1

=

n X @f (x1 ; : : : ; xn ) @xi i=1

dxi dt (zi

x0i ):

x0n ))

2.4 Main Results

47

That is, gz0 (t) =

n X @f (x1 ; : : : ; xm ) @xi i=1

(zi

x0i ):

Next we see d2 Z dt2

=

gz00 (t)

=

n X

d = dt

(zi

n X @f (x1 ; : : : ; xn ) @xi i=1

x0i )

i=1

=

n X

(zi

i=1

=

x0i )

(zi

d @f (x1 ; : : : ; xn ) dt @xi 2 n X @ 2 f (x1 ; : : : ; xn ) 4 @xj @xi j=1

n n X X @ 2 f (x1 ; : : : ; xn ) @xj @xi i=1 j=1

!

x0i )

3

x0j )5

(zj

(zi

x0i ) (zj

x0j ):

(zi

x0i ) (zj

x0j ):

That is gz00 (t) =

n n X X @ 2 f (x1 ; : : : ; xm ) @xj @xi i=1 j=1

The last is true by Theorem 2.19 (2.11) under the additional assumptions 2 f that fxi ; @x@j @x , i; j = 1; 2; : : : ; n exist and are fuzzy continuous. i Working similarly we …nd 0 1 n n 2 X X d3 Z d @ f (x ; : : : ; x ) 1 n @ = gz000 (t) = (zi x0i ) (zj x0j )A dt3 dt i=1 j=1 @xj @xi =

n n X X i=1

(zi

x0i ) (zj

x0j )

j=1

n

=

n X X i=1 n

=

(zi

x0i ) (zj

x0j )

j=1 n

d @ 2 f (x1 ; : : : ; xn ) dt @xj @xi " n X @ 3 f (x1 ; : : : ; xn ) k=1

n X X X @ 3 f (x1 ; : : : ; xn ) @xk @xj @xi i=1 j=1

(zi

@xk @xj @xi

x0i ) (zj

#

(zk

x0k )

x0j ) (zk

x0k ):

k=1

That is gz000 (t)

n n n X X X @ 3 f (x1 ; : : : ; xn ) = @xk @xj @xi i=1 j=1 k=1

(zi

x0i ) (zj

x0j ) (zk

x0k ):

48

2. About H-Fuzzy Di¤erentiation

That last is true by Theorem 2.19 (2.11) under the additional assumptions that @ 3 f (x1 ; : : : ; xn ) ; i; j; k = 1; : : : ; n @xk @xj @xi do exist and are fuzzy continuous. Etc. In general one obtains that for N = 1; : : : ; m 2 N, gz(N ) (t) =

n X

n n X X

iN

i1 =1 i2 =1

N Y

@ N f (x1 ; : : : ; xn ) @xiN @xiN 1 @xi1 =1

(zir

x0ir );

r=1

which by Theorem 2.21 is the same as (2.12) for the case zi > x0i , see also (2.15), (2.16), and (2.17). The last is true by Theorem 2.19 (2.11) under the assumptions that all H-partial derivatives of f up to order m exist and they are all fuzzy continuous including f itself. Next let tm ~ ! +1, tm ~ ! t~, as m ~ , t~ 2 [0; 1]. Consider xim ~ := x0i + tm ~ (zi

x0i )

and x ~i := x0i + t~(zi

x0i );

i = 1; 2; : : : ; n:

That is xm ~ = (~ x1 ; : : : ; x ~n ) in U . ~ = (x1m ~ ; x2m ~ ; : : : ; xnm ~ ) and x Then xm ~, as m ~ ! +1. Clearly using the properties of D-metric and ~ ! x under the theorem’s assumptions, we obtain that gz(N ) (t) is fuzzy continuous for N = 0; 1; : : : ; m: Then by Theorem 2.1, from the univariate fuzzy Taylor formula, we obtain gz0 (0)

gz (1) = gz (0)

(m 1)

gz00 (0) 2!

gz (m

(0) 1)!

Rm (0; 1);

where Rm (0; 1) := (F R)

Z

0

1

Z

s1

0

Z

sm

1

gz(m) (sm )dsm

0

!

dsm

1

!

!

ds1 :

By Lemma 4, [11] and Corollary 13.2, p. 644, [66], the remainder Rm (0; 1) exist in RF . I.e., we get the multivariate fuzzy Taylor formula f (z) = f (x0 )

gz0 (0)

when zi > x0i , i = 1; 2; : : : ; n.

gz00 (0) 2!

(m 1)

gz (m

(0) 1)!

Rm (0; 1);

2.4 Main Results

49

Finally we would like to take care of the case that some x0i = zi . Without loss of generality we may assume that x01 = z1 , and zi > x0i , i = 2; : : : ; n. In this case we de…ne Z~ := g~z (t) := f (x01 ; x02 + t(z2

x02 ); : : : ; x0n + t(zn

x0n )):

Therefore one has g~z0 (t) =

n X @f (x01 ; x2 ; : : : ; xn ) @xi i=2

(zi

x0i );

and in general we …nd g~z(N ) (t) =

n X

i2 =2;:::;iN

@ N f (x01 ; x2 ; : : : ; xn ) @xiN @xN 1 @xi2 =2 (N )

for N = 1; : : : ; m 2 N. Notice that all g~z continuous and g~z (0) = f (x01 ; x02 ; : : : ; x0n );

N Y

(zir

x0ir );

r=2

, N = 0; 1; : : : ; m are fuzzy

g~z (1) = f (x01 ; z2 ; z3 ; : : : ; zn ):

Then one can write down a fuzzy Taylor formula, as above, for g~z . But (N ) (N ) g~z (t) coincides with gz (t) formula at z1 = x01 = x1 . That is both Taylor formulae in that case coincide. At last we remark that if z = x0 , then we de…ne Z := gz (t) := f (x0 ) =: c 2 RF a constant. Since c = c + o~, that is c c = o~, we obtain the H-fuzzy derivative (c)0 = o~. Consequently we have that gz (N ) (t) = o~;

N = 1; : : : ; m:

(N )

The last coincide with the gz formula, established earlier, if we apply there z = x0 . And, of course, the fuzzy Taylor formula now can be applied trivially for gz . Furthermore in that case it coincides with the Taylor formula proved earlier for gz . We have established a multivariate fuzzy Taylor formula for the case of zi x0i , i = 1; 2; : : : ; n. That is (2.12)–(2.14) are true. At last we give the useful Corollary 2.23. Let U be an open convex subset of Rn , n 2 N, and f : U ! RF be a fuzzy continuous function. Assume that all the …rst Hfuzzy partial derivatives fxi of f exist and are fuzzy continuous. Let z := (z1 ; : : : ; zn ), x0 := (x01 ; : : : ; x0n ) 2 U such that zi x0i , i = 1; : : : ; n. Let 0 t 1, we de…ne xi := x0i + t(zi x0i ), i = 1; 2; : : : ; n and gz (t) := f (x0 + t(z x0 )). Then gz0 (t) =

n X @f (x1 ; : : : ; xn ) @xi i=1

(zi

x0i ):

(2.18)

2. About H-Fuzzy Di¤erentiation

50

Furthermore it holds f (z)

= f (x0 )

(F R)

Z

1

gz0 (s)ds

0

= f (x0 )

n X

i=1

(zi

x0i )

(F R)

(2.19) Z

0

1

@f (x1 (s); : : : ; xn (s)) ds: @xi

Proof. By Theorem 2.22, case of m = 1. The second part of (2.19) is valid by Theorem 2.6 of [53]. Here xi (s) = x0i + s(zi x0i ), s 2 [0; 1], i = 1; : : : ; n with zi x0i . Comment . Theorem 2.22 and Corollary 2.23 are still valid when U is a compact convex subset of Rn such that U W , where W is an open subset of Rn . Now f : W ! RF and it has all the properties of f as in Theorem 2.22 and Corollary 2.23. Clearly here x0 ; z 2 U .

3 ON FUZZY TAYLOR FORMULAE

We present Fuzzy Taylor formulae with integral remainder in the univariate and multivariate cases, analogs of the real setting. This chapter is based on [19].

3.1 Main Results We present the following fuzzy Taylor theorem in one dimension. Theorem 3.1. Let f 2 C n ([a; b]; RF ), n f( ) = f( )

f 0(n

1)

( )

1 (n

1)!

(

1, [ ; ]

[a; b]

R. Then

)n 1 1)!

(n Z (F R) (

t)n

1

f (n) (t) dt: (3.1)

The integral remainder is a fuzzy continuous function in . (r)

(r)

Proof. Let r 2 [0; 1]. We have here [f ( )]r = [f ( ); f+ ( )], and by (r) Theorem 5.2 ([71]) f is n-times continuously di¤erentiable on [a; b]. By (1.7) we get (f

(i)

( ))(r) = (f

(r)

( ))(i) ;

and [f (i) ( )]r = (f

(r)

all i = 0; 1; : : : ; n;

(3.2)

(r)

( ))(i) ; (f+ ( ))(i) :

G.A. Anastassiou: Fuzzy Mathematics: Approximation The., STUDFUZZ 251, pp. 51–63. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

52

3. On Fuzzy Taylor Formulae

Thus by Taylor’s theorem we obtain f

(r)

( )

= f

(r)

( ) + (f

+

+ (f

(r)

(r)

( ))0 (

)

(n 1) (

( ))

(n

)n 1 1 + 1)! (n 1)!

Z

(

t)n

1

(f

(r) (n)

)

(t)dt:

)n 1 1 + 1)! (n 1)!

Z

(

t)n

1

(f

(n) (r)

(t)dt:

(

)n 1 1)!

Furthermore by (3.2) we have f

(r)

( )

= f

(r)

( ) + (f 0 ( ))(r) (

+

+ (f

Here it holds

(n 1)

(r) (

( ))

0, (f

(i)

)

t

(n

0 for t 2 [ ; ], and (i)

(f+ (t))(r) ;

(t))(r)

)

8t 2 [a; b]

all i = 0; 1; : : : ; n, and any r 2 [0; 1]. We see that f

(r)

(r)

( ); f+ ( )]

= [f

(r)

+

( ) + (f 0 ( ))(r) ( 1

(n

1)!

Z

(

0 + (f+ ( ))(r) (

+

1 (n

1)!

t)n

)+ 1

(

t)n

(n) (r)

)

+ (f+ 1

(n 1)

( ))(r)

(n

(r)

(t)dt; f+ ( )

(n 1)

)+

Z

(f

+ (f

( ))(r)

( (n

)n 1 1)!

(n)

(f+ )(r) (t) dt :

To split the above closed interval into a sum of smaller closed intervals is where we use 0. So we get [f ( )]r

=

[f

(r)

+ + =

(r)

( ); f+ ( )] = [f + [(f 1

(n

1)!

(n 1)

"Z

(r)

(r)

0 ( ))(r) ]( ( ); f+ ( )] + [(f 0 ( ))(r) ; (f+

) (n 1)! Z 1 (n) (r) (f ) (t)dt; ( t)n (n 1)

( ))(r) ; (f+ (

n

t)

( ))(r) ]

(

( [f ( )]r + [f 0r ( )+ + [f (n 1) ( )]r (n "Z 1 (r) + (( t)n 1 f (n) (t)) dt; (n 1)! # Z (r) n 1 (n) (( t) f (t))+ dt :

)

n 1

)n 1 1)!

1

(n) (f+ )(r) (t)dt

#

3.1 Main Results

53

By Theorem 3.2 ([67]) we next get [f ( )]r

=

[f ( )]r + [f 0r ( )+ " Z 1 + (F R) ( (n 1)!

+ [f (n t)n

1

1)

( )]r

)n 1 1)!

( (n #r

f (n) (t)dt

:

Finally we obtain [f ( )]r

=

f 0(n

f( ) 1 (n

1)!

1)

(

( )

(F R)

Z

(n

)n 1 1)! r

(

t)n

1

f (n) (t)dt ;

all r 2 [0; 1]:

By Theorem 3.2 of [67] and Lemma 1.5 we get that the remainder of (3.1) is in RF , and by Lemma 3 ([11]) is a fuzzy continuous function in . The theorem has been proved. Next we present another multivariate fuzzy Taylor theorem. We need the following multivariate fuzzy chain rule. Here the H-fuzzy partial derivatives are de…ned according to the De…nition 3.3 of [53], see Section 1 there, and the analogous way to the real case. Theorem 3.2 ([10]). Let i : [a; b] R ! i ([a; b]) := Ii R, i = 1; : : : ; n, n 2 N, are strictly increasing and di¤ erentiable functions. Denote xi := xi (t) := i (t), t 2 [a; b], i = 1; : : : ; n. Consider U an open subset of Rn such that ni=1 Ii U . Consider f : U ! RF a fuzzy continuous function. Assume that fxi : U ! RF , i = 1; : : : ; n, the H-fuzzy partial derivatives of f , exist and are fuzzy continuous. Call z := z(t) := f (x1 ; : : : ; xn ). Then dz dt exists and n dz X dz dxi = ; 8t 2 [a; b] (3.3) dt dx dt i i=1

dz where dz dt , dxi , i = 1; : : : ; n are the H-fuzzy derivatives of f with respect to t, xi , respectively.

The interchange of the order of H-fuzzy di¤erentiation is needed too. Theorem 3.3 ([10]). Let U be an open subset of Rn , n 2 N, and f : U ! RF be a fuzzy continuous function. Assume that all H-fuzzy partial derivatives of f up to order m 2 N exist and are fuzzy continuous. Let x := (x1 ; : : : ; xn ) 2 U . Then the H-fuzzy mixed partial derivative of order k, Dx`1 ;:::;x`k f (x) is unchanged when the indices `1 ; : : : ; `k are permuted. Each `i is a positive integer n. Here some or all of `i ’s can be equal. Also k = 2; : : : ; m and there are nk partials of order k. We present

54

3. On Fuzzy Taylor Formulae

Theorem 3.4. Let U be an open convex subset of Rn , n 2 N and f : U ! RF be a fuzzy continuous function. Assume that all H-fuzzy partial derivatives of f up to order m 2 N exist and are fuzzy continuous. Let z := (z1 ; : : : ; zn ), x0 := (x01 ; : : : ; x0n ) 2 U such that xi x0i , i = 1; : : : ; n. Let 0 t 1, we de…ne xi := x0i + t(zi z0i ), i = 1; 2; : : : ; n and gz (t) := f (x0 + t(z x0 )). (Clearly x0 + t(z x0 ) 2 U .) Then for N = 1; : : : ; m we obtain 2 !N 3 n X @ f 5 (x1 ; x2 ; : : : ; xn ): (3.4) gz(N ) (t) = 4 (zi x0i ) @xi i=1 Furthermore it holds the following fuzzy multivariate Taylor formula m 1 (N ) X gz (0) N!

f (z) = f (x0 )

Rm (0; 1);

N =1

(3.5)

where Rm (0; 1) :=

1 (m

(F R)

1)!

Z

1

s)m

(1

1

gz(m) (s)ds:

(3.6)

0

Comment. (Explaining formula (3.4)). When N = n = 2 we have (zi x0i , i = 1; 2) gz (t) = f (x01 + t(z1

x01 ); x02 + t(z2

x02 ));

0

t

1:

We apply Theorems 3 and 4 of [10] repeatedly, etc. Thus we …nd gz0 (t) = (z1

@f (x1 ; x2 ) @x1

x01 )

(z2

x02 )

@f (x1 ; x2 ): @x2

Furthermore it holds gz00 (t)

=

(z1

@2f (x1 ; x2 ) @x21

x01 )2

@ 2 f (x1 ; x2 ) @x1 @x2

(z2

2(z1

x01 ) (z2

x02 ) (3.7)

@2f (x1 ; x2 ): @x22

x02 )2

When n = 2 and N = 3 we obtain gz000 (t)

=

(z1

x01 )3

@ 3 f (x1 ; x2 ) @x21 @x2 (z2

x02 )3

@3f (x1 ; x2 ) @x31 3(z1

3(z1

x01 )(z2

@3f (x1 ; x2 ): @x32

x01 )2 (z2 x02 )2

x02 )

@ 3 f (x1 ; x2 ) @x1 @x22 (3.8)

3.1 Main Results

When n = 3 and N = 2 we get (zi gz00 (t)

55

x0i , i = 1; 2; 3)

@2f @2f (x1 ; x2 ; x3 ) (z2 x02 )2 (x1 ; x2 ; x3 ) 2 @x1 @x22 @2f (z3 x03 )2 (x1 ; x2 ; x3 ) 2(z1 x01 )(z2 x02 ) @x23 @ 2 f (x1 ; x2 ; x3 ) 2(z2 x02 )(z3 x03 ) (3.9) @x1 @x2 @2f @ 2 f (x1 ; x2 ; x3 ) 2(z3 x03 )(z1 x01 ) (x1 ; x2 ; x3 ); @x2 @x3 @x3 @x1

=

x01 )2

(z1

etc. Proof of Theorem 3.4. Let z := (z1 ; : : : ; zn ), x0 := (x01 ; : : : ; x0n ) 2 U , n 2 N, such that zi > x0i , i = 1; 2; : : : ; n. We de…ne xi := Thus

dxi dt

= zi

i (t)

:= x0i + t(zi

x0i );

0

t

1;

i = 1; 2; : : : ; n:

x0i > 0. Consider

Z := gz (t) := f (x0 + t(z

x0 ))

= f (x01 + t(z1 = f ( 1 (t); : : : ;

x01 ); : : : ; x0n + t(zn n (t)):

Since by assumptions f : U ! RF is fuzzy continuous, also fxi exist and are fuzzy continuous, by Theorem 3 (10) of [10] we get dZ(x1 ; : : : ; xn ) dt

= =

Thus gz0 (t) Next we observe that d2 Z dt2

n X

(zi

i=1

=

n X

i=1

(zi

(zi

n X @f (x1 ; : : : ; xn ) @xi i=1

x0i )

x0i )

dxi dt

n X @f (x1 ; : : : ; xn ) @xi i=1

n X @f (x1 ; : : : ; xn ) = @xi i=1

d = gz00 (t) = dt =

n X @Z(x1 ; : : : ; xn ) @xi i=1

(zi

x0i ):

(zi

d @f (x1 ; : : : ; xn ) dt @xi 2 n X @ 2 f (x1 ; : : : ; xn ) 4 @xj @xi j=1

n n X X @ 2 f (x1 ; : : : ; xn ) = @xj @xi i=1 j=1

(zi

x0i ):

!

x0i )

(zj

x0i ) (zj

3

x0j )5 x0j ):

x0n ))

56

3. On Fuzzy Taylor Formulae

That is gz00 (t)

n n X X @ 2 f (x1 ; : : : ; xn ) = @xj @xi i=1 j=1

(zi

x0i ) (zj

x0j ):

The last is true by Theorem 3 (10) of [10] under the additional assumptions 2 f , i; j = 1; 2; : : : ; n exist and are fuzzy continuous. that fxi ; @x@j @x i Working the same way we …nd 0 1 n n 2 3 X X @ f (x1 ; : : : ; xn ) d Z d @ = gz000 (t) = (zi x0i ) (zj x0j )A dt3 dt i=1 j=1 @xj @xi =

n n X X i=1

(zi

x0i ) (zj

x0j )

j=1

n

=

n X X i=1 n

=

(zi

x0i ) (zj

x0j )

j=1 n

d @ 2 f (x1 ; : : : ; xn ) dt @xj @xi " n X @ 3 f (x1 ; : : : ; xn ) @xk @xj @xi

k=1

n X X X @ 3 f (x1 ; : : : ; xn ) @xk @xj @xi i=1 j=1

(zi

x0i ) (zj

#

(zk

x0k )

x0j ) (zk

x0k ):

k=1

Therefore, gz000 (t) =

n n n X X X @ 3 f (x1 ; : : : ; xn ) @xk @xj @xi i=1 j=1

(zi

x0i ) (zj

x0j ) (zk

x0k ):

k=1

That last is true by Theorem 3 (10) of [10] under the additional assumptions that @ 3 f (x1 ; : : : ; xn ) ; i; j; k = 1; : : : ; n @xk @xj @xi do exist and are fuzzy continuous. Etc. In general one obtains that for N = 1; : : : ; m 2 N, gz(N ) (t) =

n n X X

i1 =1 i2 =1

n X

iN

@ N f (x1 ; : : : ; xn ) @xiN @xiN 1 @xi1 =1

N Y

(zir

x0ir );

r=1

which by Theorem 4 of [10] is the same as (3.4) for the case zi > x0i , see also (3.7), (3.8), and (3.9). The last is true by Theorem 3 (10) of [10] under the assumptions that all H-partial derivatives of f up to order m exist and they are all fuzzy continuous including f itself. Next let tm ~ ! +1, tm ~ ! t~, as m ~ , t~ 2 [0; 1]. Consider xim ~ := x0i + tm ~ (zi

x0i )

3.1 Main Results

57

and x ~i := x0i + t~(zi

x0i );

i = 1; 2; : : : ; n:

That is xm ~ = (~ x1 ; : : : ; x ~n ) in U . ~ = (x1m ~ ; x2m ~ ; : : : ; xnm ~ ) and x Then xm ~, as m ~ ! +1. Clearly using the properties of D-metric and ~ ! x under the theorem’s assumptions, we obtain that gz(N ) (t) is fuzzy continuous for N = 0; 1; : : : ; m: Then by Theorem 3.1, from the univariate fuzzy Taylor formula (3.1), we …nd gz0 (0)

gz (1) = gz (0)

(m 1)

gz00 (0) 2!

gz (m

(0) 1)!

Rm (0; 1);

where Rm (0; 1) comes from (3.6). By Theorem 3.2 of [67] and Lemma 1.5 we get that Rm (0; 1) 2 RF . That is we get the multivariate fuzzy Taylor formula gz0 (0)

f (z) = f (x0 )

(m 1)

gz00 (0) 2!

gz (m

(0) 1)!

Rm (0; 1);

when zi > x0i , i = 1; 2; : : : ; n. Finally we would like to take care of the case that some x0i = zi . Without loss of generality we may assume that x01 = z1 , and zi > x0i , i = 2; : : : ; n. In this case we de…ne Z~ := g~z (t) := f (x01 ; x02 + t(z2

x02 ); : : : ; x0n + t(zn

x0n )):

Therefore one has g~z0 (t)

n X @f (x01 ; x2 ; : : : ; xn ) = @xi i=2

(zi

x0i );

and in general we …nd g~z(N ) (t) =

n X

i2 =2;:::;iN

@ N f (x01 ; x2 ; : : : ; xn ) @xiN @xN 1 @xi2 =2 (N )

for N = 1; : : : ; m 2 N. Notice that all g~z continuous and g~z (0) = f (x01 ; x02 ; : : : ; x0n );

N Y

(zir

x0ir );

r=2

, N = 0; 1; : : : ; m are fuzzy

g~z (1) = f (x01 ; z2 ; z3 ; : : : ; zn ):

58

3. On Fuzzy Taylor Formulae

Then one can write down a fuzzy Taylor formula, as above, for g~z . But (N ) (N ) g~z (t) coincides with gz (t) formula at z1 = x01 = x1 . That is both Taylor formulae in that case coincide. At last we remark that if z = x0 , then we de…ne Z := gz (t) := f (x0 ) =: c 2 RF a constant. Since c = c + o~, that is c c = o~, we obtain the H-fuzzy derivative (c)0 = o~. Consequently we have that gz (N ) (t) = o~;

N = 1; : : : ; m:

(N )

The last coincide with the gz formula, established earlier, if we apply there z = x0 . And, of course, the fuzzy Taylor formula now can be applied trivially for gz . Furthermore in that case it coincides with the Taylor formula proved earlier for gz . We have established a multivariate fuzzy Taylor formula for the case of zi x0i , i = 1; 2; : : : ; n. That is (3.4)–(3.6) are true. Note. Theorem 3.4 is still valid when U is a compact convex subset of Rn such that U W , where W is an open subset of Rn . Now f : W ! RF and it has all the properties of f as in Theorem 3.4. Clearly here we take x0 ; z 2 U .

3.2 Addendum As related material we give Theorem 3.5. Let f : [a; b]

R ! RF be fuzzy continuous. Then Z t G(t) = (F R) f (s)ds 2 RF ; a

any t 2 [a; b] : Also G(t) is H di¤erentiable and G0 (t) = f (t); t 2 [a; b]: Proof. Clearly G(t) is fuzzy continuous in t; see Lemma 1.15. Here Z t+h G(t + h) = (F R) f (s)ds a

=

(F R)

Z

t

f (s)ds

a

= G(t)

(F R)

(F R)

Z

t+h

f (s)ds

t

Z

t+h

f (s)ds:

t

Thus the H

di¤erence G(t + h)

G(t) = (F R)

Z

t

t+h

f (s)ds 2 RF ;

3.2 Addendum

59

exists. And furthermore we have G(t + h) h

G(t)

=

1 h

(F R)

Z

t+h

f (s)ds 2 RF :

t

Thus G(t + h) h

D

G(t)

; f (t)

1 D (F R) h

=

1 D (F R) h

=

Z

!

t+h

f (s)ds; h

f (t)

t

Z

t+h

f (s)ds;

Z

t

t

t+h

1ds

!

f (t)) Z t+h 1 D (F R) f (s)ds; h t ! Z t+h (F R) f (t)ds

=

t

Z

t+h

1 D (f (s); f (t)) ds h t (the integrand just above is continuous in s) 1 h sup D (f (s); f (t)) h s2[t;t+h] = D (f (s ) ; f (t)) ! 0; where s 2 [t; t + h] ; and as h ! 0+ we get s ! t; and we assumed that f is fuzzy continuous. Similarly we have G(t

h) = (F R)

Z

t h

f (s)ds; (h > 0);

a

and G(t) = (F R)

Z

t h

f (s)ds

(F R)

a

Z

t

f (s)ds:

t h

Also it holds G(t)

h ! 0+ :

G(t h

h)

=

(F R)

Rt

t h

h

f (s)ds

;

60

3. On Fuzzy Taylor Formulae

Hence we have D

G(t)

G(t h

h)

; f (t)

= =

1 D (F R) h

Z

t

f (s)ds; h

f (t)

t h Z t

1 f (s)ds; D (F R) h t h Z t (F R) f (t)ds t h

Z 1 t D (f (s); f (t)) ds h t h 1 h sup D (f (s); f (t)) h s2[t h;t] = D (f (s ) ; f (t)) ! 0; where s 2 [t h; t] ; by f continuity; as h ! 0+ ; then s ! t: So that G is fuzzy di¤erentiable and G0 (t) = f (t): Comment 3.6. Let f : [a; b] ! RF be fuzzy continuous. Then G1 (t1 ) = Rt (F R) a 1 f (s)ds is H di¤erentiable and G01 (t1 ) = f (t1 ): Here G1 (t1 ) is Rt fuzzy continuous. Let G2 (t2 ) := (F R) a 2 G1 (t1 )dt1 2 RF ; it is also fuzzy R t continuous, and G02 (t2 ) = G1 (t2 ) = a 2 f (s)ds: 0 0 Hence (G02 (t2 )) = (G1 (t2 )) = f (t2 ): I.e. G002 (t2 ) = f (t2 ); etc. That is, there are non-trivial higher order H fuzzy derivatives. We continue with Theorem 3.7. Let f be fuzzy continuous on R := [a; b] [c; d] : Then Z x1 Z x2 F (x1 ; x2 ) := (F R) (F R) f (x; y)dy dx 2 RF ; a

c

any (x1 ; x2 ) 2 R; and D2;1 F (x1 ; x2 ) = f (x1 ; x2 ) : Proof. The function G(x; x2 ) = (F R)

Z

x2

f (x; y)dy;

c

is fuzzy continuous in x2 ; and belongs to RF for each x 2 [a; b] : We have Z x1 F (x1 ; x2 ) = (F R) G(x; x2 )dx: a

We would like to prove that G(x; x2 ) is fuzzy continuous in x 2 [a; b] : Here f (x; y) is fuzzy continuous, and thus uniformly fuzzy continuous on R:

3.2 Addendum

61

Let h be such that x + h 2 [a; b] : Uniform fuzzy continuity implies 8 > 0; 9 > 0 such that, whenever k(x + h; y) (x; y)k < ; where k k is a norm in R2 ; we get D (f (x + h; y) ; f (x; y)) < : Notice that k(x + h; y)

(x; y)k = k(h; 0)k = jhj k(1; 0)k :

Thus

D (G (x + h; x2 ) ; G (x; x2 ))

Z x2 = D (F R) f (x + h; y)dy; c Z x2 f (x; y)dy (F R) c

(both integrands just above are fuzzy continuous in y; hence by Lemma 1.13 we get) Z x2 D (f (x + h; y); f (x; y)) dy c

(x2

c) ;

proving fuzzy continuity of G (x; x2 ) in x 2 [a; b] : Clearly then F (x1 ; x2 ) 2 RF ; by Corollary 1.11. Therefore Z x2 @F (x1 ; x2 ) = G (x1 ; x2 ) = (F R) f (x1 ; y)dy; @x1 c by Theorem 3.5. Thus

@G (x1 ; x2 ) @ 2 F (x1 ; x2 ) = = f (x1 ; x2 ); @x2 @x1 @x2 proving the claim. We …nish with Theorem 3.8. Let f be fuzzy continuous on R = [a; b] [c; d] : Then Z x1 Z x2 Z y M (x1 ; x2 ) = (F R) (F R) (F R) f (x; s)ds dy dx 2 RF ; a

c

c

any (x1 ; x2 ) 2 R; and D2;2;1 M (x1 ; x2 ) = f (x1 ; x2 ) : Proof. Call '(x; y) = (F R)

Z

c

y

f (x; s)ds 2 RF :

62

3. On Fuzzy Taylor Formulae

Notice ' is fuzzy continuous in x and y: Then M (x1 ; x2 ) = (F R)

Z

x1

(F R)

Z

x2

'(x; y)dy dx:

c

a

Call G(x; x2 ) = (F R)

Z

x2

'(x; y)dy 2 RF ;

c

notice G is fuzzy continuous in x2 and M (x1 ; x2 ) = (F R)

Z

x1

G(x; x2 )dx:

a

We need to prove that '(x; y) is (simultaneously) fuzzy continuous in (x; y): Indeed for letting yn y we observe D ('(xn ; yn ); '(x; y))

= D (F R)

Z

yn

f (xn ; s)ds; (F R)

c

= D (F R) Z

Z

y

f (x; s)ds

c

y

f (xn ; s)ds

(F R)

c

y

Z

Z

yn

f (xn ; s)ds;

y

f (x; s)ds Z y Z y D (F R) f (xn ; s)ds; (F R) f (x; s)ds + c c Z yn D (F R) f (xn ; s)ds; oe (F R)

c

y

( >0 sm all)

+D (F R)

Z

yn

f (xn ; s)ds; (F R)

y

Z

y

yn

oeds

(since the integrands just above are fuzzy continuous in s; by Lemma 1.13 we get) Z yn + D (f (xn ; s); oe) ds y

+ (yn

(here D (f (xn ; s); oe)

y) ! 0;

; > 0; as in Lemma 2, [11]); i.e. D ('(xn ; yn ); '(x; y)) ! 0;

as (xn ; yn ) ! (x; y); with yn

y:

3.2 Addendum

Let now yn

63

y; then

D ('(xn ; yn ); '(x; y))

= D (F R)

Z

yn

f (xn ; s)ds; (F R)

c

= D (F R) (F R)

Z

Z

yn

f (xn ; s)ds

f (xn ; s)ds

yn

(F R)

(F R)

D (F R)

y

D (F R)

y

f (xn ; s)ds; (F R)

D (F R) +

Z

y

+

f (xn ; s)ds; oe

y

f (xn ; s)ds; (F R)

yn

y

yn

+ (y So as (xn ; yn ) ! (x; y); with yn

Z

f (xn ; s)ds;

y

f (x; s)ds

y

f (x; s)ds Z

y

f (x; s)ds +

c

yn

( >0 sm all)

y

c

c

Z

Z

c

Z Z

f (x; s)ds

yn

Z

= D (F R) f (xn ; s)ds; c Z Z y f (xn ; s)ds (F R) (F R) yn

y

c

c

y

Z

D (f (xn ; s); oe) ds

Z

y

yn

oeds

yn ) ! 0:

y; we get again

D ('(xn ; yn ); '(x; y)) ! 0: Therefore we have proved that '(x; y) is (jointly) fuzzy continuous in (x; y) over R: Consequently G(x; x2 ) is fuzzy continuous in x 2 [a; b] ; i.e. M (x1 ; x2 ) 2 RF : Thus @ 2 M (x1 ; x2 ) @G (x1 ; x2 ) = = '(x1 ; x2 ); @x2 @x1 @x2 and

@ 3 M (x1 ; x2 ) @ 2 G (x1 ; x2 ) @'(x1 ; x2 ) = = = f (x1 ; x2 ) : @x22 @x1 @x22 @x2

Conclusion. There are higher order nontrivial H fuzzy partial derivatives.

4 FUZZY OSTROWSKI INEQUALITIES

We present optimal upper bounds for the deviation of a fuzzy continuous function from its fuzzy average over [a; b] R, error is measured in the Dfuzzy metric. The established fuzzy Ostrowski type inequalities are sharp, in fact attained by simple fuzzy real number valued functions. These inequalities are given for fuzzy Hölder and fuzzy di¤erentiable functions and these facts are re‡ected in their right-hand sides. This chapter relies on [13].

4.1 Introduction Ostrowski inequality (see [92]) has as follows 1 b

a

Z

a

b

f (y)dy

f (x)

x 1 + 4 (b

a+b 2 2 a)2

!

(b

a)kf 0 k1 ;

where f 2 C 1 ([a; b]), x 2 [a; b]. The last inequality is sharp, see [5]. Since 1938 when A. Ostrowski proved his famous inequality, see [92], many people have been working about and around it, in many di¤erent directions and with a lot of applications in Numerical Analysis and Probability, etc. One of the most notable works extending Ostrowski’s inequality is the work of A.M. Fink, see [64]. The author in [5] continued that tradition. G.A. Anastassiou: Fuzzy Mathematics: Approximation The., STUDFUZZ 251, pp. 65–73. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

66

4. Fuzzy Ostrowski Inequalities

This chapter is mainly motivated by [5], [64], [92], [103] and extends Ostrowski type inequalities into the fuzzy setting, as fuzziness is a natural reality genuine feature di¤erent than randomness and determinism.

4.2 Background We use the Fuzzy Taylor formula. Theorem 4.1 (Theorem 1 of [11]). Let T := [x0 ; x0 + ] R, with > 0. We assume that f (i) : T ! RF are H-di¤ erentiable for all i = 0; 1; : : : ; n 1, for any x 2 T . (I.e., there exist in RF the H-di¤ erences f (i) (x + h) f (i) (x), f (i) (x) f (i) (x h), i = 0; 1; : : : ; n 1 for all small h : 0 < h < . Furthermore there exist f (i+1) (x) 2 RF such that the limits in D-distance exist and f (i+1) (x) = lim+ h!0

f (i) (x + h) h

f (i) (x)

= lim+

f (i) (x)

f (i) (x h

h!0

h)

;

for all i = 0; 1; : : : ; n 1.) Also we assume that f (n) , is fuzzy continuous on T . Then for s a, s; a 2 T we obtain f (s)

f 0 (a)

= f (a)

f (n

(s 1)

f 00 (a)

a)

(s a)n 1 (n 1)!

(a)

(s

a)2 2!

Rn (a; s);

where Rn (a; s) := (F R)

Z

s

a

Z

a

s1

Z

sn

1

f (n) (sn )dsn dsn

1

ds1 :

a

Here Rn (a; s) is fuzzy continuous on T as a function of s. We use Proposition 4.2 (Proposition 1 of [10]). Let F (t) := tn and u 2 RF be …xed. Then (the H-derivative) F 0 (t) = ntn

1

u, t

0, n 2 N,

u:

In particular when n = 1 then F 0 (t) = u. We mention Proposition 4.3 (Proposition 6 of [10]). Let I be an open interval of R and let f : I ! RF be H-fuzzy di¤ erentiable, c 2 R. Then (c

f )0 exists and (c

f )0 = c

We use the “Fuzzy Mean Value Theorem".

f 0 (x):

4.3 Results

67

Theorem 4.4 (Theorem 1 of [10]). Let f : [a; b] ! RF be a fuzzy di¤ erentiable function on [a; b] with H-fuzzy derivative f 0 which is assumed to be fuzzy continuous. Then D(f (d); f (c))

c) sup D(f 0 (t); o~);

(d

t2[c;d]

for any c; d 2 [a; b] with d c. We …nally need the “Univariate Fuzzy Chain Rule". Theorem 4.5 (Theorem 2 of [10]). Let I be a closed interval in R. Here g : I ! := g(I) R is di¤ erentiable, and f : ! RF is H-fuzzy di¤ erentiable. Assume that g is strictly increasing. Then (f g)0 (x) exists and g)0 (x) = f 0 (g(x))

(f

g 0 (x);

8x 2 I:

4.3 Results We present the following Theorem 4.6. Let f 2 C([a; b]; RF ), the space of fuzzy continuous functions, x 2 [a; b] be …xed. We assume that f ful…lls the Hölder condition D(f (y); f (z))

Lf jy

zj ;

0<

1; 8y; z 2 [a; b];

for some Lf > 0. Then D

1 b

(F R)

a

Z

!

b

f (y)dy; f (x)

a

D

b

(F R)

a

=

Z

!

b

1 b 1

b

a

(F R)

a

b

a

b

f (y)dy;

b

f (y)dy;

a

Z

a

1 b

b

jy

Z

a

b

f (x)dy

Z

xj dy =

1

b

a

(by Lemma 1.13) Lf

Z

a

Z

D

+1

:

f (y)dy; f (x)

a

= D

a) +1 + (b x) ( + 1)(b a)

(4.1)

Proof. We have that 1

(x

Lf

a

(F R)

Z

b

f (x)dy

a

!

!

b

D(f (y); f (x))dy

a

Lf b

(x a

a)

+1

+ (b +1

x)

+1

:

68

4. Fuzzy Ostrowski Inequalities

Optimality of (4.1) comes next. Proposition 4.7. Inequality (4.1) is sharp, in fact, attained by f (y) := jy xj u, 0 < 1, with u 2 RF …xed. Here x; y 2 [a; b]. Proof. Clearly f 2 C([a; b]; RF ): for letting yn ! y, yn 2 [a; b], then

D(f (yn ); f (y)) = D(jyn xj u; jy xj u) (by Lemma 1.2) jjyn xj jy xj j D(u; o~) ! 0; as n ! +1: Furthermore D(f (y); f (z)) = D(jy xj (by Lemma 1.2) jjy

xj

jz

u; jz

xj

u)

jz xj j D(u; o~) jjy xj xjj D(u; o~) jy zj D(u; o~):

That is, for Lf := D(u; o~) we get D(f (y); f (z))

Lf jy

zj ;

0<

1; any y; z 2 [a; b]:

So that f is a Hölder function. Finally we have ! Z b 1 (F R) f (y)dy; f (x) D b a a ! Z b 1 (F R) (jy xj u)dy; o~ = D b a a ! Z b 1 D (F R) (jy xj u)dy; o~ = b a a ! ! Z b 1 = D jy xj dy u; o~ b a a =

1

b a Lf = b a

(x

D (x

a)

+1

+ (b x) +1 +1 a) + (b x) +1 +1

+1

u; o~ :

Next comes the basic Ostrowski type fuzzy result in Theorem 4.8 let f 2 C 1 ([a; b]; RF ), the space of one time continuously di¤ erentiable functions in the fuzzy sense. Then for x 2 [a; b], ! ! Z b 1 (x a)2 + (b x)2 D (F R) f (y)dy; f (x) sup D(f 0 (t); o~) b a 2(b a) t2[a;b] a (4.2)

:

4.3 Results

Inequality (4.2) is sharp at x = a, in fact attained by f (y) := (y a) u, u 2 RF being …xed.

69

a)(b

Proof. We observe that D

1 b

(F R)

a

1 b 1

b

!

b

f (y)dy; f (x)

a

= D =

Z

a

a

Z

(F R)

f (y)dy;

a

Z

D (F R)

1 b

b

f (y)dy; (F R)

b

a

(F R) f (x)dy

a !

sup D(f 0 (t); o~) t2[a;b]

Z

!

b

f (x)dy

a

!

b

D(f (y); f (x))dy

a

1 b

b

Z

1

(by Theorem 4.4)

a Z b a

a

(by Lemma 1.13)

=

b

Z

!

b

a

jy

0

xj

sup D(f (t); o~) dy t2[a;b]

a)2 + (b 2

(x

a

x)2

;

proving (4.2). By Propositions 4.2, 4.3 and Theorem 4.5 we get that f 0 (y) = (b a) u. We have that ! Z b 1 (F R) ((y a)(b a) u)dy; o~ L:H:S:(4:2) = D b a a ! Z b = D (F R) ((y a) u)dy; o~ a

= D

Z

b

(y

a

= D

a)2

(b 2

a)dy

!

!

u; o~

u; o~ =

a)2

(b 2

D(u; o~):

And R:H:S:(4:2) = sup D((b t2[a;b]

a)

u; o~)

(b

a) 2

=

a)2

(b 2

D(u; o~):

That is equality in (4.2) is attained. We conclude with the following Ostrowski type inequality fuzzy generalization in

70

4. Fuzzy Ostrowski Inequalities

Theorem 4.9. Let f 2 C n+1 ([a; b]; RF ), n 2 N, the space of (n + 1) times continuously di¤ erentiable functions on [a; b] in the fuzzy sense. Call n X (b

M :=

i=1

a)i D(f (i) (a); o~): (i + 1)!

Then 1

D "

b

(F R)

a

M+

Z

f (x)dx; f (a)

a

sup D(f

!

b

(n+1)

!

(t); o~)

t2[a;b]

(4.3)

# (b a)n+1 : (n + 2)!

If f (i) (a) = o~, i = 1; : : : ; n. Then D

1 b

(F R)

a

Z

!

b

f (x)dx; f (a)

a

sup D(f

(n+1)

!

(t); o~)

t2[a;b]

(b a)n+1 : (n + 2)! (4.4)

Inequalities (4.3) and (4.4) are sharp, in fact attained by f (x) := (b

a)n+1

a)(x

u;

u 2 RF being …xed:

Corollary 4.10. Let f 2 C 2 ([a; b]; RF ). Then ! Z b 1 D (F R) f (x)dx; f (a) (4.5) b a a " ! # (b a) (b a)2 0 00 D(f (a); o~) + sup D(f (t); o~) : 2 6 t2[a;b] When f 0 (a) = o~, then D

1 b

a

Z

(F R)

!

b

00

f (x)dx; f (a)

a

!

sup D(f (t); o~) t2[a;b]

a)2

(b 6

:

(4.6)

Proof of Theorem 4.9. Let x 2 [a; b], then by Theorem 4.1 we get f (x) =

n 1 X

a)i

(x

f (i) (a)

i!

i=1

Rn (a; x);

where Rn (a; x) := (F R)

Z

a

x

Z

a

x1

Z

a

xn

1

f (n) (xn )dxn dxn

1

dx1

4.3 Results

(here we need x 1

D

b =

=

a 1

b

Z

!

b

f (x)dx; f (a)

a

1 b

a). We observe that

(F R)

a

a

D (F R)

Z

b

f (x)dx; (F R)

a

D (F R)

Z

(F R)

Z

n 1 X

b

=

b 1 b

a

D (F R)

Z

n 1 X

b

Z

X

b

b

f

(n)

=

b

a

n X

D

(x

(a)

(F R)

Z

1 b

a

Z

(F R)

Z

(a)

i!

Z

(x

f (n) (a)

a

f

(n)

= M+

1 b

a

D (F R)

a)n dx n!

(x

(a)

a

(F R)

Z

b

Rn (a; x)dx;

(x

f (n) (a)

a

Z

a

b

Rn (a; x)dx; (F R)

Z

a

!

b

a

! a)n dx n!

n X (b

a

dx

b

a)i+1 D(f (i) (a); o~) b a i=1 (i + 1)! Z b Z + D (F R) Rn (a; x)dx; (F R) 1

a)i

(b a)i+1 (i + 1)!

b

Rn (a; x) dx;

i!

(x

f (i) (a)

f (i) (a)

!

a)i

(x

!

Rn (a; x) dx; o~

i!

! a)n dx n!

Rn (a; x)dx; (F R)

n X

!

a)i

(x

(a)

b

i=1

"

(i)

b

D

(i)

a

i=1

a

=

(F R)

f

f

i=1

a

1

Rn (a; x) dx;

i!

i=1 n

a

(F R)

f (i) (a)

!

a)i

(x

f (a)dx

a

Z

f (a)dx

!

b

D (F R)

a

!

b

i=0

a

=

Z

a

a

1

71

!# a)n dx n!

b

f

(n)

(a)

(x

! a)n dx n!

72

4. Fuzzy Ostrowski Inequalities

= M+

1 b

a

D (F R)

Z

b

a

Z

Z

x

a

Z

x1

a

xn

1

f (n) (xn )dxn dxn

a

dx1 dx; (F R)

Z

Z

b

x

Z

x1

xn

1

(by Lemmas 1.13, 1.15) M+ D(f

(n)

(xn ); f

(n)

(by Theorem 4.4) M+ sup D(f

1 b

a

(a) dxn dxn

(n+1)

f (n) (a)dxn dxn

dx1 dx

1

a

a

a

a

Z

1 b

!

a

Z

a

Z

a

b

Z

x

a

Z

Z

x1

xn

1

a

a

dx1 dx

1

Z

(t); o~) dxn dxn

t2[a;b]

b

x

a

Z

x1

a

Z

xn

1

(xn

a)

a

dx1 dx

1

sup D(f (n+1) (t); o~)

= M+ = M+

(b a)n+2 b a (n + 2)! ! (b a)n+1 sup D(f (n+1) (t); o~) : (n + 2)! t2[a;b]

t2[a;b]

We have established inequalities (4.3) and (4.4). Consider g(x) := c(x a)` u, x 2 [a; b], c > 0, ` 2 Z+ , u 2 RF …xed. We prove that g is fuzzy continuous. Let xn 2 [a; b] such that xn ! x as n ! +1. Then D(g(xn ); g(x))

= D(c(xn a)` u; c(x a)` u) cj(xn a)` (x a)` jD(u; o~) ! 0:

Hence by the last argument, Propositions 4.2, 4.3 and Theorem 4.5 we obtain that f 2 C n+1 ([a; b]; RF ). We see that f

(i)

(a) = o~; for i = 1; : : : ; n:

That is M = 0. Furthermore it holds f

(n+1)

(x) = (b

a)(n + 1)!

u:

1

4.3 Results

73

Finally, we notice that L:H:S:((4:3); (4:4))

= D

1 b

a Z

= D u

(F R)

Z

b n+1

a)

a

=

(b

((b

a

(x

!

b

a)(x !

dx; o~

n+1

a)

=D u

u)dx; o~ (b

a)n+2 ; o~ n+2

n+2

a) n+2

D(u; o~):

Also we …nd R:H:S:((4:3); (4:4)) = (b a)(n+1)!D(u; o~)

(b a)n+2 (b a)n+1 = D(u; o~): (n + 2)! n+2

Proving (4.3) and (4.4) sharp, in fact attained inequalities.

5 A FUZZY TRIGONOMETRIC APPROXIMATION THEOREM OF WEIERSTRASS-TYPE

In this chapter we show that any 2 -periodic fuzzy continuous function from R to the fuzzy number space RF , can be uniformly approximated by some fuzzy trigonometric polynomials. This chapter is based on [31].

5.1 Introduction A fuzzy valued function f : [a; b] ! RF is said to be continuous at x0 2 [a; b], if for each " > 0 there is > 0 such that D(f (x); f (x0 )) < ", whenever x 2 [a; b] and jx x0 j < . We say that f is fuzzy continuous on [a; b] if f is continuous at each x0 2 [a; b], and denote the space of all such functions by CF [a; b]. For f 2 CF [0; 1], let us consider the Bernstein-type fuzzy polynomials Bn(F ) (f )(x) =

n X

k=0

f (k=n)

pk;n (x);

n 2 N; x 2 [0; 1]

(5.1)

P where pk;n (x) = nk xk (1 x)n k and means addition with respect to in RF . It is obvious that pn;k (x) 0, 8x 2 [0; 1] and pn;0 (x), pn;1 (x); : : : ; pn;n (x) are linearly independent algebraic polynomials of degree n. Concerning these fuzzy polynomials recently was proved the following G.A. Anastassiou: Fuzzy Mathematics: Approximation The., STUDFUZZ 251, pp. 75–82. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

76

5. A fuzzy trigonometric approximation theorem of Weierstrass-type

Theorem 5.1 (see [49]). If f 2 CF [0; 1], then for any " > 0, there exists n0 2 N, such that D Bn(F0 ) (f )(x); f (x) < ";

8x 2 [0; 1]:

(5.2)

Moreover, in [66, p. 642, Theorem 13.13] it is proved the following quantitative estimate D Bn(F ) (f )(x); f (x)

3 (F ) 1 ! f; p 2 1 n

(F )

where ! 1 (f ; ) = supfD(f (x); f (y)); jx modulus of continuity of f .

;

8n 2 N; 8x 2 [0; 1]; yj

(5.3)

, x; y 2 [0; 1]g is the

It is the main aim of this chapter to present an analogue of Theorem 5.1 in the case of approximation of 2 -periodic fuzzy continuous functions, by some trigonometric fuzzy polynomials.

5.2 Preliminaries The proof of the main result requires some auxiliary results and concepts. A function f : R ! RF will be called 2 -periodic if f (x + 2 ) = f (x), 8x 2 R. A generalized fuzzy trigonometric n, will be P polynomial of degree de…ned as a …nite sum of the form Tk (x) ck , where ck 2 RF and Tk (x) are trigonometric polynomials of degree n. Here, from approximation theory’s point of view, that is to approximate a function by other simpler functions; the polynomials Tk (x), are not necessarily supposed to be linearly independent on R. We need a particular case of the Henstock integral introduced in [52]. De…nition 5.2 ([66, p. 644]). Let f : [a; b] ! RF . We say that f is Riemann integrable to I 2 RF , if for any " > 0, there exists > 0, such that for any division P = f[u; v]; g of [a; b] with the norm (P ) < , we have ! X D (v u) f ( ); I < ": (5.4) P

Rb We write I = (R) f (x)dx. a

We also need the following:

Lemma 5.3. For any a; b 2 R, a; b D(a

u; b

u)

0, and any u 2 RF we have jb

aj D(u; o~);

(5.5)

5.3 Main Result

where o~ 2 RF is de…ned by o~ =

77

f0g .

Proof. If a = b then the inequality becomes o = o. Let us suppose 0 a < b (the case 0 b < a is similar). We get (denoting = a=(b a) 0) D(a

u; b

u)

a b u ; (b a) u b a b a a b u (b a) D u; b a b a (b a)D( u; (1 ) u) (b a)[D( u; u) + D(u; (1 ) u)] (b a)[D( u o~; u (1 ) u) + D( u (1 ) u; (1 ) u o~)] (b a)[D(~ o; (1 ) u) + D( u; o~)] (b a)[(1 )D(~ o; u) + D(~ o; u)] = (b a)D(u; o~):

= D (b = = = = =

a)

Remark 5.4. If a; b are not both 0 then Lemma 5.3 is not in general valid, because the property ( + ) u = u u, ( ; 2 R, u 2 RF ) is valid only when ; 0, but in general it is not valid for arbitrary ; 2 R.

5.3 Main Result (F )

Let us denote by C2 (R) = ff : R ! RF ; f is fuzzy continuous and 2 periodic on Rg. Theorem 5.5. For any " > 0, there exists a generalized fuzzy trigonometric polynomial T (x), such that D(f (x); T (x)) < ";

8x 2 R:

(5.6)

Proof. Let us de…ne the fuzzy Jackson-type operators (see [66, p. 646]) Z Z Jn (f )(x) = (R) Kn (t) f (x + t)dt = (R) Kn (u x) f (u)du; h i4 R where Kn (t) = Ln0 (t), n0 = n2 , Ln (t) = n 1 sin(nt=2) , sin(t=2) According to [66, p. 647, Theorem 13.14], we have D(Jn (f )(x); f (x))

(F )

C! 1

f;

1 n

;

Ln (t)dt = 1.

8n 2 N; 8x 2 R;

78

5. A fuzzy trigonometric approximation theorem of Weierstrass-type (F )

where ! 1 (f ; ) = supfD(f (x); f (y)); jx yj , x; y 2 Rg represents the modulus of continuity of f . Then, for …xed " > 0, there exists n0 2 N such that

(F )

D(Jn0 (f )(x); f (x))

C! 1

f;

1 n0

<

" : 2

Now, divide [ ; ] into m equal parts and consider the Riemann sum of the integral Jn0 (f )(x), i.e., X

m

(f )(x) =

m 2 X K n0 m

+

k=1

2 k m

x

f

+

2 k : m

P Obviously m (f )(x) represents a generalized fuzzy trigonometric polynomial of degree n0 . By [52, Theorem 2.5] we can write

Jn0 (f )(x)

Z

=

(R)

=

Z m X

k=1

Kn0 (u

x)

f (u)du

+k 2m

Kn0 (u

x)

f (u)du:

+(k 1)2 =m

For any …xed x 2 R we obtain X D Jn0 (f )(x); (f )(x) m Z m +k 2m X Kn0 (u =D m X

k=1 m X

D

k=1

2 K n0 m

x)

f (u)du;

+(k 1) 2m

k=1

2 m Z

K n0

+k

2 m

x

f

+k

+k 2m +(k 1) 2m

+k

2 m

Kn0 (u x

x) f

f (u)du; +k

2 m

2 m

5.3 Main Result

=

m X

Z

D

k=1

Z

+k 2m +(k 1) 2m

Kn0 (u

+k 2m 1) 2m

+(k

79

K n0

+k

x)

2 m

f (u)du; x

f

+k

2 m

(see [52, Remarks 3.2] or [66, p. 644, Lemma 13.2, (ii)]) m Z +k 2m X D(Kn0 (u x) f (u); k=1

+(k 1) 2m

K n0

+k (F )

2 !1 Let u1 ; u2 2 [

2 m

K n0 (

; ], ju1

x x)

u2 j

f f ( );

2 m

+k 2 m

2 m

du

:

be …xed. We have

D(Kn0 (u1 x) f (u1 ); Kn0 (u2 x) f (u2 )) D(Kn0 (u1 x) f (u1 ); Kn0 (u1 x) f (u2 )) +D(Kn0 (u1 x) f (u2 ); Kn0 (u2 x) f (u2 )) = Kn0 (u1 x)D(f (u1 ); f (u2 )) + D(Kn0 (u1 x) f (u2 ); Kn0 (u2 x) f (u2 )) 2 (F ) Mn 0 ! 1 f; + D(Kn0 (u1 x) f (u2 ); Kn0 (u2 x) f (u2 )) m (by Lemma 5.3) 2 (F ) Mn 0 ! 1 f; + jKn0 (u1 x) Kn0 (u2 x)j D(f (u2 ); o~) m 2 2 (F ) + ! 1 K n0 ; Mn 0 ! 1 f; m m supfD(f (u); o~); u 2 [0; 2 ]g 2 2 (F ) Mn 0 ! 1 f; + C! 1 Kn0 ; ; m m because by e.g., the proof of Lemma 13.2 in [66, p. 644], we get that F (u) = D(f (u); o~), u 2 R is 2 -periodic and continuous on R, that is 0 F (u) C, 8u 2 R. Now, choose m0 2 N su¢ ciently large such that (F )

M !1

f;

2 m

+ C! 1 Kn0 ;

2 m

<

" 2

(this is possible because (F )

lim ! 1

m!+1

f;

2 m

=

lim ! 1 Kn0 ;

m!+1

2 m

= 0):

80

5. A fuzzy trigonometric approximation theorem of Weierstrass-type

Finally, we obtain X D (f )(x); f (x)

X

D

m0

m0

(f )(x); Jn0 (f )(x)

+ D(Jn0 (f )(x); f (x)) <

i.e., the theorem is proved with T (x) =

" " + ; 2 2

8x 2 R;

P

m0 (f )(x).

5.4 General Remarks In this section we present some properties in the space (RF ; D) that can be useful in the study of approximation of fuzzy valued functions. They can be summarized by the following: Lemma 5.6. (i) If we denote o~ = respect to , i.e. u

o~ = o~

f0g

then o~ 2 RF is neutral element with

u = u;

for all u 2 RF :

(ii) With respect to o~, none of u 2 RF , u 6= o~, has opposite in RF . (iii) For any a; b 2 R with a; b 0 or a; b 0, and any u 2 RF , we have (a + b)

u=a

u

b

u:

For general a; b 2 R, the above property does not hold. (iv) For any 2 R and any u; v 2 RF , we have (u (v) For any ;

v) =

u

v:

2 R and any u 2 RF , we have (

u) = (

)

u:

(vi) If we denote kukF = D(u; o~), 8u 2 RF , then k kF has the properties of a usual norm on RF , i.e., kukF = 0 i¤ u = o~, k ukF = j j kukF , ku vkF kukF + kvkF , kukF kvkF D(u; v). Proof. (i) It is immediate by the de…nitions of o~ and . (ii) Let u 2 RF be …xed and let us suppose that there exists v 2 RF , such that u v = o~. We get [u]r + [v]r = [~ o]r ; 8r 2 [0; 1]; r r = ] = [0; 0] = f0g, and …nally ur +v r = 0, ur+ +v+ that is [ur ; ur+ ]+[v r ; v+ r r 0, 8r 2 [0; 1]. Because u < u+ at least for one r (since u 6= o~), the previous r relation implies that for r the contradiction v+ < vr .

5.4 General Remarks

(iii) Let [u]r = [ur ; ur+ ], r 2 [0; 1]. For a; b [ur ; ur+ ]

(a + b)

0 we get

= [(a + b)ur ; (a + b)ur+ ] = [aur + bur ; aur+ + bur+ ] = [aur ; aur+ ] + [bur ; bur+ ] = a [u]r + b [u]r ; 2 [0; 1]:

8r If a; b 2 R, a; b

81

0, we get (a + b)

(a + b) [ur ; ur+ ]

0 and

= [(a + b)ur+ ; (a + b)ur ] = [aur+ + bur+ ; aur + bur ] = [aur+ ; aur ] + [bur+ ; bur ] = a[ur ; ur+ ] + b[ur ; ur+ ]; 2 [0; 1]:

8r

If a; b 2 R are arbitrary, we cannot in general apply the above reasonings. (iv) For any r 2 [0; 1] we get by the de…nitions of and [

(u

Now, if

= =

[u v]r = ([u]r + [v]r ) r r ([ur ; ur+ ] + [v r ; v+ ]) = ([ur + v r ; ur+ + v+ ]):

0, then [

If

v)]r

(u

v)]r

r = [ ur + v r ; ur+ + v+ ] r r r r = [ u ; u+ ] + [ v ; v+ ] = [u]r + [v]r :

0, then [

(u

v)]r

r = [ (ur+ + v+ ); (ur + v r )] r r = [ ur+ + v+ ; ur + v r ] = [ ur+ ; ur ] + [ v+ ; vr ] = [u]r + [v]r :

(v) By de…nition of [

, for any r 2 [0; 1] we get (

u)]r = [

u]r =

[u]r :

(vi) The …rst three properties are immediate consequences of the properties of D. For the last property, by e.g. [69, p. 21, Exercise 1.2], in any metric space we have jD(u; w) D(v; w)j D(u; v). Taking w = o~ we get the proof. Remark 5.7. The properties (ii) and (iii) in Lemma 5.6 show us that (RF ; ; ) is not a linear space over R, and consequently (RF ; k kF ) is not a normed space. But the properties of D and Lemma 5.6, (iv)–(vi), have as an e¤ect that the metric properties of a function de…ned on R with values in a Banach space could be translated also to fuzzy functions in a similar manner. Remark 5.8. The fact that the above Bernstein-type and Jackson-type (see the proof of Theorem 5.5) generalized fuzzy polynomials are of the

82

form

5. A fuzzy trigonometric approximation theorem of Weierstrass-type n P

ck

Pk (x) and

k=1

n P

ck

Tk (x) (respectively), where Pk (x) and

k=1

Tk (x) are positive polynomials on [0; 1] and R (respectively), is essential in applying the properties of the metric D. In case Pk (x) (or Tk (x)) are not positive, by Lemma 5.6, (iii), the properties of D cannot be applied. The concept of fuzzy polynomials can also be de…ned as in e.g. [66, p. 621, De…nition 13.2] n X

xk ;

ck

x 2 [a; b]; ck 2 RF ;

k=0

and

n X

(ak

cos kx + bk

ak ; bk 2 R; k = 0; n:

sin kx);

k=0

Unfortunately, because of Lemma 5.6, (iii), we cannot apply the properties of D in order to obtain approximation results by such polynomials, so the problem of closure (in the uniform metric D (f; g) = supfD(f (x); g(x))g) x

of the set of such polynomials remains open. Obviously the generalized fuzzy polynomials of Bernstein-type and of Jackson-type mentioned earlier are not of the latter kind. Remark 5.9. The property (iv) of Lemma 5.6 is useful to show the linearity of some approximation operators, like that of Bernstein-type Bn(F ) (f )(x) =

n X

k n

f

k=0

pn;k (x);

for example. Indeed, we have Bn(F ) (

f

g)(x) =

n X

[

f

g]

k=0

=

n X

f

k=0

k n

g

k n

= (by Lemma 5.6, (iv)) n X k = pn;k (x) f n

+

=

+

k=0

=

n X

pn;k (x)

k=0 Bn(F ) (f )(x)

+

f

k n

n X

k n

pn;k (x)

pn;k (x)

pn;k (x)

g

k n

pn;k (x)

g

k n

k=0 n X

k=0

Bn(F ) (g)(x):

6 ON BEST APPROXIMATION AND JACKSON-TYPE ESTIMATES BY GENERALIZED FUZZY POLYNOMIALS

In [31] was proved that any 2 -periodic continuous fuzzy-number-valued function can be uniformly approximated by sequences of generalized fuzzy trigonometric polynomials, but without giving any estimates for the approximation error. In this chapter, connected to the best approximation problem we present Jackson-type estimates. For the algebraic case we also give a Jackson-type estimate, using the Szabados-type polynomials. Finally, as an application we study the convergence of fuzzy Lagrange interpolation polynomials. This chapter relies on [41].

6.1 Introduction A function f : R !RF is called 2 periodic if f (x + 2 ) = f (x); 8x 2 R. A generalized fuzzy trigonometric polynomial of degree n is de…ned as Pn a …nite sum of the form T (x) = k=0 tk (x) ck ; where ck 2RF and tk (x) are usual trigonometric polynomials of degree n: Let us denote C2F (R) = ff : R !RF ; f is 2 periodic and continuous on Rg: In [31], the following Weierstrass-type result is proved. Theorem 6.1. For any f 2 C2F (R); there exists a sequence of generalized fuzzy trigonometric polynomials (Tn (x))n2N such that lim sup D(Tn (x); f (x)) = 0:

n!1 x2R

G.A. Anastassiou: Fuzzy Mathematics: Approximation The., STUDFUZZ 251, pp. 83–98. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

84

6. Jackson-Type Estimates by Generalized Fuzzy Polynomials

Other results concerning approximation and interpolation of fuzzy-numbervalued functions can be found in: [11], [59], [57], [81], [84], [72], [80]. But the problems of existence of best approximation by fuzzy polynomials and of convergence of fuzzy Lagrange polynomials, were not discussed much in the fuzzy mathematical literature. In Section 6.2 we consider some problems of best approximation by generalized fuzzy trigonometric polynomials (of degree n) and a Jackson-type estimate is proved. Section 6.3 contains the case of best approximation by generalized fuzzy algebraic polynomials. In Section 6.4, as an application, we prove the convergence of Lagrange interpolating polynomials for the class of fuzzy Lipschitz functions of order > 12 :

6.2 Best approximation, trigonometric case On C2F (R) let us consider the uniform distance D (f; g) = supfD(f (x); g(x); x 2 Rg = supfD(f (x); g(x); x 2 [

; ]g;

8f; g 2 C2F (R): For an interval I R P and a subset K RF ; K 6= ;; let us consider n VnK;I = fTn ; Tn (x) = k=0 tk (x) ck ; where all ck 2 K and each tk (x) is an usual trigonometric polynomial of degree n with all its coe¢ cients belonging to Ig: RF and I R and for each n 2 N; it is For …xed f 2 C2F (R); K natural to consider the following problem of best approximation. EnK;I (f ) = inffD (f; Tn ); Tn 2 VnK;I g: In the study of this problem, it is essential the following Theorem 6.2. If K RF ; K 6= ;; is a compact and I = [A; B] is compact subinterval of R; then the set VnK;I is sequentially compact in the metric space (C2F (R); D ); for all n 2 N. Proof. Let us denote by TnI = ftk ; tk is usual trigonometric polynomial of degree n; with all coe¢ cients belonging to IgPand de…ne ' : K n+1 n (TnI )n+1 ! C2F (R); by '(c0 ; :::; cn ; t0 ; :::; tn )(x) = k=0 tk (x) ck : First let us prove that ' is continuous. Indeed, Pn let another generalized fuzzy trigonometric polynomial of degree n; k=0 sk (x) dk : By the properties of D in Introduction and by [31, Lemma 2.2], we get D

n X

k=0

tk (x)

ck ;

n X

k=0

sk (x)

dk

!

6.2 Best approximation, trigonometric case n X

k=0

jtk (x)jD(ck ; dk ) +

n X

k=0

jtk (x)

sk (x)j

85

D(ck ; e 0);

where e 0 = f0g 2 RF : Pn Because each tk (x) is of the form 0 + j=0 ( j cos jx + j sin jx); with j ; j 2 I = [A; B]; it immediately follows that jtk (x)j (2n + 1) maxfjAj; jBjg = M; for all k = 0; n and all x 2 R: 0g is Also, because ck 2 K compact, 8k = 0; n; we get that K 0 = K [ fe compact too (in the metric space RF ), which implies that it is bounded and therefore there exists a constant M 0 > 0 such that D(ck ; e 0) M; 8ck 2 K: As a conclusion, it follows ! n n X X D tk (x) ck ; sk (x) dk k=0

M

n X

k=0

D(ck ; dk ) + M 0

k=0

n X

k=0

ktk (x)

sk (x)k

(here k k denotes the usual uniform norm on the set of real valued, 2 periodic functions, denoted by C2 ). This last inequality immediately shows that ' is continuous, if K n+1 (TnI )n+1 is endowed with the box metric given by [(c0 ; :::; cn ; t0 ; :::; tn ); (d0 ; :::; dn ; s0 ; :::; sn )] = max fD(ck ; dk ); ktk k=0;n

sk kg:

Now we claim that TnI is compact in (C2 ; k k). Indeed, if we consider : I 2n+1 ! C2 de…ned by (

0 ; :::;

n;

1 ; :::;

n )(x)

=

0

+

n X

[

k

cos kx +

k

sin kx];

k=0

then it easily follows that is continuous and therefore TnI = (I 2n+1 ) is compact. As a conclusion, K n+1 (TnI )n+1 is compact which implies that VnK;I = '(K n+1 (TnI )n+1 ) is compact, and therefore as a compact subset of a metric space, VnK;I is sequentially compact. As an immediate consequence of Theorem 6.2, we get Corollary 6.3. Let f 2 C2F (R): If K RF ; K 6= ; is compact and I = [A; B] is a compact interval of R, then for each n 2 N, there exists T 2 VnK;I such that EnK;I (f ) = D (f; T ); i.e. T is a generalized fuzzy trigonometric polynomial (of degree n) of best approximation for f: Proof. Since K 6= ;; it follows that VnK;I 6= ;: 1 ; there exists Tm 2 VnK;I such that EnK;I (f ) D (f; Tm ) For " = m 1 K;I En (f ) + m ; m = 1; 2; :::. Since by Theorem 6.2, VnK;I is sequentially

86

6. Jackson-Type Estimates by Generalized Fuzzy Polynomials

compact, the sequence (Tm )m has a convergent subsequence (Tmk )k to an element T 2 VnK;I : Passing above to limit, we get EnK;I (f ) = D (f; T ); i.e. T is of best approximation, which proves the corollary. Note. If f 2 C2F (R); then by f ([ in Corollary 6.3 we can take K = f ([

; ]) = K compact, it follows that ; ]) (i.e. depending on f ).

In what follows we will derive a Jackson-type estimate for EnK;I (f ) with K = f ([ ; ]): Theorem 6.4. If f 2 C2F (R) and [ 1; 1] [A; B]; then there exists a constant C > 0 (independent of f and n) and an index n0 2 N (independent of f ) such that for K = f ([ ; ]) we have C! F 1

EnK;[A;B] (f )

f;

1 n

; 8n

n0

where ! F yj g: 1 (f; ) = supfD(f (x); f (y); x; y 2 R, jx Proof. In [66, p.646] was introduced the following fuzzy Jackson operator Z Z Jn (f )(x) = (R) Kn (t) f (x + t)dt = (R) Kn (u x) f (u)du; where Kn (t) = Ln0 (t); n0 = [n=2] + 1; Ln0 (t) =

sin(n0 t=2) 3 0 0 2 2 n [2(n ) + 1] sin(t=2)

4

;

Z

Ln0 (t)dt = 1;

and it was proved [66, p.647, Theorem 13.14] the estimate D(Jn (f )(x); f (x)

C! F 1

f;

1 n

; 8n 2 N; x 2 R:

On the other hand by taking the Riemann sum of Jn (f )(x) (on an equidistant division of [ ; ]), we get 0

n 2 X Tn (x) = 0 Ln0 n

+

k=0

2k n0

x

f

+

2k n0

:

Obviously Tn (x) is a generalized fuzzy trigonometric polynomial of degree n (since n0 n) and by [40, Corollary 3], for all x 2 [ ; ] and n 2 N; we have 2 D(Jn (f )(x); Tn (x)) 2 ! F f; 0 1 n [ ; ] 2 (2 + 1)! F 1

f;

1 n0

4 (2 + 1)! F 1 [

; ]

f;

1 n

[

; ]

6.2 Best approximation, trigonometric case

87

(since n0 = [n=2] + 1 > n=2). But reasoning exactly as in the usual case (see [30, p.75, Lemma 2.2.1], we have ! F !F 2! F 1 (f; )[ 1 (f; ) 1 (f; )[ ; ] ; ]: As a consequence, we obtain the following Jackson-type estimate D(Tn (x); f (x)) C! F 1

D(Tn (x); Jn (f )(x)) + D(Jn (f )(x); f (x)) f;

1 n

; for all n 2 N; x 2 [

; ]:

1 1 F too). (Note that above ! F 1 f; n can be replaced by ! 1 f; n [ ; ] To …nish the proof, we have to calculate the bounds for the coe¢ cients of the usual trigonometric polynomials Ln0 + 2k x in the expression n0 of Tn (x): First, it is well known the identity 2

sin mx 2 sin x2 It follows

=m+2

m X1

(m

k) cos kx:

k=1

Pm 1 Pm 1 = m + 2 k=1 (m k) cos kx m + 2 k=1 (m k) cos kx Pm 1 Pm 1 Pm 1 = m +4m k=1 (m k) cos kx+4 i=1 j=1 (m i)(m j) cos(ix) cos(jx) = P Pm 1 P m 1 m 1 1 2 = m +4m k=1 (m k) cos kx+4 i=1 j=1 (m i)(m j)f 2 [cos x(i+ j) + cos x(i j)]g = Pm 1 Pm 1 Pm 1 = m2 + 4m k=1 (m k) cos kx + 2 i=1 j=1 (m i)(m j) cos x(i + Pm 1 Pm 1 j) + 2 i=1 (m i)(m j) cos x(i j) = Pj=1 Pm 1 m 1 2 = m + 4m k=1 (m k) cos kx + 2 i)(m j) cos x(i + j) + i;j=1 (m m Pm 1 Pi+j m 1 2 i)(m j) cos x(i + j) + 2 i;j=1 (m i)(m j) cos x(i j) := i;j=1 (m sin mx 2 sin x 2 2

4

i+j>m

m2 + S 1 + S 2 + S 3 + S 4 : By simple calculation we can write S2 =

m k X X1 [2 (m

S3 =

m X2

[2

p=1

i)(m

mX p 1

(m

(p + i))(m

1)2 ) +

m X2 k=1

By the relations

i=1

i))]cos(kx);

(m

i))]cos((m + p)x);

i=1

S4 = 4(12 + 22 + ::: + (m

k X1

(k

i=1

k=2

(m i)(m (k i)) =

k X1 i=1

[4

mX 1 k

(m

i)(m

(k + i))]cos(kx):

i=1

[ i2 +ik+m(m k)]

k X1 i=1

[k 2 +4m(m k)]=2

88

6. Jackson-Type Estimates by Generalized Fuzzy Polynomials

1)[k 2 + 4m(m

(k mX p 1

k)]=2

1)[m2 + 4m(m

(m

(m (p+i))(m (m i)) =

i=1

mX p 1

[ i2 +i(m p)]

i=1

2

(m

p) (m

mX 1 k

p

(m

1)=2

i)(m

i

(m

mX 1 k

i=1

(m

(m p)2 =2

2)=2; p = 1; :::; m

(m

1)(m

1

2;

k)

i=1

1)(m 2

mX p 1 i=1

2

1) (m

k)

2)]=2; k = 2; :::; m;

1

k)2

2

1 + 2 + ::: + (m

(m

1)(m 2

1) = m(m

2)2 ; k = 1; :::; m 1)(2m

2;

1)=6; sin

mx

4

it follows that for k 2 f0; :::; 2m 2g, the coe¢ cients of cos kx in sin x2 2 are all positive and bounded by an algebraic polynomial of degree 3, with constant coe¢ cients, independent of f , let us denote it by H3 (m) (in S1 , obviously all the coe¢ cients of cos(kx) are bounded by 4m(m 1)). As a conclusion, it easily follows that in (2 =n0 )Ln0 + 2k x (which n0 contains terms in cos kx and sin kx), all the coe¢ cients are bounded, in absolute value, by F (n0 ) = 3H3 (n0 )=[(n02 (2(n02 + 1)], that is an n0 2 N (independent of f ) can be found (constructively), such that for all n0 n0 we have F (n0 ) 1 (since F (n0 ) converges to 0 when n0 converges to in…nity). K;[ 1;1] . Now, Therefore, for n 3n0 , it follows that Tn (x) belongs to Vn K;[ 1;1] K;[A;B] (f ); which (f ) En for [ 1; 1] [A; B] it is obvious that En proves the theorem. Remark 6.5. From the proof it is easily seen that an interval [A; B] (independent of f and n) can be constructively determined such that the Jackson kind estimate in Theorem 6.4 holds for all n = 1; 2; :::

6.3 Best approximation, algebraic case Let CF [a; b] = ff : [a; b] ! RF ; f continuous on [a; b]g where [a; b] is a compact subinterval of R. If we de…ne the concept of generalized fuzzy algebraic n as in [49], i.e. as a …nite sum of the Pn polynomial of degree form k=0 pk (x) ck ; where ck 2 RF and pk (x) are algebraic polynomials of degree n; we can repeat the reasonings in the above Theorem 6.2 and Corollary 6.3 simply by replacing [ ; ] there by [a; b]; C2F (R) by CF [a; b] and the generalized fuzzy trigonometric polynomials by generalized fuzzy algebraic polynomials. But if we consider probably the simplest generalized fuzzy algebraic polynomials, given by the fuzzy Bernstein polynomials Bn (f )(x) =

n X

k=0

pn;k (x)

f (k=n); x 2 [0; 1]

6.3 Best approximation, algebraic case

89

where pn;k (x) = nk xk (1 x)n k ; we easily see that the coe¢ cients of xs in pn;k (x) are in general unbounded, for k = 1; n 1 , while however jpn;k (x)j 1; 8x 2 [0; 1]; n 2 N; k = 0; n: Therefore, in algebraic case, would be more natural to consider the problem of best approximation as follows. For a constant M >P 0 and a subset K RF ; K 6= ;; let us consider n AK;M [a; b] = fP ; P (x) = ck ; where all ck 2 RF and pk (x) n n n k=0 pk (x) are algebraic polynomials of degree n; satisfying jpk (x)j M; for all k and all x 2 [a; b]g: For …xed f 2 CF [a; b]; K RF ; K 6= ; and M > 0 and for each n 2 N, we can consider the following problem of best approximation EnK;M (f ) = inffD (f; Pn ); Pn 2 AK;M [a; b]g; n where D (f; g) = supfD(f (x); g(x); x 2 [a; b]g; for f; g 2 CF [a; b]: We have: Theorem 6.6. If K RF ; K 6= ; is compact and M > 0; then the set AK;M [a; b] is sequentially compact in the metric space (CF [a; b]; D ); for n all n 2 N. Proof. Let us denote by PnM = fp; p usual algebraic polynomials of n+1 degree n; satisfying jp(x)j M; for all x 2 [a; b]g and Pn de…ne ' : K M n+1 (Pn ) ! CF [a; b]; by '(c0 ; :::; cn ; p0 ; :::; pn )(x) = k=0 pk (x) ck : Reasoning exactly as in the proof of Theorem 6.2, we get that ' is continuous and because PnM is compact in C[a; b] (endowed with the uniform norm k k), see e.g. [56, p16 Lemma 1], we get the desired conclusion. Consequently, we obtain the following Corollary 6.7. Let f 2 CF [a; b]: If K RF ; K 6= ; is compact and M > 0; then for all n 2 N, there exists T 2 AK;M [a; b] such that EnK;M (f ) = n D (f; T ); i.e. T is a generalized fuzzy algebraic polynomial (of degree n), of best approximation for f: Remark 6.8. By [66, p. 642, Theorem 13.13], we immediately obtain EnK;1 (f )

C! F 1

1 f; p n

[0;1]

; 8n 2 N, f 2 CF [0; 1]; K = f ([0; 1]):

In what follows we deduce Jackson-type estimate for EnK;M (f ) by using some fuzzy analogous of Szabados-type polynomials (see e.g.[101]). For this aim we need the following lemmas. Lemma 6.9. Let f :

1 1 4; 4

Rn (f; x) =

! RF ; be continuous and n X

k= n

rn;k (x)

f (xk );

90

6. Jackson-Type Estimates by Generalized Fuzzy Polynomials 4

xk ) with rn;k (x) = Pn (x (x xj ) j= n estimate holds true:

4

k 4n ;

and xk =

5! F 1

D (f; Rn )

1 n [

f;

n; n: Then the following

k=

: 1 1 4;4

]

Proof. We follow the proof of Lemma 1 in [101] for r = 0 and s = 4: Thus, for …xed x; let i be an index such that jx

xi j = min jx

1 : 8n

xk j

jkj n

(6.1)

Then evidently ji

kj 8n

jx

1 1 Denote I = 4 ; 4 : Since rn;k (x) 1; by the properties of D we have:

D(f (x); Rn (f; x)) = D

ji

xk j

n X

0; 8k = !

rn;k (x)

k= n n X

rn;k (x)

D(f (x); f (xk ))

(x xi )4

!F 1 (f; jx

xi j)I + (8n)

4

n X

k=

!F 1

Pn

k= n rn;k (x)

n X

rn;k (x)

f;

n

4

jx xk j

ji

kj 2n

I

5! F 1

f;

k6=i

2

6 41 +

n X

k=

n

k6=i

ji

kj

3

27

F

5 !1

f;

1 n

f (xk )

I

xk j)I

!F 1 (f; jx 8n ji kj 1 n

=

!

rn;k (x)! F 1 (f; jx

k= n

k= n

n X

(6.2)

k= n

k= n

n X D(f (x); f (xk )) (x xk )4

n; n and

f (x);

n X

k= n

(x xi )4

kj ; for i 6= k: 2n

xk j)I 4

: I

Remark 6.10. The result in the above Lemma 6.9 can be seen as a Jackson-type estimate for the error of the approximation by fuzzy generalized rational functions. For the next results we need an embedding theorem. Theorem 6.11. (see e.g. [52]) Let C[0; 1] be the class of all real valued bounded functions f on [0; 1]; such that f is left continuous on (0; 1] and f

6.3 Best approximation, algebraic case

91

has right limit for x 2 [0; 1); especially f is right continuous at 0: With the norm kf k = supx2[0;1] jf (x)j ; C[0; 1] is a Banach space. For u 2 RF ; de…ne j : RF ! C[0; 1]; j(u) = (u ; u+ ); where u (r) = ur and u+ (r) = ur+ : Then j(RF ) is a closed convex cone in the Banach space C[0; 1] C[0; 1] and: (i) j(s u t v) = s j(u) + t j(v); 8u; v 2 RF and s; t 2 R+ (here ”+” and ” ”denote the addition and scalar multiplication in C[0; 1] C[0; 1]). (ii) D(u; v) = kj(u) j(v)k ; 8u; v 2 RF : i.e. j embeds RF in C[0; 1] C[0; 1] isometrically and isomorphically (k k being the usual product norm in C[0; 1] C[0; 1]). The following lemmas give some approximation properties in Banach spaces. 1 1 Lemma 6.12. Let (B; k P k) be a Banach space and g : 4; 4 ! B n continuous. Let Rn (g; x) = k= n rn;k (x) g(xk ); with rn;k as in Lemma 6.9. Then 1 ; kRn0 (g; x)k 900n! B1 g; n [ 1;1] 4 4

Rn0 (g; x)

is the Fréchet derivative of Rn (g; x) in B and ! B1 (g; )[ 1 ; 1 ] = 4 4 1 1 sup kg(x) g(y)k ; x; y 2 ; jx yj : ; 4 4 Proof. The proof is the same as that of [101, Lemma 2], written in the case of functions with values in a Banach space. Thus by (6.1) and (6.2) we get Pn Pn 4 k= n g(xk )(x xk ) 5 k= n (x xk ) 4 0 kRn (g; x)k = + Pn 2 xk ) 4 k= n (x Pn Pn 4 k= n g(xk )(x xk ) 4 k= n (x xk ) 5 + Pn 2 xk ) 4 k= n (x

where

Pn

k= n (x

=2

2(x

xi )8

xk )

n X

k= n

8 > n <X > :j=

n

j6=i

jx

5

Pn

j= n [g(xk )

Pn

k= n (x

xk j

5

(8n)

3

g(xj )](xk

xk )

g;

1 n

[

1 1 4;4

2n 5

8n jj

ij

jj

ij2 +

n X

k=

n

k6=i

8n jk ij

5

xj )

5

4 2

1 1 B ! g; 4n 1 n [ ! B1

xj )(x

2

n X jj jx 1 1 ; ] j= n 4 4

j6=k

kj2 xj j5

]

6 4jk

ij2 +

n X

j=

jj jj n

j6=i

39 > kj2 7= 5 ij5 > ;

92

6. Jackson-Type Estimates by Generalized Fuzzy Polynomials

32n! B1 g;

8 > n <X

1 n [

1 1 > :j= 4;4]

n

3

ij

jj

+

n X

k=

n

jk

3

ij

k6=i

j6=i

900n! B1 g;

1 n [

2

6 41 + 4

n X

j=

continuous. Let Pn (x) = cn R1 P (x)dx = 1 and let 1 n Kn (f; x) =

Z

jj

j6=i

: 1 1 4;4

] 1 1 2; 2

Lemma 6.13. Let (B; k k) be a Banach space and f : cos(2n arccos x) x2 sin2 4n

n

39 > = 37 ij 5 > ;

2

! B

where cn is chosen such that

1 2

[f (t) 1 2

f (0)]Pn (t

x)dt + f (0)

be the Bojanic-DeVore operator, where the integral is considered to be the 1 1 usual Riemann integral for functions g : 2 ; 2 ! B. Then kf

Kn (f )kC ([

1 1 4;4

];B)

C4 ! B1 f;

1 n [

: 1 1 4;4

]

Proof. The proof is the same as the proof of [56, p. 275-276, Proposition 3.4] but for functions with values in a Banach space. Indeed, …rstly Pn (x) is an even algebraic polynomial of degree 4n 4; therefore Kn (f; x) is a generalized (algebraic) polynomial of degree 4n 4, with coe¢ cients in 1 1 the Banach space B: Let us denote I = [ 1; 1]; I 0 = 4 ; 4 and for 1 1 0 + x; x : If we denote g(x) = f (x) f (0) and …xed x 2 I ; Ix = 2 2 R 12 Ln (g; x) = 1 g(t)Pn (t x)dt; then by [56, p.276, relation (3.10)], it follows 2

kf

Kn (f )kC(I 0 ;B) = kg

Kn (g)kC(I 0 ;B) ;

(6.3)

where C(I 0 ; B) = ff : I 0 ! B; f continuous on I 0 g: Then for …xed x 2 I 0 ; as in [56, p.276] we get Z Z Ln (g; x) g(x) = [g(x + u) g(x)]Pn (u)du g(x) Pn (u)du Ix

where g(x) Z

R

InIx

[g(x + u)

Pn (u)du

C2 kgkC(I 0 ;B) n

C(I 0 ;B)

1 n

C1 ! B1 g;

g(x)]Pn (u)du

Ix

C(I 0 ;B)

Since kgkC(I 0 ;B) = kf kf

InIx

Kn (f )kC(I 0 ;B)

! B1 f; 14

f (0)kC(I 0 ;B) C3 ! B1 f;

1 n

I0

+ ! B1 f; I0

2

and = C1 ! B1 f;

I0

1 n

: I0

; by (6.3) we obtain

1 4

n I0

2

C4 ! B1 f;

1 n

; I0

6.3 Best approximation, algebraic case

taking into account that ! B1 f; 41 I 0 ! B1 (f; 1)I 0 = ! B1 f; n n! B1 f; n1 I 0 n2 ! B1 f; n1 I 0 : The lemma is proved.

93

1 n I0

1 1 4; 4

Now let us consider h :

! B and 8 1 1 < h 4 ; if 2 1 Rn (h; x); if Rn (h; x) = 4 : 1 1 h 4 ; if 4 x

x x 1 2

1 4 1 4

:

1 1 For f 2 CF 4 ; 4 ; let Kn (Rn (j f ); x) be the Bojanic-DeVore operator associated to Rn (j f ), where j is the embedding in Theorem 6.1, i.e.

Kn (Rn (j

f ); x) =

Z

1 2 1 2

[Rn (j

f )(t)

Rn (j

+ (j

+

n X

"

f ); x) = (j

(j

f)

f )(xk )

k= n

+(j

f ) (0) 1 1 4

f)

Z

Z

1 4

x)dt+

1 2

(t Z

Z

1 2 1 2

Pn (t

x)dt

#

1 4

1 4 1 4

f )(0)]Pn (t

f )(0):

Then we have Kn (Rn (j

Rn (j

1 2

Pn (t

x)dt

Pn (t x)dt Pn xk )4 j= n (t xj )

1 4

Pn (t

4

x)dt:

It is easy to see that all the terms in x associated to (j f )(xk ) are positive and therefore we obtain the form Kn (Rn (j

f ); x) =

n X

(j

f )(xk )pn;k (x)

k= n 1 1 with pn:k (x) 0 for all x 2 4; 4 : With the help of pn:k (x) given as above, we de…ne the Szabados-type 1 1 fuzzy generalized polynomial associated to f : 4 ; 4 ! RF , by

S(x) =

n X

pn;k (x)

f (xk ) :

k= n

The following theorem gives Jackson-type estimate for the error of approximation by Szabados-type polynomials.

94

6. Jackson-Type Estimates by Generalized Fuzzy Polynomials 1 1 4; 4

Theorem 6.14. Let f : S(x) =

n X

! RF be continuous and pn;k (x)

f (xk )

k= n

de…ned as above. Then C! F 1

D (f; S) Proof. Since all pn;k (x) n X

(j

f;

1 n [

: 1 1 4;4

]

0; we have n X

f )(xk )pn;k (x) = j

k= n

pn;k (x)

!

f (xk ) :

k= n

But j is an isometry, so we have: D(f (x); S(x)) = D f (x);

n X

pn;k (x)

!

f (xk )

k= n

= (j

f )(x)

j

n X

pn;k (x)

!

f (xk )

k= n

= (j

f )(x)

n X

(j

f )(xk )pn;k (x) ;

k= n

where k k is the norm in B =C[0; 1] C[0; 1]: We observe that the last sum is Kn (Rn (j f ); x): Also we have: (j

f )(x)

Kn (Rn (j + Rn (j

f ); x) f; x)

k(j Kn (Rn (j

f )(x)

Rn (j

f; x)k +

f ); x) :

Since the coe¢ cients rn;k (x) of Rn in Lemma 6.9 are all positive, we have ! n n X X Rn (j f; x) = rn;k (x)(j f )(xk ) = j pn;k (x) f (xk ) k= n

k= n

and taking into account that j is an isometry, we obtain for all x 2 D(f (x); S(x))

1 1 4; 4

D(f (x); Rn (f; x)) + Rn (j

f; x)

Kn (Rn (j

f ); x) =

= D(f (x); Rn (f; x)) + Rn (j

f; x)

Kn (Rn (j

f ); x)

(this last inequality is obvious by the de…nition of Rn ).

6.3 Best approximation, algebraic case

95

By Lemma 6.9 and Lemma 6.13 we obtain 5! F 1

D(f (x); S(x))

f;

1 n [

1 1 4;4

+ C4 ! B1 Rn (j

]

f );

1 n [

: 1 1 4;4

]

f ); n1 [ 1 ; 1 ] = ! B1 Rn (j f ); n1 [ 1 ; 1 ] : 4 4 4 4 By Lagrange theorem for functions with values in Banach spaces we obtain It is easy to see that ! B1 Rn (j kRn (j

f; y)

with I 0 = obtain ! B1 Rn (j

Rn (j

sup kRn0 (j

f; x)k

f; )kC(I 0 ;B) jy

2[x;y] 1 1 4; 4

f );

and for jy 1 n

taking into account Lemma 6.12 we

900n ! B1 j

f;

I0

It is easy to check that ! B1 j 5! F 1

D(f (x); S(x))

1 n;

xj

1 n

f;

f; n1

I0

xj

1 n

I0

1 = 900! B1 j n

1 = !F 1 f; n

+ C4 900! F 1 I0

f;

I0

1 n

f;

1 n

: I0

and we …nally obtain = C! F 1

f;

I0

1 n

I0

which completes the proof. As an immediate consequence we obtain the following Jackson-type estimate for the error in approximation by generalized fuzzy algebraic polynomials. Corollary 6.15. For the best approximation by algebraic polynomials we have EnK;1

C! F 1

f;

1 n [

1 1 4;4

]

1 1 ; ; K=f 4 4

; 8n 2 N, f 2 CF

1 1 ; 4 4

where C > 0 is an absolute constant independent of n and f: R 1 Proof. Since the polynomial Pn (t x) 0; 8t; x 2 [ 1; 1] and 1 Pn (t x)dt = 1; we have jpn;

n (x)j

=

Z

Z

1 4 1 2 1 4 1 2

Pn (t Pn (t

x)dt + x)dt

Z

1 4 1 4

1:

(t

x

Pn (t x)dt Pn 4 n) j= n (t

xj )

4

Also jpn;n (x)j =

Z

1 2 1 4

Pn (t

x)dt +

Z

1 4 1 4

(t

Pn (t x)dt Pn xn )4 j= n (t xj )

4

1:

:

96

6. Jackson-Type Estimates by Generalized Fuzzy Polynomials

and jpn;0 (x)j = 1 1

Z

Z

1 2 1 2 1 2 1 2

Pn (t Pn (t

x)dt + x)dt +

Z Z

1 4 1 4 1 4 1 4

(t Pn (t

For k 2 = f n; 0; ng we have Z 14 Pn (t x)dt Pn jpn;k (x)j = 4 1 (t xj ) x ) k j= n (t 4

Pn (t x)dt Pn x0 )4 j= n (t xj )

4

x)dt

1:

Z

Pn (t

4

1 4 1 4

x)dt

1:

and the proof is complete.

Remark 6.16. We can obtain the above results in any interval [a; b] in1 1 stead of 4 ; 4 by mapping this interval in [a; b] through a linear function which maps 14 to a and 14 to b:

6.4 Application to fuzzy interpolation In this section we prove the convergence of fuzzy Lagrange polynomials for some classes of fuzzy functions. The fuzzy Lagrange polynomial is de…ned in [84], [72] as follows (see also Pn ):::=:::(x xn ) [66, p.651]) Ln (x) = i=0 li (x) f (xi ); where li (x) = (x(xi xx00 ):::=:::(x i xn ) are the usual fundamental Lagrange interpolation polynomials and the sign ”/” means that the ith operand is missing. Theorem 6.17. Let f : [ 1; 1] ! RF be a Lipschitz mapping of order > 12 (i. e. there exists L such that D(f (x); f (y)) L jx yj for all x; y 2 [ 1; 1]: Let (xn;i )i=1;n ; n 2 N be a normal matrix of nodes and Ln (x) the fuzzy Lagrange polynomial which interpolates f on fxn;0 ; :::; xn;n g: Then lim Ln (x) = f (x); 8x 2 [ 1; 1]: n!1

The convergence is uniform in any interval [ 1 + h; 1p h] ; 0 < h < 1: n Proof. By Corollary 6.15 if we take M = max 1; 2h ; 8h > 0, then 1 K;M F En (f ) C! 1 f; n [ 1;1] ; where K = f ([ 1; 1]) and also the best ap-

proximation polynomial in AK;M (denoted n ) exists. By Lemma 8.3.2, n pn P[43, n p. 351] for a normal matrix of nodes we have jli (x)j jl i=0 i (x)j 2h for x 2 [ 1 + h; 1 h] and so Ln 2 AK;M : Then D (Ln ; f ) D (Ln ; n ) + n 1 D ( n ; f ): By Corollary 6.15 we obtain D ( n ; f ) C! F 1 f; n [ 1;1] : Let Ln ( n ) be the fuzzy Lagrange polynomial associated to n at fx Pnn:0 ; :::; xn;n g: We prove that Ln ( n ) = n : We observe that n (xn;j ) = (xn;i ) since li (xn;j ) = i;j (Kronecker symbol i;j ). i=0 li (x)

6.4 Application to fuzzy interpolation

97

So Ln ( n )(xn;j ) = n (xn;j ); j = 0; n: Since the Lagrange polynomial is unique (see [66, p.650]), we get Ln ( n ) = n : Then using the properties of the metric D we obtain: ! n n X X D (Ln (x); n (x)) = D li (x) f (xn;i ) ; li (x) (xn;i ) i=0

n X

D(li (x)

f (xn;i ) ; li (x)

i=0

(xn;i ))

i=0

n X i=0

Using again Corollary 6.15 we obtain D (Ln (x);

n (x))

n X i=0

jli (x)j C! F 1

jli (x)j D(f (xn;i ) ; (xn;i )):

f;

1 n

: [ 1;1]

By [43, Lemma 8.3.2, p. 351], for a normal matrix of nodes we have r n X (b a)n jli (x)j ; for x 2 [a + h; b h]: h i=0 Then D (Ln ; f )

C! F 1

1 f; n

1+ [ 1;1]

1 Since f is of Lipschitz-type, we have ! F 1 f; n

D (Ln ; f ) which completes the proof.

r

! 2p n : h L n1 ;

[ 1;1]

r 1 2 1 + CL CL n hn

1 2

;

> 12 : Then

7 BASIC FUZZY KOROVKIN THEORY

We present the basic fuzzy Korovkin theorem via a fuzzy Shisha– Mond inequality given here. This determines the degree of convergence with rates of a sequence of fuzzy positive linear operators to the fuzzy unit operator. The surprising fact is that only the real case Korovkin assumptions are enough for the validity of the fuzzy Korovkin theorem, along with a natural realization condition ful…lled by the sequence of fuzzy positive linear operators. The last condition is ful…lled by almost all operators de…ned via fuzzy summation or fuzzy integration. This chapter relies on [18].

7.1 Basics Motivation for this chapter are the references [4], [60], [57], [66], [78], [96], [102]. References [60], [102] are the …rst articles dealing with the fuzzy Korovkin issue and only one reference [102] provides a fuzzy Shisha–Mond inequality but to a very di¤erent and specialized direction for fuzzy random variables. Of course pre-existed some Korovkin type set valued literature not related at all to this chapter. The results of Theorems 7.7 and 7.8 are simple, basic and very general, directly transferring the real case of the convergence with rates of positive linear operators to the unit, to the fuzzy one. The same real assumptions are kept here in the fuzzy setting, and they are the only assumptions we make along with the very natural and general G.A. Anastassiou: Fuzzy Mathematics: Approximation The., STUDFUZZ 251, pp. 99–104. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

100

7. Basic Fuzzy Korovkin Theory

realization condition (7.5). Condition (7.5) is ful…lled by almost all example — fuzzy positive operators, that is, by most fuzzy summation and fuzzy integration operators. At each step of the chapter we provide an example to justify our method. We use the following De…nition 7.1. Let f : [a; b] ! RF be a fuzzy real number valued function. We de…ne the (…rst) fuzzy modulus of continuity of f by (F )

! 1 (f; ) :=

any 0 <

b

sup

D(f (x); f (y));

x;y2[a;b] jx yj

a.

De…nition 7.2. Let f : [a; b] R ! RF . We say that f is fuzzy continuous at x0 2 [a; b] i¤ whenever xn ! x0 , then D(f (xn ); f (x0 )) ! 0, as n ! 1, n 2 N. We call f fuzzy continuous i¤ it is fuzzy continuous 8x 2 [a; b] and we denote the space of fuzzy continuous functions by CF ([a; b]). (r) (r) Denote [f ]r = [f ; f+ ] and we mean [f (x)]r = f

(r)

(r)

(x); f+ (x) ;

8x 2 [a; b]; all r 2 [0; 1]:

Let f; g 2 CF ([a; b]) we say that f is fuzzy larger than g pointwise and we (r) (r) (r) (r) denote it by f % g i¤ f (x) % g(x) i¤ f (x) g (x) and f+ (x) g+ (x), (r) (r) (r) (r) g+ , 8r 2 [0; 1]. g , f+ 8x 2 [a; b], 8r 2 [0; 1], i¤ f Let L be a map from CF ([a; b]) into itself, we call it a fuzzy linear operator i¤ L c1 f1 c2 f2 = c1 L(f1 ) c2 L(f2 ); for any c1 ; c2 2 R, f1 ; f2 2 CF ([a; b]). We say that L is a fuzzy positive linear operator i¤ for f; g 2 CF ([a; b]) with f % g we get L(f ) % L(g) i¤ (r) (r) (r) (r) (L(g))+ on [a; b] for all r 2 [0; 1]. (L(g)) and (L(f ))+ (L(f )) Example 7.3. Let f 2 CF ([0; 1]), we de…ne the fuzzy Bernstein operator Bn(F ) (f ) (x) =

n X

k=0

n k x (1 k

x)n

k

f

k n

; 8x 2 [0; 1]; n 2 N:

This is a fuzzy positive linear operator. We mention the very interesting with rates approximation motivating this chapter. Theorem 7.4 (see p. 642, [66], S. Gal). If f 2 CF ([0; 1]), then D Bn(F ) (f ); f

3 (F ) 1 !1 f; p 2 n

;

8n 2 N

7.1 Basics

101

i.e. lim D Bn(F ) (f ); f = 0;

n!1 (F )

that is Bn f ! f , n ! 1 in fuzzy uniform convergence. (F )

The last fact comes by the property that ! 1 (f; ) ! 0 as whenever f 2 CF ([a; b]). We need to use

! 0,

~ n )n2N Theorem 7.5 (Shisha and Mond (1968), [96]). Let [a; b] R. Let (L be a sequence of positive linear operators from C([a; b]) into itself. For n = ~ n (1) is bounded. Let f 2 C([a; b]). Then for n = 1; 2; : : :, 1; 2; : : :, suppose L we have ~nf kL

~n1 kf k1 kL

f k1

~ n (1) + 1k1 ! 1 (f; 1k1 + kL

n );

where ! 1 is the standard real modulus of continuity and n

~ n ((t := (L

x)2 ))(x)

1=2 ; 1

and k k1 stands for the sup-norm over [a; b]. In particular, if Ln (1) = 1 then ~ n f f k1 2! 1 (f; n ): kL Note. One can easily see ([96]), for n = 1; 2; : : :, 2 n

~ n (t2 ))(x) (L

x2

1

~ n (t))(x) + 2c (L

x

1

~ n (1))(x) + c2 (L

1

1

;

where c := max(jaj; jbj). ~ n (1) u! 1, L ~ n (id) u! id, L ~ n (id2 ) u! id2 (id is the Assuming that L identity map), n ! 1, uniformly, then from Theorem 7.5’s main inequality ~ n (f ) u! f , 8f 2 C([a; b]), that is the famous Korovkin theorem we get L in the real case. We …nally need Lemma 7.6. Let f 2 CF ([a; b]), [a; b] (F )

! 1 (f; ) = sup max ! 1 (f r2[0;1]

for any 0 <

b

=

(r)

(r)

; ); ! 1 (f+ ; ) ;

a.

Proof. Let x; y 2 [a; b] : jx D(f (x); f (y))

R. Then it holds

yj

,0< (r)

sup max j(f (x))

b

a. Then we have (r)

(f (y))

r2[0;1]

sup max ! 1 (f r2[0;1]

(r)

(r)

(r)

j; j(f (x))+

; ); ! 1 (f+ ; ) :

(r)

(f (y))+ j

102

7. Basic Fuzzy Korovkin Theory

Thus

(F )

! 1 (f; )

sup max ! 1 (f

(r)

r2[0;1]

For any r 2 [0; 1] and any x; y 2 [a; b] : jx (F )

! 1 (f; )

(r)

D(f (x); f (y))

Therefore ! 1 (f

(r)

(f (x))

Hence sup max ! 1 (f r2[0;1]

yj (r)

! 1 (f; ); (r)

we see that (r)

(f (y))

(F )

; )

(r)

; ); ! 1 (f+ ; ) :

; (f (x))+

(r)

(f (y))+ :

8r 2 [0; 1]:

(r)

(F )

; ); ! 1 (f+ ; )

! 1 (f; );

proving the claim. (F )

Note. For f 2 CF ([a; b]) we get that f is fuzzy bounded and ! 1 (f; ) is (r) (r) …nite for all 0 < b a. Also f are continuous on [a; b] and ! 1 (f ; ) are …nite too, all r 2 [0; 1].

7.2 Main Results We present the fuzzy analog of Shisha–Mond inequality of Theorem 7.5. Theorem 7.7. Let fLn gn2N be a sequence of fuzzy positive linear operators from CF ([a; b]) into itself, [a; b] R. We assume that there exists a ~ n gn2N of positive linear operators from C([a; b]) corresponding sequence fL into itself with the property Ln (f )

(r)

~ n (f (r) ); =L

(7.1)

~ n (1)gn2N is respectively, 8r 2 [0; 1], 8f 2 CF ([a; b]). We assume that fL bounded. Then for n 2 N we have D (Ln f; f )

~n1 kL

(F )

~ n (1) + 1k1 ! (f; 1k1 D (f; o~) + kL 1

where n

:=

~ n ((t (L

x)2 ))(x)

1=2 1

8f 2 CF ([a; b]), o~ := Xf0g the neutral element for then (F ) D (Ln f; f ) 2! 1 (f; n ): (F )

;

n );

(7.2) (7.3)

~ n 1 = 1, n 2 N, . If L (7.4)

Note. The fuzzy Bernstein operators Bn and the real corresponding ones Bn acting on CF ([0; 1]) and C([0; 1]), respectively, ful…ll assumption (7.5). We present now the Fuzzy Korovkin Theorem.

7.2 Main Results

103

Theorem 7.8. Let fLn gn2N be a sequence of fuzzy positive linear operators from CF ([a; b]) into itself, [a; b] R. We assume that there exists a ~ n gn2N of positive linear operators from C([a; b]) corresponding sequence fL into itself with the property Ln (f )

(r)

~ n (f =L

(r)

);

(7.5)

respectively, 8r 2 [0; 1], 8f 2 CF ([a; b]). Furthermore assume that ~ n (1) u! 1; L ~ n (id) u! id; L ~ n (id2 ) u! id2 ; L as n ! 1, uniformly. Then D (Ln f; f ) ! 0;

as n ! 1;

D

for any f 2 CF ([a; b]), i.e. Ln f ! f , that is Ln ! I unit operator in the fuzzy sense, as n ! 1. Proof. Use of (7.2), property of (7.3), etc. (F )

Example for Theorem 7.8 the fuzzy Bernstein operators Bn . Proof of Theorem 7.7. We would like to estimate D (Ln f; f )

=

sup D (Ln f )(x); f (x) x2[a;b]

=

sup

(r)

sup max j(Ln f )

(x)

(r)

(f )

(x)j;

x2[a;b] r2[0;1] (r)

j(Ln f ))+ (x) =

sup

(r)

(f )+ (x)j

~ n (f (r) (x) sup max jL

(r)

(f )

(x)j;

x2[a;b] r2[0;1] (r)

~ n (f )(x) jL + =

(r)

(f )+ (x)j

~nf sup max kL

(r)

f

r2[0;1]

(r)

~ n f (r) k1 ; kL +

(r)

f+ k1

(by Theorem 7.5) (r) ~ n 1 1k1 + kL ~ n (1) + 1k1 ! 1 (f (r) ; sup max kf k1 kL

r2[0;1]

~n1 kL

(r) ~n1 kf+ k1 kL

~ n (1) + 1k1 ! 1 (f (r) ; 1k1 + kL +

1k1 sup max kf r2[0;1]

(r)

~ n (1) + 1k1 sup max ! 1 (f (r) ; + kL r2[0;1]

~n1 = kL

n)

(r) k1 ; kf+ k1

(r) n ); ! 1 (f+ ;

(by Lemma 7.6) ~ n (1) + 1k1 ! (F ) (f; 1k1 D (f; o~) + kL 1

n );

n)

n)

;

10 4

7. Basic Fuzzy Korovkin Theory

proving (7.2). Application 7.9. Let f 2 CF ([0; 1]) then by applying (7.2) we obtain D (Bn(F ) f; f )

(F )

2! 1

1 f; p 2 n

;

8n 2 N:

(7.6)

8 FUZZY TRIGONOMETRIC KOROVKIN THEORY

We present the fuzzy Korovkin trigonometric theorem via a fuzzy Shisha–Mond trigonometric inequality presented here too. This determines the degree of approximation with rates of a sequence of fuzzy positive linear operators to the fuzzy unit operator. The astonishing fact is that only the real case trigonometric assumptions are enough for the validity of the fuzzy trigonometric Korovkin theorem, along with a very natural realization condition ful…lled by the sequence of fuzzy positive linear operators. The latter condition is satis…ed by almost all operators de…ned via fuzzy summation or fuzzy integration. This chapter is based on [32].

8.1 Basics Motivation for this chapter are the references [102], [18], [24], [97], [96]. References [102], [18], [24] are the …rst articles dealing with the fuzzy Korovkin matter and inequalities, however [102] is very specialized and restrictive though very interesting dealing with fuzzy random variables and positive linearity. The main results here are Theorems 8.12 and 8.13. They are simple, basic and very general directly transferring the real trigonometric case, of the convergence with rates of positive linear operators to the unit under trigonometric assumptions, to the fuzzy one. The same real trigonometric G.A. Anastassiou: Fuzzy Mathematics: Approximation The., STUDFUZZ 251, pp. 105–113. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

106

8. Fuzzy Trigonometric Korovkin Theory

assumptions are kept here in the fuzzy setting and they are the only convergence assumptions we make, along with the general realization condition (8.5). Condition (8.5) is satis…ed by almost all example-fuzzy positive linear operators of fuzzy summation or fuzzy integration form. At each step of the development of our method we present an example that satis…es our theory. Let f; g : I R ! RF be fuzzy real number valued functions. The distance between f; g is de…ned by D (f; g) := sup D(f (x); g(x)): x2I

The P function f : R ! RF is 2 -periodic if f (x) = f (x + 2 ), 8x 2 R. Here stands for the fuzzy summation and o~ = Xf0g . We use the following De…nition 8.1. Let f : R ! RF be a fuzzy real number valued function. We de…ne the (…rst) fuzzy modulus of continuity of f by (F )

! 1 (f; ) :=

D(f (x); f (y));

sup x;y2R jx yj

for any > 0. We have a similar obvious de…nition for subsets of R. De…nition 8.2. Let f : R ! RF . We say that f is fuzzy continuous at x0 2 R i¤ whenever xn ! x0 , then D(f (xn ); f (x0 )) ! 0, as n ! 1, n 2 N. We call f fuzzy continuous i¤ it is fuzzy continuous 8x 2 R and we denote the space of fuzzy continuous functions by CF (R). We call f fuzzy uniformly continuous i¤ 8" > 0 9 > 0: whenever x; y 2 R with jx yj U (R). Denote then D(f (x); f (y)) " and we denote the related space by CF [f ]r = [f

(r)

(r)

; f+ ]

and we mean [f (x)]r = [f

(r)

(F )

(r)

(x); f+ (x)];

8x 2 R;

all r 2 [0; 1]. Let us denote C2 (R) = ff : R ! RF ; f is fuzzy continuous and 2 -periodic on Rg. Let f; g 2 CF (R) we say that f is fuzzy larger than g pointwise and we (r) (r) (r) (r) denote it by f % g i¤ f (x) % g(x) i¤ f (x) g (x) and f+ (x) g+ (x), (r) (r) (r) (r) 8x 2 R, 8r 2 [0; 1], i¤ f g , f+ g+ , 8r 2 [0; 1]. Let L be a map from CF (R) into itself, we call it a fuzzy linear operator i¤ L(c1 f1 c2 f2 ) = c1 L(f1 ) c2 L(f2 );

8.1 Basics

107

for any c1 ; c2 2 R, f1 ; f2 2 CF (R). We say that L is a fuzzy positive linear (r) operator i¤ for f; g 2 CF (R) with f % g we get L(f ) % L(g), i¤ (Lf ) (r) (r) (r) (Lg) and (Lf )+ (Lg)+ on R for all r 2 [0; 1]. We need Lemma 8.3. Let the fuzzy trigonometric polynomial Qn (x) =

n X

(cos kx)

ak

(sin kx)

bk ;

k=0

where x; y 2 R; ak ; bk 2 RF , k = 0; 1; : : : ; n. Then Qn (x) is a fuzzy 2 periodic continuous function in x 2 R. Proof. Clear. We present Lemma 8.4. Let f : R ! RF be a 2 -periodic and fuzzy continuous func(F ) tion, i.e. f 2 C2 (R). Then for all 2 [0; ] we have (F )

!1

f

[0;2 ]

(F )

;

(F )

! 1 (f; )

2! 1

f

[0;2 ]

;

:

Proof. The left hand side inequality is obvious. Now, let us denote Ik = [2k ; 2(k + 1) ], 8k 2 Z. For x; y 2 R, jx yj , there exist two possibilities: (1) 9k 2 Z such that x; y 2 Ik , (2) 9k 2 Z such that x 2 Ik , y 2 Ik+1 , or x 2 Ik+1 , y 2 Ik . Case (1). We have: x0 = x jx yj and

2k , y 0 = y (F )

D(f (x); f (y)) = D f (x0 ); f (y 0 )

!1

f

2 [0; 2 ], jx0

2k

[0;2 ]

(F )

;

2! 1

f

[0;2 ]

y0 j = ;

:

Case (2). Let x 2 Ik , y 2 Ik+1 (the case y 2 Ik , x 2 Ik+1 , as symmetric, is similar). We have: x0 = x 2k 2 [0; 2 ], y 0 = y 2k 2 [2 ; 4 ], jx0 y 0 j , 0 x 2 y 0 . Thus D(f (x); f (y))

= D(f (x0 ); f (y 0 )) (F )

!1

f

[0;2 ]

;

D(f (x0 ); f (2 )) + D(f (2 ); f (y 0 )) (F )

+ !1

f

[2 ;4 ]

;

(F )

(since f 2 C2 (R) we obviously have (F )

!1

f

[0;2 ]

;

(F )

= !1

f

[2 ;4 ]

;

(F )

) = 2! 1

f

[0;2 ]

;

:

108

8. Fuzzy Trigonometric Korovkin Theory

Taking the supremum in the above with x; y 2 R, jx the claim.

yj

we establish

We give (F )

Lemma 8.5. Let f 2 C2 (R), then f is fuzzy bounded and fuzzy uni(F ) U formly continuous. I.e. C2 (R) = 2 CF (R); the space of fuzzy uniformly continuous 2 -periodic functions. Proof. By Lemma 2 of [11] we have D(f (x); o~)

M;

8x 2 [0; 2 ]; M > 0:

For any z 62 [0; 2 ] there exists x 2 [0; 2 ] such that z = x+2k , k 2 Z f0g. Hence we have D(f (z); o~) = D(f (x); o~)

M;

8z 2 R

[0; 2 ];

proving that f is fuzzy bounded on R. (F ) By Proposition 2 of [24] we have that lim ! 1 f !0

[0;2 ]

;

= 0 be-

cause f is fuzzy uniformly continuous on [0; 2 ]. Thus by Lemma 8.4 (F ) we get lim ! 1 (f; ) = 0, equivalently by Proposition 2 of [11] we have !0

U (R), proving the claim. f 2 2 CF

We also need

Proposition 8.6 ([24]). Let f : R ! RF be a fuzzy real number valued (F ) (r) (r) function. Assume that ! 1 (f; ), ! 1 (f ; ), ! 1 (f+ ; ) are …nite for > 0, where ! 1 is the usual real modulus of continuity. Then it holds (F )

! 1 (f; ) = sup max ! 1 (f r2[0;1]

(r)

(r)

; ); ! 1 (f+ ; ) :

De…nition 8.7 ([66]). Let f : [a; b] ! RF . We say that f is fuzzy-Riemann integrable to I 2 RF if, for any " > 0, 9 > 0: for any division P = f[u; v]; g of [a; b] with the norms (P ) < , we have ! X D (v u) f ( ); I < ": P

We write I := (F R)

Z

b

f (x) dx;

a

we also call an f , as above, (F R)-integrable. By Corollary 13.2 of [66], p. 644 we have that if f 2 CF ([a; b]) (fuzzy continuous on [a; b]), then f is fuzzy-Riemann integrable on [a; b]. Also,

8.1 Basics

109

by Lemma 13.2 of [66], p. 644 for f : R ! RF which fuzzy continuous 2 -periodic function we have that Z 2 Z a+2 Z (F R) f (x)dx = (F R) f (x)dx = (F R) f (x)dx ; 8a 2 R: 0

a

We need the following. Theorem 8.8 (see [67]). Let f : [a; b] ! RF be fuzzy continuous function. Then #r " Z # " Z b Z b b (r) (r) (f ) (x)dx; (f )+ (x)dx ; 8r 2 [0; 1]: (F R) f (x)dx = a

a

a

(r)

Clearly f : [a; b] ! R are continuous functions. We are motivated by (F )

De…nition 8.9 (see [66], p. 646). Let f 2 C2 (R) we give the fuzzy Jackson operator Z (Jn (f ))(x) = (F R) Kn (t) f (x + t)dt; where n0 =

Kn (t) = Ln0 (t); [ ] the integral part, Lm (t) = with

m

1 m

hni 2

sin(mt=2) sin(t=2)

being determined by Z Lm (t)dt = 1;

+ 1;

4

;

m 2 N:

It is noticed that Kn (t) 0 being even trigonometric polynomial of degree n. By [66], p. 647 it is shown that (Jn (f ))(x) is a fuzzy continuous trigonometric polynomial (also by Lemma 8.3). Note by Z (J~n (g))(x) = Kn (t)g(x + t)dt; g 2 C2 (R): the corresponding real Jackson operator. We mention Theorem 8.10 ([66], p. 647). There exists a constant C > 0 (independent (F ) of n and f ), such that for all f 2 C2 (R) we have D((Jn (f ))(x); f (x))

(F )

C! 1

f;

1 n

;

8n 2 N; x 2 R:

110

8. Fuzzy Trigonometric Korovkin Theory

By Lemma 8.5 and Proposition 2 of [11] we notice that as (F )

n ! 1; ! 1

f;

1 !0 n

and we get D (Jn f; f ) ! 0. Based on Theorem 3.4 of [67] we trivially have that Jn is a fuzzy linear operator. By Theorem 8.8 we have Z r Kn (t) f (x + t)dt [(Jn (f ))(x)]r = (F R) Z Z (r) (r) Kn (t)f (x + t)dt; Kn (t)f+ (x + t)dt ; = (r)

(r)

(r)

8r 2 [0; 1], 8x 2 R. I.e. (Jn f ) = J~n (f ), 8r 2 [0; 1]. Here f 2 C2 (R), (r) (F ) (r) g , 8r 2 [0; 1], 8r 2 [0; 1]. Let f; g 2 C2 (R) such that f % g i¤ f respectively. Then Z Z (r) (r) Kn (t)f (x + t)dt Kn (t)g (x + t)dt; 8r 2 [0; 1]; respectively, 8x 2 R, i.e. (r)

(Jn f ) respectively, i¤

(r)

(Jn g)

;

(Jn f ) % (Jn g);

8r 2 [0; 1]; n 2 N:

That is proving that Jn is a fuzzy positive operator. In fact almost all fuzzy operators de…ned via fuzzy summation or fuzzy integration are fuzzy positive linear operators. We further need to use ~1; L ~ 2 ; : : :, be linTheorem 8.11 (Shisha and Mond (1968), [97]). Let L ear positive operators, whose common domain K consists of real functions with domain ( 1; 1). Suppose 1, cos x, sin x, f belong to K, where f is an everywhere continuous, 2 -periodic function, with usual modulus of con~ n (1) tinuity ! 1 . Let 1 < a < b < 1, and suppose that for n = 1; 2; : : : ; L is bounded in n over [a; b]. Then for n = 1; 2; : : :, ~nf kL

f k1

~ n (1) kf k1 kL

~ n 1 + 1k1 ! 1 (f; 1k1 + kL

where n

=

~ n sin2 L

t

x 2

n );

(8.1)

1=2

(x)

;

(8.2)

1

~ n (1) = 1, where k k1 is the supremum norm over [a; b]. In particular, if L as is often the case, the last inequality (8.1) reduces to ~nf kL

f k1

2! 1 (f;

n ):

(8.3)

8.2 Main Results

~ n sin2 Remarks ([97]). (1) In forming L able. (2) We have that 2

2 n

2

~ n (1) kL

t x 2

, t is the independent vari-

~ n (cos t))(x) 1k1 + k cos xk1 k(L

~ n (sin t))(x) + k sin xk1 k(L

111

cos xk1

sin xk1 :

(8.4)

~ n F converges uniformly to F in [a; b] for F (t) 1, cos t, sin t, then So if L as n ! 1, we get by (8.4) that n ! 0 and ! 1 (f; n ) ! 0, thus by (8.1) ~ n f ! f uniformly on [a; b], 8f continuous 2 -periodic function giving us L on R. Hence we get with rates in an inequality form, quantitatively, the famous trigonometric Korovkin theorem, see [78].

8.2 Main Results We present the fuzzy analog of Trigonometric Shisha–Mond inequality of Theorem 8.11. Theorem 8.12. Let fLn gn2N be a sequence of fuzzy positive linear oper(F ) ators on C2 (R). We assume that there exists a corresponding sequence ~ fLn gn2N of positive linear operators on C2 (R) with the property (r)

(Ln (f ))

~ n (f (r) ); =L

(8.5)

(F )

~ n (1)gn2N is respectively, 8r 2 [0; 1], 8f 2 C2 (R). We assume that fL bounded in n over [a; b] R. Then for n 2 N we have D (Ln f; f )

~n1 kL

~ n (1) + 1k1 ! (F ) (f; 1k1 D (f; o~) + kL 1

n );

(8.6)

where n

=

~ n sin2 t L

1=2

x

(x)

2

;

(8.7)

1

~ n (1) = where D and k k1 -sup norm are taken over [a; b]. In particular, if L 1, then we get D (Ln f; f ) (F )

Here ! 1

(F )

2! 1 (f;

n );

n 2 N:

is the fuzzy modulus of continuity over R.

(8.8)

112

8. Fuzzy Trigonometric Korovkin Theory

Proof. We would like to estimate D (Ln f; f )

=

sup D (Ln f )(x); f (x) x2[a;b]

=

sup

(r)

sup max j(Ln f )

(r)

(x)

(f )

(x)j;

x2[a;b] r2[0;1] (r)

(r)

j(Ln f )+ (x) =

sup

(f )+ (x)j

~ n (f sup max j(L

(r)

(r)

)(x)

(f )

(x)j;

x2[a;b] r2[0;1] (r)

(r)

~ n (f )(x) jL + =

(f )+ (x)j

~nf sup max kL

(r)

f

(r)

r2[0;1]

~ n f (r) k1 ; kL +

(r)

f+ k1

(by Theorem 8.11) (r) ~ n 1 1k1 sup max (kf k1 kL

r2[0;1]

~ n (1) + 1k1 ! 1 (f (r) ; + kL

~ n (1) + + kL ~n1 kL

(r) 1k1 ! 1 (f+ ;

1k1 sup max(kf r2[0;1]

(r) ~ n )); (kf+ k1 kLn 1 n )) (r)

(r)

k1 ; kf+ k1 )

~ n (1) + 1k1 sup max ! 1 (f + kL r2[0;1]

~n1 = kL

1k1

(r)

;

(r) n ); ! 1 (f+ ;

(by Proposition 8.6) ~ n (1) + 1k1 ! (F ) (f; 1k1 D (f; o~) + kL 1

n)

n );

proving (8.6). We present now the …rst Fuzzy Trigonometric Korovkin theorem. Theorem 8.13. Let fLn gn2N be a sequence of fuzzy positive linear oper(F ) ators on C2 (R). We assume that there exists a corresponding sequence ~ n gn2N of positive linear operators on C2 (R) with the property fL (r)

(Ln (f ))

~ n (f (r) ); =L

(8.9)

(F )

respectively, 8r 2 [0; 1], 8f 2 C2 (R). u u u ~ n (1) ! ~ n (sin x) ! ~ n (cos x) ! Furthermore assume that L 1, L sin x, L cos x, as n ! 1, uniformly over x 2 [a; b] R. Then D (Ln f; f ) ! 0, (F )

D

as n ! 1, over [a; b], 8f 2 C2 (R). I.e. Ln f ! f , over [a; b], that is Ln ! I fuzzy unit operator, over [a; b], as n ! 1.

Proof. From (8.4) we get n ! 0, as n ! 1. By Lemma 8.5 any f 2 (F ) (F ) ~n1 C2 (R) is uniformly continuous on R and thus ! 1 (f; n ) ! 0. Also L are bounded in n over [a; b]. Hence by (8.6) we get D (Ln f; f ) ! 0 as n ! 1, over [a; b].

8.2 Main Results

113

Application 8.14. As an example for Theorem 8.13 we mention the fuzzy Jackson operator Jn we discussed earlier, see Theorem 8.10, etc. Notice that J~n (1) = 1, 8n 2 N and by Theorem 2.2, p. 204, [56] of Jackson, we have that 1 kJ~n g gk1 C! 1 g; ; 8g 2 C2 (R); n u u C > 0 universal constant, giving us also that J~n (sin x) ! sin x, J~n (cos x) ! C cos x over R, as n ! 1. Furthermore, by [96] we get n n, C > 0 constant independent of f and n. Then by Theorem 8.12, inequality (8.8) we obtain 1 (F ) D (Jn f; f ) C! 1 ; n 2 N; f; n

where C > 0 universal constant. That is recon…rming Theorem 8.10.

9 FUZZY GLOBAL SMOOTHNESS PRESERVATION

Here we present the property of global smoothness preservation for fuzzy linear operators acting on spaces of fuzzy continuous functions. Basically we transfer the property of real global smoothness preservation into the fuzzy setting, via some natural realization condition ful…lled by almost all example-fuzzy linear operators. The derived inequalities involve fuzzy moduli of continuity and we give examples. This chapter relies on [21].

9.1 Basics Motivation to this chapter are [26], [30]. In general and in applications, we prefer to have nice and …t approximations. That is the approximants, in this case operators, e.g. see [24], [32] should not wiggle more than the approximated functions. This feature is expressed with the property of global smoothness preservation by the approximating operators. In general let L be a fuzzy linear operator acting on a space of fuzzy continuous functions T de…ned on a metric space (X; d) and taking values in RF , the set of fuzzy real numbers. We say that L preserve the property of global smoothness preservation, i¤ (F ) (F ) ! 1 (Lf; ) c! 1 (f; ); 8f 2 T; 8 > 0; G.A. Anastassiou: Fuzzy Mathematics: Approximation The., STUDFUZZ 251, pp. 115–124. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

116

9. Fuzzy Global Smoothness Preservation (F )

where ! 1 is the fuzzy …rst modulus of continuity and c > 0 a universal constant possibly depended only on L. In the real setting the analogous to the above inequality is true under certain conditions, see [26], [25], [30], [33]. So here we establish related inequalities in the fuzzy setting for various spaces T , by transferring from the real case under a minimal and natural realization assumption on L, see Theorems 9.10, 9.12, 9.14 and Proposition 9.15. This assumption is ful…lled by almost all fuzzy operators de…ned via fuzzy summation or fuzzy integration. We provide also some interesting examples that motivate and ful…ll our general theory. On the way to prove the main results we prove other important side results. We need De…nition 9.1. Let f : X ! RF , we call it fuzzy bounded, i¤ D(f (x); o~) M , M > 0, 8x 2 X and we denote the related space by BF (X). We d

call f fuzzy continuous, i¤ whenever xn ; x 2 X with xn ! x we have D(f (xn ); f (x)) ! 0, as n ! 1, and this is true for all x 2 X, and we U (X) = ff : X ! denote the related space by CF (X). We denote by CF RF j 8" > 0, 9 > 0 so that whenever x; y 2 X with d(x; y) , then D(f (x); f (y)) "g, the space of fuzzy uniformly continuous functions on X. Set B CF (X) = BF (X) \ CF (X); BU U CF (X) = BF (X) \ CF (X): B U BU (X), (X) = CF If X is compact metric space then CF (X) = CF (X) = CF (r) (r) see [2], pp. 52–53. Denote [f ] = [f ; f+ ], which means

[f (x)]r = [f

(r)

(r)

(x); f+ (x)];

8x 2 X; 8r 2 [0; 1]:

We use De…nition 9.2. Let f : (X; d) ! RF . We de…ne the (…rst) fuzzy modulus of continuity of f by (F )

! 1 (f; ) :=

sup

D(f (x); f (y));

x;y2X d(x;y)

8 > 0:

It holds Proposition 9.3 (see [24]). Let f : (X; d) ! RF . Assume (F )

! 1 (f; ); ! 1 (f

(r)

(r)

; ); ! 1 (f+ ; );

r 2 [0; 1]

that they are all …nite 8 > 0. Here ! 1 is the usual real modulus of continuity. Then (F )

! 1 (f; ) = sup maxf! 1 (f r2[0;1]

(r)

(r)

; ); ! 1 (f+ ; )g;

8 > 0:

(9.1)

9.1 Basics

117

We need De…nition 9.4. Let L be a map on CF (X), we call it a fuzzy linear operator, i¤ L(c1 f1 c2 f2 ) = c1 L(f1 ) c2 L(f2 );

8c1 ; c2 2 R; f1 ; f2 2 CF (X): (9.2) The linear operator L is called fuzzy bounded, i¤ D (Lf; o~)

B 8f 2 CF (X);

AD (f; o~);

(9.3)

where A > 0 depending only on L. We denote kLk = inf A, the norm of BU the operator. Similarly (9.3) is de…ned on CF (X). We need B (X), or Proposition 9.5. Let L be a fuzzy linear operator acting on CF BU CF (X), respectively. We assume that to L there corresponds a bounded ~ 6= 0 acting on CB (X) (bounded continuous real valued linear operator L functions on X), or CUB (X) (bounded uniformly continuous real valued functions on X), respectively, with the property (r)

(Lf )

~ (r) ) (x); (x) = L(f

B BU 8r 2 [0; 1], 8x 2 X, 8f 2 CF (X) or CF (X), respectively. Then L is a B BU fuzzy bounded linear operator on CF (X), or CF (X), respectively, with ~ kLk kLk.

~ < 1 by assumption. We observe that Proof. Here 0 < kLk D (Lf; o~)

=

(r)

sup sup maxfj(Lf ) x2X r2[0;1]

=

~ sup sup maxfjL(f

(r)

x2X r2[0;1]

~ sup maxfkL(f

(r)

r2[0;1]

~ sup maxfkf kLk r2[0;1]

(r)

(x)j; j(Lf )+ (x)jg

~ (r) )(x)jg )(x)j; jL(f +

~ (r) )k1 g )k1 ; kL(f +

(r)

(r) ~ k1 ; kf+ k1 g = kLkD (f; o~):

~ I.e. we got that D (Lf; o~) kLkD (f; o~). Hence L is a fuzzy bounded ~ operator. Clearly then kLk is …nite and kLk kLk. We require also

~ be a linear Proposition 9.6. Let [a; b] R and t0 2 [a; b] …xed. Let L operator that maps C([a; b]) into Z := fg 2 C([a; b]) j g(t0 ) = 0g, i.e. ~ (Lg)(t 0 ) = 0; 8g 2 C([a; b]). Furthermore assume that ~ ! 1 (Lg; )

c! 1 (g; );

8 > 0; 8g 2 C([a; b]);

118

9. Fuzzy Global Smoothness Preservation

~ i.e. L ~ has the property where c is universal constant possibly depended on L, ~ is a bounded linear operator. of preservation of global smoothness. Then L Proof. By ! 1 (g; )

2kgk1 we have

~ ! 1 (Lg; )

2ckgk1 ; all 0 <

b

a:

Thus sup x2[a;b] jx t0 j

Setting

~ j(Lg)(t 0) =b

~ (Lg)(x)j

sup x;y2[a;b] jx yj

~ j(Lg)(x)

~ (Lg)(y)j

2ckgk1 :

a we have ~ 1 kLgk

2ckgk1 ;

~ is a bounded operator. proving that L We present Example 9.7. Let f 2 CF ([0; 1]), we de…ne the fuzzy Bernstein operator (Bn(F ) f )(x)

=

n X

k=0

n k x (1 x)n k

k

f

k n

;

8x 2 [0; 1]; n 2 N: (9.4)

(F )

Notice that Bn is a fuzzy linear operator mapping CF ([0; 1]) into itself. Furthermore we easily see that D (Bn(F ) f; o~)

1 D (f; o~) < 1;

(F )

i.e. (see also [24]) Bn is a fuzzy bounded operator. By p. 642, [66], we have that 3 (F ) 1 !1 f; p 2 n

D (Bn(F ) f; f )

;

8n 2 N:

(9.5)

Let the real Bernstein operator n X n k (Bn g)(x) = x (1 k

x)n

k

g

k=0

k n

;

8x 2 [0; 1]; g 2 C([0; 1]); (9.6)

which converges uniformly to g with rates. We notice that (r)

(Bn(F ) f )

= Bn (f

(r)

);

8r 2 [0; 1];

(9.7)

respectively in , 8f 2 CF ([0; 1]). That is an important property motivating our theory next. Also from [26] and [30], p. 244 we have that ! 1 (Bn g; )

2! 1 (g; t);

8g 2 C([0; 1]);

(9.8)

9.1 Basics

119

and number 2 is the best constant. I.e. real Bernstein operators possess the global smoothness preservation property. We also present (F )

Example 9.8. Let f 2 C2 (R) (space of 2 -periodic fuzzy continuous functions on R). We de…ne (see [66], p. 646) the fuzzy Jackson operator Z Kn (t) f (x + t) dt; 8n 2 N; (9.9) Jn (f )(x) = (F R) R where ((FR) ) the Fuzzy–Riemann integral is as in [67], [66], p. 644. Here Kn (t) = Ln0 (t), n0 = n2 + 1, and Lm (t) = such that

Z

1 m

sin(mt=2) sin(t=2)

Lm (t)dt = 1;

4

;

(9.10)

8m 2 N:

R That is Kn (t)dt = 1. It is known that Kn (t) 0 being an even trigonometric polynomial of order n. The fuzzy operator Jn is linear, by the linearity of Fuzzy–Riemann integral, see [67]. By Lemma 13.2, p. 644 of [66] we get Z D((Jn f )(x); o~) = D (F R) Kn (t) f (x + t)dt; o~ Z D(Kn (t) f (x + t); o~)dt Z = Kn (t)D(f (x + t); o~)dt Z D (f; o~) Kn (t)dt = D (f; o~) < 1: I.e. we have D (Jn f; o~)

1 D (f; o~);

(9.11)

proving Jn a fuzzy bounded operator, 8n 2 N. By Theorem 13.14, p. 647 of [66] we have that D((Jn f )(x); f (x))

(F )

C! 1

f;

1 n

;

8n 2 N; x 2 R;

C > 0 universal constant, that is convergence with rates. Next we mention the real Jackson operator Z ~ (Jn (g))(x) = Kn (t)g(x + t)dt; 8n 2 N; g 2 C2 (R):

(9.12)

(9.13)

120

9. Fuzzy Global Smoothness Preservation

From Theorem 2.2, p. 204, [56] we have that kJ~n g

gk1

C! 1 g;

1 n

;

8n 2 N; g 2 C2 (R);

(9.14)

C > 0 universal constant, that is convergence with rates. By [67] we observe that (r) (r) (Jn f (x)) = J~n (f )(x); 8x 2 R; r 2 [0; 1]; (9.15) respectively in Next we see j(J~n g)(x)

(F )

, 8f 2 C2 (R). Z

(J~n g)(y)j =

Z Z

Z

Kn (t)g(x + t)dt Kn (t)jg(x + t) Kn (t)! 1 (g; jx

Kn (t)g(y + t)dt

g(y + t)jdt yj)dt = ! 1 (g; jx

yj);

i.e. giving us ! 1 (J~n g; jx

yj)

! 1 (g; jx

yj);

and ! 1 (J~n g; )

! 1 (g; ); 8 > 0; 8g 2 C2 (R);

(9.16)

that is proving that property of preservation of global smoothness for the real Jackson operators. Similarly we see that D((Jn f )(x); (Jn f )(y)) Z = D (F R) Kn (t) Z =

Z Z

f (x + t)dt; (F R)

(by Lemma 13.2, p. 644 of [66]) D(Kn (t)

f (x + t); Kn (t)

Z

Kn (t)

f (y + t)dt

f (y + t))dt

Kn (t)D(f (x + t); f (y + t))dt (F )

Kn (t)! 1 (f; jx

(F )

yj)dt = ! 1 (f; jx

yj);

i.e. giving us (F )

! 1 (Jn f; )

(F )

! 1 (f; );

(F )

8 > 0; 8f 2 C2 (R):

(9.17)

That is proving the property of fuzzy global smoothness preservation for the fuzzy Jackson operators.

9.2 Main Results

121

Next let X := (X; d) be a compact metric space, and C(X) the space of continuous real-valued functions on X endowed with k k1 . Let f 2 C(X), we de…ne the …rst modulus of continuity of it by ! 1 (f; ) :=

sup

x;y d(x;y)

jf (x)

f (y)j;

8 > 0:

(9.18)

Let Lip(X) the sub-space of all g 2 C(X) with …nite semi-norm jgjLip :=

sup d(x;y)>0

jg(x) g(y)j : d(x; y)

(9.19)

It is known Lip(X) is a dense subset of C(X). Let ! ~ 1 (f; ) the least concave majorant of ! 1 (f; ), this also measures smoothness of f . We would like to mention the following related fundamental result. Theorem 9.9 (see [26] and [30], p. 234). Let X be a compact metric space, and L : C(X) ! C(X), L 6= 0, be a bounded linear operator mapping Lip(X) to Lip(X) such that for all g 2 Lip(X), jLgjLip

cjgjLip ;

(9.20)

with constant c possibly depending on L, but independent of g. Then for all f 2 C(X) and > 0 we have ! 1 (Lf; )

kLk~ ! 1 f;

c kLk

;

(9.21)

where kLk denotes the operator norm of L on C(X). It is known that over [a; b] R we have ! 1 ! ~ 1 2! 1 , etc.

9.2 Main Results Examples 9.7, 9.8 and Theorem 9.9 motivate the main results next. Theorem 9.10. Let (X; d) be a compact metric space. Let L be a fuzzy linear operator acting on CF (X). Assume for L there corresponds a real ~ on C(X) with the property linear operator L (r)

(L(f )) respectively in

~ (r) )(x); (x) = L(f

8r 2 [0; 1]; 8x 2 X;

(9.22)

, 8f 2 CF (X). Furthermore it holds ~ ! 1 (Lg; )

c! 1 (g; );

8 > 0; 8g 2 C(X);

(9.23)

122

9. Fuzzy Global Smoothness Preservation

~ Then where c > 0 is a universal constant may be depended only on L. (F )

! 1 (Lf; )

(F )

c! 1 (f; );

8 > 0; 8f 2 CF (X):

(9.24)

That is L possesses the property of preservation of fuzzy global smoothness. (r)

(F )

(r)

Proof. Notice here that all f 2 C(X). Also ! 1 (f; ); ! 1 (f ; ), 8r 2 [0; 1], 8 > 0, are all …nite, easily observed since X is compact. Then by Proposition 9.3 we see that (F )

(r)

! 1 (Lf; )

sup max ! 1 ((Lf ) r2[0;1] (9:22)

~ sup max ! 1 (L(f

=

(r)

r2[0;1] (9:23)

sup max c! 1 (f

(r)

r2[0;1]

=

c sup max ! 1 (f

(r)

~ ); ); ! 1 (L(f + ); ) (r)

; ); c! 1 (f+ ; )

(r)

r2[0;1]

(r)

; ); ! 1 ((Lf )+ ; )

(r)

; ); ! 1 (f+ ; )

(9:1)

(F )

= c! 1 (f; ):

That is proving (9.24). We give Example 9.11. Using Theorem 9.10 and (9.8) we obtain (F )

(F )

! 1 (Bn(F ) (f ); )

2! 1 (f; );

8 > 0; 8f 2 CF ([0; 1]):

(9.25)

We present also Theorem 9.12. Let open convex subset of a real normed vector space B ( ). Assume for L (V; k k). Let L be a fuzzy linear operator acting on CF ~ on CB ( ) with the property there corresponds a real linear operator L (r)

(L(f )) respectively in

~ (x) = L(f

(r)

)(x);

8r 2 [0; 1]; 8x 2

;

(9.26)

B ( ). Furthermore it holds , 8f 2 CF

~ ! 1 (Lg; )

c! 1 (g; );

8 > 0; 8g 2 CB ( );

(9.27)

~ Then where c > 0 is a universal constant may be depended only on L. (F )

! 1 (Lf; )

(F )

c! 1 (f; );

B 8 > 0; 8f 2 CF ( ):

(9.28)

That is L possesses the property of preservation of fuzzy global smoothness. (r)

B Proof. Since f 2 CF ( ), this implies that f 2 CB ( ), 8r 2 [0; 1]. (r) (F ) Consequently we get that all ! 1 (f; ), ! 1 (f ; ), 8r 2 [0; 1] are …nite, 8 > 0. Then the proof goes the same way as the proof of Theorem 9.10.

9.2 Main Results

123

We give (F )

U Example 9.13. In [32] we proved that C2 (R) = 2 CF (R), the space of (F ) (F ) fuzzy uniformly continuous 2 -periodic functions. Also C2 (R) = 2 CB (R) the space of fuzzy bounded continuous 2 -periodic functions on R. Clearly, working as in Theorem 9.12 we can derive again (9.17). We give also

Theorem 9.14. Let open convex subset of a real normed vector space U (V; k k). Let L a fuzzy linear operator acting on CF ( ). Assume for L ~ on CU ( ) (the space of uniformly there corresponds a real linear operator L continuous real valued functions on ) with the property (r)

(L(f )) respectively in

~ (r) )(x); (x) = L(f

8r 2 [0; 1]; 8x 2

;

(9.29)

U ( ). Furthermore it holds , 8f 2 CF

~ ! 1 (Lg; )

c! 1 (g; );

8 > 0; 8g 2 CU ( );

(9.30)

~ Then where c > 0 is a universal constant may be depended only on L. (F )

! 1 (Lf; )

(F )

c! 1 (f; );

U 8 > 0; 8f 2 CF ( ):

(9.31)

That is L possesses the property of preservation of fuzzy global smoothness. (r)

U Proof. Since f 2 CF ( ), this implies that f 2 CU ( ), 8r 2 [0; 1]. (F ) Consequently, by [24] and as in [30], p. 281, 298, we get that all ! 1 (f; ), (r) ! 1 (f ; ), 8r 2 [0; 1] are …nite, 8 > 0. Then the proof goes the same way as the proof of Theorem 9.10. We …nish with

Proposition 9.15. Let [a; b] R, t0 2 [a; b] …xed. Let L be a fuzzy linear operator acting on CF ([a; b]). Assume for L there corresponds a real linear ~ from C([a; b]) into Z = fg 2 C([a; b]) j g(t0 ) = 0g, with the operator L property (r)

(Lf ) respectively in

~ (r) )(x); (x) = L(f

8r 2 [0; 1]; 8x 2 [a; b];

(9.32)

, 8f 2 CF ([a; b]). Furthermore it holds

~ ! 1 (Lg; )

c! 1 (g; );

8 > 0; 8g 2 C([a; b]);

(9.33)

~ Then where c > 0 is a universal constant may be depended only on L. (1) (F )

! 1 (Lf; )

(F )

c! 1 (f; );

8 > 0; 8f 2 CF ([a; b]):

(9.34)

I.e. L possesses the property of preservation of fuzzy global smoothness.

124

9. Fuzzy Global Smoothness Preservation

(2) L is a fuzzy bounded linear operator with kLk

~ < 1. kLk

Proof. (1) Same as in Theorem 9.10. ~ is a bounded real linear opera(2) By Proposition 9.6 we get that L ~ acting on CF ([a; b]), tor. And by Proposition 9.5, for X = [a; b] and L, L C([a; b]), respectively, we establish the claim,that L is a fuzzy bounded linear operator.

10 FUZZY KOROVKIN THEORY AND INEQUALITIES

Here we study the fuzzy positive linear operators acting on fuzzy continuous functions. We prove the fuzzy Riesz representation theorem, the fuzzy Shisha–Mond type inequalities and fuzzy Korovkin type theorems regarding the fuzzy convergence of fuzzy positive linear operators to the fuzzy unit in various cases. Special attention is paid to the study of fuzzy weak convergence of …nite positive measures to the unit Dirac measure. All convergences are with rates and are given via fuzzy inequalities involving the fuzzy modulus of continuity of the engaged fuzzy valued function. The assumptions for the Korovkin theorems are minimal and of natural realization, ful…lled by almost all example –fuzzy positive linear operators. The surprising fact is that the real Korovkin test functions assumptions carry over here in the fuzzy setting and they are the only enough to impose the conclusions of fuzzy Korovkin theorems. We give a lot of examples and applications to our theory, namely: to fuzzy Bernstein operators, to fuzzy Shepard operators, to fuzzy Szasz–Mirakjan and fuzzy Baskakov-type operators and to fuzzy convolution type operators. We work in general, basically over real normed vector space domains that are compact and convex or just convex. On the way to prove the main theorems we establish a lot of other interesting and important side results This chapter relies on [24]. G.A. Anastassiou: Fuzzy Mathematics: Approximation The., STUDFUZZ 251, pp. 125–157. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

126

10. Fuzzy Korovkin Theory and Inequalities

10.1 Background Motivation for this chapter are [102], [18]. In [102] the authors establish some Korovkin type theory for fuzzy random variables under some specialized assumptions. In [18] the author proves the …rst fuzzy Shisha–Mond type inequalities and the …rst basic fuzzy Korovkin theorem over [a; b] R. The surprising fact is that the basic assumptions of real Korovkin theory for the test functions 1, id, id2 carry over there and here and they are the only ones needed. Of course a natural realization condition is needed in the fuzzy setting to prove the fuzzy convergence. Here we continue the …rst study [18], see also Chapter 7, now over compact convex subsets of real normed vector spaces (V; k k) or just convex subsets. On the way there we prove the interesting fuzzy Riesz representation theorem and its implications. Also we prove a lot of other needed interesting results which by themselves independently have their own merit, e.g. about fuzzy continuity and boundedness and about abstract fuzzy modulus of continuity properties, etc. So this chapter is essentially the study with rates and quantitatively of the fuzzy convergence of a sequence of fuzzy positive linear operators to the fuzzy unit operator. In duality one can see it as the study with rates and quantitatively of the fuzzy weak convergence of a sequence of …nite positive measures to the unit Dirac measure. The concept of positivity we use in our operators is the natural analog of the real case. The same thing with linearity, in our case is over R, in [102] is only over R+ . Our fuzzy integral here is de…ned according to the excellent and great article [75]. Finally a lot of examples are given. The application of the fuzzy Korovkin theory to fuzzy Szasz–Mirakjan and fuzzy Baskakov-type operators is very elaborate. It takes a great deal of probabilistic and fuzzy real analysis tools and work to accomplish the results. Also it reveals a lot as these operators are de…ned on R+ , they are in…nite fuzzy sums, and still ful…ll our basic realization condition de…ned over compact sets, see Assumption 10.20 and (10.46). Remark P 1 10.1. (1) Let (uk )k2N 2 RF . We denote the fuzzy in…nite P n series by lim D k=1 uk and we say that it converges to u 2 RF i¤ n!1 k=1 uk ; u P 1 = 0. We denote the last by k=1 uk = u. Let (uk )k2N , (vk )k2N , u; v 2 RF such that 1 X

uk = u;

k=1

1 X

vk = v:

k=1

Then 1 X

k=1

(uk

vk ) = u

v=

1 X

k=1

uk

1 X

k=1

vk :

10.1 Background

127

The last is true since lim D

n!1

n X

(uk

lim

P

1 k=1

n X

D

n!1

Let

vk ); u

v

k=1

uk ; u

k=1

!

!

n X

= lim D n!1

n X

+D

vk ; v

k=1

!!

n X

vk

k=1

!

;u

= 0:

uk = u 2 RF then one has that 1 X

(r)

(uk )

=u

(r)

1 X

=

k=1

1 X

(r) (uk )+

uk

k=1

and =

(r) u+

1 X

=

uk

k=1

k=1

!(r) +

n!1

k=1

(

n X

lim sup max

n!1 r2[0;1]

lim

n!1

(

n X

(uk )

;

u

n X

(r) (uk )+

k=1 (r)

;

n X

(r) (uk )+

(r) u+

k=1

k=1

proving the claim. Also we need: let (uk )k2N 2 RF with has for any

u

(r)

k=1

(r)

(uk )

(r)

!(r)

; 8r 2 [0; 1]:

We prove the last claim: We have that ! n X 0 = lim D uk ; u =

uk

k=1

!

2 R that

1 P

uk = u.

1 P

k=1

)

(r) u+

)

; 8r 2 [0; 1];

uk = u 2 RF , then clearly one

k=1 (r)

(r)

2) From [75] we see: Let un = (un ; un+ ) j 0 r 1 2 RF such that 1 1 P P (r) (r) (r) (r) un = u and un+ = u+ converge uniformly in r 2 [0; 1], then

n=1

u = (u 1 X

n=1

n=1

(r)

(r)

; u+ ) j 0

(r) (r) (un ; un+ )

r

j0

We use the following

1 2 RF and u =

r

1 =

(

1 X

n=1

1 P

un . I.e. we have

n=1

(r) un ;

1 X

n=1

(r) un+

!

j0

r

)

1 :

v

!

128

10. Fuzzy Korovkin Theory and Inequalities

De…nition 10.2. Let U (M; d) a metric space and let f : U ! RF . We de…ne the (…rst) fuzzy modulus of continuity of f by (F )

! 1 (f ; ) :=

for 0 <

diameter(U ). If

sup

D(f (x); f (y))

x;y2U d(x;y)

> diam(U ) then we de…ne

(F )

(F )

! 1 (f ; ) := ! 1 (f ; diam(U )): Proposition 10.3. Let U (M; d) metric space and f : U ! RF . Assume that (F ) (r) (r) ! 1 (f; ); ! 1 (f ; ); ! 1 (f+ ; ) are …nite for any Then it holds

> 0. Here ! 1 is the usual real modulus of continuity.

(F )

(r)

! 1 (f ; ) = sup max ! 1 (f r2[0;1]

Proof. Let x; y 2 U : d(x; y) D(f (x); f (y))

. We have (r)

(r)

=

(r)

; ); ! 1 (f+ ; ) :

(f (y))

sup max j(f (x))

r2[0;1]

sup max ! 1 (f

(r)

r2[0;1]

Hence

(F )

! 1 (f; )

(F )

(r)

sup max ! 1 (f

(r)

r2[0;1]

D(f (x); f (y))

Therefore ! 1 (f

(r)

; )

Hence sup max ! 1 (f r2[0;1]

(r)

j(f (x))

(r)

; ); ! 1 (f+ ; ) : we see that (r)

(f (y))

(F )

! 1 (f ; ); (r)

(r)

(f (y))+ j

; ); ! 1 (f+ ; ) :

For any r 2 [0; 1] and any x; y 2 U : d(x; y) ! 1 (f ; )

(r)

j; j(f (x))+

(r)

(r)

j; j(f (x))+

(r)

(f (y))+ j:

8r 2 [0; 1]:

; ); ! 1 (f+ ; )

(F )

! 1 (f ; )

proving the claim. We need De…nition 10.4. Let U open or compact (M; d) metric space and f : U ! RF . We say that f is fuzzy continuous at x0 2 U i¤ whenever xn ! x0 , then D(f (xn ); f (x0 )) ! 0. If f is continuous for every x0 2 U ,

10.1 Background

129

we then call f a fuzzy continuous real number valued function. We denote the related space by CF (U ). Similarly one de…nes CF ([a; b]), [a; b] R. De…nition 10.5. Let f : K ! RF , K open or compact (M; d) metric space. We call f a fuzzy uniformly continuous real number valued function, i¤ 8" > 0, 9 > 0: whenever d(x; y) , x; y 2 K, implies that U D(f (x); f (y)) ". We denote the related space by CF (K). De…nition 10.6. Let f : U ! RF , U (M; d) metric space. If D(f (x); o~) M , 8x 2 U , M 0, we call f a fuzzy bounded real number valued function. In particular if f 2 CF ([a; b]), [a; b] R, then f is a fuzzy bounded (F ) function, also ! 1 (f ; ) < 1 for any 0 < b a. Also notice that U CF (K) = CF (K), for K compact (V; k k) real normed vector space. We use Proposition 10.7. Let K

(V; k k) a real normed vector space and

(F )

! 1 (f ; ) =

sup

D(f (x); f (y));

> 0;

x;y2K kx yk

the fuzzy modulus of continuity for f : K ! RF . Then U (1) If f 2 CF (K), K open convex or compact convex (F ) ! 1 (f ; ) < 1, 8 > 0.

(V; k k), then

(2) Assume that K is open convex or compact convex (V; k k), then (F ) U ! 1 (f ; ) is continuous on R+ in for f 2 CF (K). (3) Assume that K is convex, then (F )

(F )

! 1 (f; t1 + t2 )

(F )

! 1 (f; t1 ) + ! 1 (f; t2 );

t1 ; t 2

0;

that is the subadditivity property is true. Also it holds (F )

! 1 (f; n )

(F )

n! 1 (f; );

and (F )

! 1 (f; where n 2 N,

> 0,

(F )

)

d e! 1 (f; )

(F )

( + 1)! 1 (f; );

> 0, d e is the ceiling of the number. (F )

(4) Clearly in general ! 1 (f ; ) (F ) ! 1 (f ; 0) = 0. (5) Let K be open or compact U i¤ f 2 CF (K).

0 and is increasing in

(F )

> 0 and

(V; k k). Then ! 1 (f ; ) ! 0 as

#0

130

10. Fuzzy Korovkin Theory and Inequalities

(6) It holds (F )

! 1 (f for

(F )

g; )

(F )

! 1 (f ; ) + ! 1 (g; );

> 0, any f; g : K ! RF , K

(V; k k) is arbitrary.

U Proof. (1) Here K is open convex. Let here f 2 CF (K), i¤ 8" > 0, 9 > 0 : kx yk implies D(f (x); f (y)) < ". Let "0 > 0 then 9 0 > 0 : kx yk "0 < 1. 0 with D(f (x); f (y)) < "0 , hence ! 1 (f; 0 ) Let > 0 arbitrary and x; y 2 K such that kx yk . Choose n 2 N : n 0 > , and set xi = x + ni (y x), 0 i n. Notice that all xi 2 K. Then ! n 1 n 1 X X D(f (x); f (y)) = D f (xi ); f (xi+1 ) i=0

i=0

D(f (x); f (x1 )) + D(f (x1 ); f (x2 )) + D(f (x2 ); f (x3 )) + + D(f (xn 1 ); f (y)) (F )

n! 1 (f ;

0)

n"0 < 1; (F )

1 n"0 < 1, Since kxi xi+1 k = n1 kx yk n < 0 . Thus ! 1 (f ; ) proving the claim. If K is compact, then claim is obvious. (2) Let x; y 2 K and let kx yk t1 + t2 , then there exists a point z 2 xy, z 2 K : kx zk t1 and ky zk t2 , where t1 ; t2 > 0. Notice that

D(f (x); f (y)) Hence

D(f (x); f (z)) + D(f (z); f (y)) (F )

! 1 (f ; t1 + t2 )

(F )

(F )

(F )

! 1 (f; t1 ) + ! 1 (f; t2 ): (F )

! 1 (f; t1 ) + ! 1 (f; t2 );

proving (3). Then by the obvious (4), we get 0 and

(F )

! 1 (f ; t1 + t2 ) (F )

j! 1 (f ; t1 + t2 ) (F )

(F )

! 1 (f ; t1 ) (F )

! 1 (f ; t1 )j

(F )

! 1 (f ; t2 ); (F )

! 1 (f ; t2 ): (F )

U Let f 2 CF (K), then lim ! 1 (f; t2 ) = 0 by (5). Hence ! 1 (f; ) is contint2 #0

uous on R+ . (F ) (5) ()) Let ! 1 (f ; ) ! 0 as # 0. Then 8" > 0, 9 > 0 with (F ) ! 1 (f ; ) ". I.e. 8x; y 2 K : kx yk we get D(f (x); f (y)) ". U That is f 2 CF (K). U (() Let f 2 CF (K). Then 8" > 0, 9 > 0: whenever kx yk , (F ) x; y 2 K, it implies D(f (x); f (y)) ". I.e. 8" > 0, 9 > 0 : ! 1 (f ; ) ". (F ) That is ! 1 (f ; ) ! 0 as # 0.

10.1 Background

131

(6) Notice that D f (x)

g(x); f (y)

g(y)

D(f (x); f (y)) + D(g(x); g(y)):

That is (6) now is clear. We also need Lemma 10.8. Let K be a compact subset of the real normed vector space (V; k k) and f 2 CF (K). Then f is a fuzzy bounded function. Proof. Let xn ; x 2 K such that xn ! x, as n ! 1, then D(f (xn ); f (x)) ! 0 by continuity of f . But (r)

(r)

D(f (xn ); f (x)) = sup max j(f (xn ))

(r)

(f (x))

j; j(f (xn ))+

r2[0;1]

(r)

(r)

(f (x))+ j :

(r)

Hence j(f (xn )) (f (x)) j ! 0, all 0 r 1, as n ! +1. That is (r) (r) (r) (f (xn )) ! (f (x)) , all 0 r 1, as n ! +1. Hence (f ) 2 C(K), (r) all 0 r 1. Consequently, (f ) are bounded over K, all 0 r 1. Here (r)

D(f (x); o~) = sup max j(f (x)) r2[0;1]

(r)

j; jf (x))+ j :

From basic fuzzy theory we get that (1)

(r)

(f (x))

(r)

(f (x))+ ;

(0)

(f (x))

(1)

(f (x))+

(f (x))

;

and (f (x))+

(0)

for all r 2 [0; 1] and for any x 2 K. Thus (r)

j(f (x))

(0)

j

max j(f (x))

(1)

j; j(f (x))

j ;

and (r)

j(f (x))+ j

(0)

(1)

all 0

(0)

(1)

max kf

max j(f (x))+ j; j(f (x))+ j ;

r

1; for any x 2 K:

Therefore D(f (x); o~)

max j(f (x))

j; j(f (x))

j

(0)

k1 ; kf

(1)

k1 =: M;

8x 2 K, where M 0, that is proving the claim. I.e. for all 0 r 1, (r) M (f (x)) M , 8x 2 K ) Xf M g f (x) XfM g , where f (x) 2 RF .

132

10. Fuzzy Korovkin Theory and Inequalities

10.2 Related Results We start with Proposition 10.9. Let U open or compact (M; d) metric space, f 2 (r) CF (U ). Then f are equicontinuous with respect to r 2 [0; 1] over U , respectively in . Proof. Easy. For the reverse we have (r)

Proposition 10.10. Let f [0; 1] on U -open or compact f 2 CF (U ).

(r)

, f+ be equicontinuous with respect to r 2 (M; d)-metric space, respectively in , then (r)

(r)

Proof. We have 8" > 0, 9 1 > 0 : jf (x) f (x0 )j < ", 8x 2 U : d(x; x0 ) < 1 , all r 2 [0; 1], x0 2 U . (r) (r) Similarly, we have 8" > 0, 9 2 > 0 : jf+ (x) f+ (x0 )j < ", 8x 2 U : d(x; x0 ) < 2 , all r 2 [0; 1]. Taking := min( 1 ; 2 ) > 0 we get for D(f (x); f (x0 )) = sup max jf

(r)

(x)

f

(r)

r2[0;1]

that if x 2 U with d(x; x0 ) < jf

(r)

(x)

f

(r)

(r)

(x0 )j; jf+ (x)

(r)

f+ (x0 )j ;

then (r)

(x0 )j; jf+ (x)

(r)

f+ (x0 )j < ";

for all r 2 [0; 1], and so is the max of both over all r 2 [0; 1]. That is D(f (x); f (x0 )) < ", proving the claim. We need De…nition 10.11. Let L : CF (U ) ,! CF (U ), where U is open or compact (M; d) metric space, such that L(c1 f + c2 g) = c1 L(f ) + c2 L(g);

8c1 ; c2 2 R:

We call L a fuzzy linear operator. We give the following example of a fuzzy linear operator, etc. De…nition 10.12. Let f : [0; 1] ! RF be a fuzzy real function. The fuzzy algebraic polynomial de…ned by Bn(F ) (f )(x)

=

n X

k=0

n k x (1 k

x)n

k

will be called the fuzzy Bernstein operator. We do have

f

k n

;

8x 2 [0; 1];

10.2 Related Results

133

Theorem 10.13 (see p. 642, [66], S. Gal). If f 2 CF ([0; 1]), then 3 (F ) 1 ! f; p 2 1 n

D Bn(F ) (f )(x); f (x)

;

n 2 N; 8x 2 [0; 1];

i.e., lim D Bn(F ) (f ); f = 0;

n!+1 (F )

that is Bn f !n!+1 f , fuzzy uniform convergence. We also need

De…nition 10.14. Let f; g : U ! RF , U (M; d) metric space. We denote (r) (r) (r) (r) f % g, i¤ f (x) % g(x), 8x 2 U , i¤ f+ (x) g+ (x) and f (x) g (x), (r) (r) (r) (r) 8x 2 U , 8r 2 [0; 1], i¤ f+ g+ and f g , 8r 2 [0; 1]. We give

De…nition 10.15. Let L : CF (U ) ,! CF (U ) be a fuzzy linear operator, U open or compact (M; d) metric space. We say that L is positive, i¤ whenever f; g 2 CF (U ) are such that f % g then L(f ) % L(g), i¤ (r)

(r)

(L(g))+

(L(f ))+ and

(r)

(L(f ))

(r)

(L(g))

;

8r 2 [0; 1]:

Here we denote (r)

[L(f )]r = (L(f ))

(r)

; (L(f ))+ ;

8r 2 [0; 1]:

An example of a fuzzy positive linear operator is the fuzzy Bernstein operator on the domain [0; 1], etc. For the de…nition of general fuzzy integral we follow [75] next. De…nition 10.16. Let ( ; ; ) be a complete -…nite measure space. We call F : ! RF measurable i¤ 8 closed B R the function F 1 (B) : ! [0; 1] de…ned by F

1

(B)(!) := sup F (!)(x); all ! 2 x2B

is measurable, see [75]. Theorem 10.17 ([75]). For F : ! RF , F (!) = 0 r 1 , the following are equivalent. (1) F is measurable, (2) 8r 2 [0; 1], F

(r)

(r)

, F+ are measurable.

(F

(r)

(r)

(!); F+ (!)) j

134

10. Fuzzy Korovkin Theory and Inequalities (r)

(r)

Following [75], given that for each r 2 [0; 1], F , F+ are integrable we have that the parametrized representation Z Z (r) (r) F d ; F+ d j0 r 1 A

A

is a fuzzy real number for each A 2 . The last fact leads to De…nition 10.18 ([75]). A measurable function F : F (!) = (F

(r)

(r)

(!); F+ (!)) j 0

! RF ,

r

1

(r)

is called integrable if for each r 2 [0; 1], F are integrable, or equivalently, (0) if F are integrable. In this case, the fuzzy integral of F over A 2 is de…ned by Z Z Z (r) (r) j0 r 1 : F+ d F d := F d ; A

A

A

By [75], F is integrable i¤ ! ! kF (!)kF is real-valued integrable. We need also Theorem 10.19 ([75]). Let F; G :

! RF be integrable. Then

(1) Let a; b 2 R, then aF + bG is integrable and for each A 2 , Z Z Z (aF + bG) d = a F d +b Gd ; A

A

A

(2) D(F; G) is a real-valued integrable function and for each A 2 , Z Z Z D Fd ; Gd D(F; G) d : A

In particular,

A

Z

A

Fd

A

F

Z

A

kF kF d :

We need to state the following. Assumption 10.20. Let L be a fuzzy positive linear operator from CF (K), K compact (M; d) metric space, into itself. Here we assume that there ~ from C(K) into itself with the property exists a positive linear operator L (r)

(Lf )

~ (r) ); = L(f

respectively, for all r 2 [0; 1], 8f 2 CF (K).

10.2 Related Results

135

As an example again we mention the fuzzy Bernstein operator and the real Bernstein operator ful…lling the above assumption on [0; 1], etc. Remark 10.21 (following Assumption 10.20). Then by Riesz representation theorem we have Z (r) ~ (r) )(x) = L(f f (t) x (dt); K

(the last is true for all real continuous functions on K) respectively, 8r 2 [0; 1], where x is the unique positive …nite Borel measure on K, 8x 2 K. Furthermore in the last integral we can take as x the unique positive …nite completed Borel measure of the same mass on K as the initial one. Hence

= =

Z

~ (L(f Z

(r)

K

(r)

~ )(x); L(f + )(x)) j r 2 [0; 1] Z (r) (r) f (t) x (dt); f+ (t) x (dt)

f (t)

K

j r 2 [0; 1]

x (dt);

K

the general fuzzy integral with respect to But we also have (L(f ))(x)

(r)

=

((Lf )

=

~ (L(f

(r)

x,

8x 2 K, see De…nition 10.18. (r)

(x); (Lf )+ (x)) j r 2 [0; 1] (r)

~ )(x); L(f + )(x)) j r 2 [0; 1] :

Based on the above we have proved the following Fuzzy Riesz Representation Theorem. Theorem 10.22. Let L be a fuzzy positive linear operator from CF (K) into itself as in Assumption 10.20, K compact (M; d) metric space. Then for each x 2 K there exists a unique positive …nite completed Borel measure x on K such that Z (Lf )(x) = f (t) x (dt); 8f 2 CF (K): K

The other way around follows. ~ be a positive linear operator from C(K) into itself, Remark 10.23. Let L K compact (M; d) metric space. Then by the basic Riesz representation theorem we have Z ~ (L(g))(x) = g(t) x (dt); 8g 2 C(K); K

where

x

is a unique positive …nite completed Borel measure on K.

136

10. Fuzzy Korovkin Theory and Inequalities

Next consider any f 2 CF (K), thus Z (r) ~ (r) )(x) = L(f f (t)

x (dt);

K

all r 2 [0; 1]:

Then ~ (L(f

(r)

(r)

~ )(x); L(f + )(x)) j r 2 [0; 1] Z Z (r) (r) = f (t) x (dt); f+ (t) K Z K = f (t) x (dt) =: L(f )(x);

x (dt)

j r 2 [0; 1]

K

by De…nition 10.18. Clearly by Theorem 10.19, L is a linear operator. (r) (r) h , respectively, all r 2 [0; 1]. Next let f % h; f; h 2 CF (K), i¤ f Thus Z Z f

(r)

(t)

x (dt)

h

(r)

(t)

x (dt);

K

K

respectively, all r 2 [0; 1]. But o n (r) (r) (Lf )(x) = (Lf ) (x); (Lf )+ (x) j r 2 [0; 1] :

That is by De…nition 10.18 we have Z (r) (r) (Lf ) (x) = f (t)

x (dt);

K

(r)

(r)

(Lh) , respectively, all r 2 [0; 1], i.e. etc. Hence we observe (Lf ) Lf % Lh. We have proved that L is a fuzzy positive linear operator. We get the important conclusion by using also Proposition 10.10. ~ from C(K) into itself, Theorem 10.24. Any positive linear operator L K compact (M; d) metric space, induces a unique fuzzy positive linear operator L on CF (K). It holds (r)

(Lf )

~ (r) ); = L(f

8f 2 CF (K); all r 2 [0; 1]:

~ (r) ) are equicontinuous with respect to r 2 [0; 1], respecIf additionally L(f tively in , then Lf 2 CF (K) whenever f 2 CF (K).

10.3 Main Results We need

10.3 Main Results

137

Lemma 10.25. Let (V; k k) be a real normed vector space and Q is a subset of V which is star-shaped relative to its …xed point x0 . Let f : Q ! RF with (F ) 0 < ! 1 (f ; ) < 1, any > 0. Then kt

D(f (t); f (x0 )) for all

x0 k

(F )

! 1 (f ; );

> 0, d e denotes the ceiling of the number, 8t 2 Q.

Clearly Lemma 10.25 is true when Q is convex and we have Lemma 10.26. Let (V; k k) be a real normed vector space and Q V , Q (F ) is convex. Let f : Q ! RF with 0 < ! 1 (f ; ) < 1, any > 0. Then D(f (x); f (y))

kx

yk

(F )

! 1 (f ; ); all

> 0; 8x; y 2 Q:

In particular we obtain (F )

Lemma 10.27. Let f : [a; b] ! RF with 0 < ! 1 (f ; ) < 1, any Then jx

D(f (x); f (y)) all

> 0, 8x; y 2 [a; b]

yj

(F )

! 1 (f ; );

R.

Proof of Lemma 10.25. Let t 6= x0 : kt

Now for another t : kt x0 k > some n 2 integer. That is

x0 k < , i.e.

we have that (n

x0 k

1) < kt

= n:

Consider the points

(t x0 ) n

j(x0 t) 2 Q; n .

= 1, then

! 1 (f ; ) =: w = 1 w:

kt

and see that

kt x0 k

(F )

D(f (t); f (x0 ))

t+

> 0.

j = 0; 1; : : : ; n

x0 k

n , for

138

10. Fuzzy Korovkin Theory and Inequalities

Then observe that D(f (t); f (x0 ))

n 1 X

= D f (t)

n 1

f t+

k=1

!

k(x0 t) X k(x0 t) ; f t+ n n k=1

f (x0 )

x0

D f (t); f t +

t n

2(x0 t) +D f t+ ;f t + n n !! 2(x0 t) 3(x0 t) +D f t+ ;f t + n n ! x0 t + + D f t + (n 1) ; f (x0 ) n (x0

Thus D(f (t); f (x0 ))

kt

nw =

t)

x0 k

!

nw:

(F )

! 1 (f ; );

proving the claim. Sometimes it is useful Lemma 10.28. Let f : Q ! RF , Q convex 1, any > 0. Then D(f (x); f (y))

1+

kx

yk

(F )

(V; k k) with 0 < ! 1 (f ; ) <

(F )

! 1 (f ; ); all

> 0; 8x; y 2 Q:

It holds Lemma 10.29. Let f : Q ! RF , Q convex 1, any > 0. Then D(f (t); f (x))

1+

xk2

kt

(F )

! 1 (f ; );

2

In particular, let f : [a; b] ! RF , [a; b] > 0. Then D(f (t); f (x)) Proof. If kt D(f (t); f (x))

1+

xk >

then

1+

kt

x)2

(t

2 kt xk

xk

(F )

(V; k k) with 0 < ! 1 (f ; ) < 8t; x 2 Q; 8 > 0: (F )

R, with 0 < ! 1 (f ; ) < 1, any

(F )

! 1 (f ; );

8t; x 2 [a; b]; 8 > 0:

> 1. Thus by Lemma 10.28 we have (F )

! 1 (f ; )

1+

kt

xk2 2

(F )

! 1 (f ; ):

10.3 Main Results

I.e. when kt

xk >

139

we get

D(f (t); f (x))

kt

1+

But the last is obviously true when kt

xk2

(F )

! 1 (f ; ):

2

xk

.

Convention 10.30. From now on, unless otherwise stated, the -…elds we consider will be the power sets of the spaces we are working on to the e¤ect that every real-valued function there is measurable and the measures considered are complete. We need De…nition 10.31. Let K be a compact convex (V; k k) real normed vector space. Let ( n )n2N be a sequence of …nite positive measures on K. We say that n converges fuzzy weakly to Dirac measure x0 , x0 2 K, i¤ Z lim D f d n ; f (x0 ) = 0; 8f 2 CF (K): n!1

K

We denote it by n

F

x0

as n ! 1:

Clearly real weak convergence cannot imply fuzzy weak convergence. We study here the degree of fuzzy weak convergence with rates. We present the …rst main result. Theorem 10.32. Let K be a convex and compact subset of the real vector space (V; k k). Let x0 2 K …xed and a …nite measure on K with (K) = m > 0. Let also f 2 CF (K). Then Z D f (t) (dt); f (x0 ) (10.1) K Z kt x0 k (F ) jm 1jD(f (x0 ); o~) + ! 1 (f ; ) (dt) ; K

for all 0 < diam(K). When m = 1, that is when measure on K we get Z Z kt x0 k (F ) D f (t) (dt); f (x0 ) ! 1 (f ; ) K

K

for all 0 <

diam(K).

Proof. By Lemma 10.8 we have that D(f (x); o~)

M;

8x 2 K; M

0;

is a probability

(dt) ;

(10.2)

140

10. Fuzzy Korovkin Theory and Inequalities

so that f is integrable. Also we see that m

f (x0 )

(r)

= = =

Z

(r)

(mf (x0 ) ; mf (x0 )+ ) j 0 r 1 Z Z (r) (r) (f (x0 )) d ; (f (x0 ))+ d K

K

j0

r

1

f (x0 ) d :

K

We then observe Z D f (t) (dt); f (x0 ) K Z f (t) (dt); f (x0 ) m + D f (x0 ) m; f (x0 ) D K Z Z = D f (t) (dt); f (x0 ) (dt) + D f (x0 ) m; f (x0 ) K K Z D f (t); f (x0 ) (dt) + jm 1jD(f (x0 ); o~) K

(the last comes by Theorem 10.19, and Lemma 1.2, respectively) Z kt x0 k (F ) (dt) ! 1 (f ; ) + jm 1jD(f (x0 ); o~); K

0< diam(K) (the last comes by Lemma 10.26). We have established (10.1) and (10.2). Remark 10.33. 1) By the use of geometric moment theory methods (method of optimal distance, see [74], [4]) we can …nd best upper bounds for R various interesting cases to K kt x0 k (dt) subject to the given moment condition Z 1=r

K

kt

x0 kr (dt)

= Dr (x0 ):

Here r > 0, Dr (x0 ) > 0 are given. For the existence of Dr (x0 ) m1=r diam(K). I.e. to …nd the optimal quantity Z 1 kt x0 k sup (dt) w K(x0 ) := m K

(10.3)

we assume

over all measures with (K) = m that ful…ll (10.3). Here we assume that for …xed 0 < diam(K) we have that (F )

! 1 (f ; )

w;

where w > 0 is given. The precise calculation of quantity K(x0 ) is given in Theorem 7.2.1, pp. 212–213 of [4] for all cases.

10.3 Main Results

141

2) Consider iF 2 RF such that (r)

iF := (i

(r)

; i+ ) j 0

(r)

r

1 with all i

=1 ;

actually iF = Xf1g . Then the fuzzy integral Z kt x0 k iF d K Z Z kt x0 k (r) kt x0 k (r) = i d ; i+ d j0 r K K Z Z kt x0 k kt x0 k = d ; d j0 r 1 K K Z Z kt x0 k kt = d f(1; 1) j 0 r 1g = K

1

x0 k

d

K

iF :

I.e. we got that Z

kt

x0 k

iF d =

K

Z

kt

x0 k

d

iF :

K

Also notice that D(iF ; o~) = 1. Then the equality sign in (10.1) cannot be attained but it can be arbitrarily closely approached by the close approxiF , where w > 0 is a prescribed value of imations to f^(t) := w kt x0 k (F ) ! 1 ( ; ). We give Corollary 10.34 (to Theorem 10.32). We get that Z D f (t) (dt); f (x0 ) jm 1jD(f (x0 ); o~) K Z 1 (F ) + ! 1 (f ; ) m + kt x0 k (dt) :

(10.4)

K

For r

1 we obtain D

Z

f (t) (dt); f (x0 )

jm

K (F )

+ ! 1 (f ; ) m + and D

Z

f (t) (dt); f (x0 )

K (F )

+ !1

f;

Z

K

Dr (x0 )

jm

1jD(f (x0 ); o~) m1

1 r

1jD(f (x0 ); o~) !

;

(10.5)

1=r

kt

x0 kr (dt)

m(1 + m

1=r

): (10.6)

142

10. Fuzzy Korovkin Theory and Inequalities

When r = 2 we get Z

D

f (t) (dt); f (x0 )

K (F )

+ !1

Z

f;

jm

1jD(f (x0 ); o~) ! 1=2

x0 k2 (dt)

kt

K

(m +

p

m): (10.7)

Proof. Obvious. We further present Theorem 10.35. Same assumptions as in Theorem 10.32. Then Z

D

f (t) (dt); f (x0 )

K (F )

+ ! 1 (f ; ) m + for all 0 < If

R

jm K

1jD(f (x0 ); o~) xk2 (dt)

kt

2

;

(10.8)

diam(K): Z

=

K

1=2

xk2 (dt)

kt

then it holds Z

D

f (t) (dt); f (x0 )

K

+

(F ) !1

f;

Z

jm

1jD(f (x0 ); o~) ! 1=2

K

xk2 (dt)

kt

(m + 1):

(10.9)

!

: (10.10)

For m = 1 we get D

Z

(F ) 2! 1

f (t) (dt); f (x0 )

K

f;

Z

1=2

kt

K

2

xk

(dt)

Corollary 10.36 (to Theorems 10.32, 10.35). It holds D

Z

f (t) (dt); f (x0 )

K (F )

+ !1

f;

Z

K

(m + 1)g:

jm

1jD(f (x0 ); o~) ! 1=2

kt

x0 k2 (dt)

minf(m +

p

m); (10.11)

10.3 Main Results

In case of K = [a; b] R we have ! Z D f (t) (dt); f (x0 )

jm

[a;b]

+

(F ) !1

f;

Z

1jD(f (x0 ); o~) 1=2

(t

2

x0 )

143

(dt)

[a;b]

!

(m + 1)g:

minf(m +

p

m); (10.12)

R

Clearly by (10.11) or (10.12), as m ! 1 and K kt x0 k2 (dt) ! 0 or R (t x0 )2 (dt) ! 0, we get the degree of approximation of fuzzy weak [a;b] F

convergence

x0

with rates, respectively.

Proof of Theorem 10.35. Most here are as in the proof of Theorem 10.32. Furthermore, we make use of Lemma 10.29 and we have Z D f (t) (dt); f (x0 ) K Z D(f (t); f (x0 )) (dt) + jm 1jD(f (x0 ); o~) K Z kt xk2 (F ) jm 1jD(f (x0 ); o~) + 1+ (dt) ! 1 (f ; ) 2 K R kt xk2 (dt) (F ) = jm 1jD(f (x0 ); o~) + ! 1 (f ; ) m + K : 2 Remark 10.37. (1) Theorems 10.32, 10.35 and Corollaries 10.34, 10.36 are true also when is the completed Borel measure, same proofs. (2) Theorems 10.32, 10.35 and Corollaries 10.34, 10.36 are true also when f is integrable in the fuzzy sense, see De…nition 10.18. So we do not need always to assume that f 2 CF (K). Also we do not need to assume always that K is compact. Again same proofs. Based on Theorem 10.22 and Remark 10.37(1) we present the following fuzzy Shisha–Mond inequalities result. Theorem 10.38. Let K be a convex and compact subset of the real normed vector space (V; k k). Let L be a fuzzy positive linear operator from CF (K) e from into itself with the property that there exists positive linear operator L (r) (r) ~ C(K) into itself with (Lf ) = L(f ), respectively for all r 2 [0; 1], 8f 2 CF (K). Then D(L(f )(x); f (x)) (F )

+ !1

~ jL(1)(x)

~ f ; (L(k

~ (L(1)(x) + 1)g:

xk2 ))(x)

1jD(f (x); o~) 1=2

(10.13) q ~ ~ minf(L(1)(x) + L(1)(x));

144

10. Fuzzy Korovkin Theory and Inequalities

Furthermore we get ~ D (f; o~)kL1 1k1 q (F ) ~ ~ ~ ~ + min kL(1) + L(1)k f ; k(L(k 1 ; kL(1) + 1k1 ! 1

D (Lf; f )

(10.14) xk2 ))(x)k1=2 1 ;

~ and k k1 stands for the sup-norm over K. In particular, if L(1) = 1 then (10.14) reduces to D (Lf; f ) When K = [a; b]

(F )

2! 1

~ f ; k(L(k

xk2 ))(x)k1=2 1 :

(10.15)

R then we obtain ~ D (f; o~)kL(1) 1k1 q ~ ~ + min kL(1) + L(1)k 1;

D (Lf; f )

(F )

~ kL(1) + 1k1 ! 1

~ f ; k(L((

(10.16)

x)2 ))(x)k1=2 1 :

Here one has that (see [96]) ~ k(L((

x)2 ))(x)k1

~ 2 )(x) x2 k1 + 2ckL(t)(x) kL(t +c2 kL(1)(x) 1k1 ;

xk1 (10.17)

where c := max(jaj; jbj). Consequently we get the following fuzzy Korovkin theorem, see also [78]. Theorem 10.39. Let (Ln )n2N be a sequence of fuzzy positive linear operators from CF ([a; b]) into itself, [a; b] R, with the property that there ~ n )n2N of positive linear operators from C([a; b]) into itexists a sequence (L (r) ~ n (f (r) ), respectively for all r 2 [0; 1], 8f 2 CF ([a; b]), self with (Ln f ) = L 8n 2 N. Furthermore assume that u ~ n (1) ! L 1;

u ~ n (id) ! L id;

u ~ n (id2 ) ! L id2 ; as n ! +1;

where id is the identity map. Then D (Ln f; f ) ! 0 as n ! 1, 8f 2 CF ([a; b]). Inequality (10.16) gives above convergence quantitatively and with rates. ~ n (1), n 2 N being bounded and the other Proof. By Theorem 10.38, L assumptions. Example 10.40. Applying (10.16) we obtain D (Bn(F ) f; f )

(F )

2! 1

1 f; p 2 n

; 8f 2 CF ([0; 1]); 8n 2 N:

Let now f be of Lipschitz type i.e. D(f (x); f (y))

M jx

yj;

M > 0; 8x; y 2 [0; 1]:

(10.18)

10.3 Main Results

145

Then by (10.18) we get the improved M p ; n

D (Bn(F ) f; f )

(10.19)

while by Theorem 10.13 we only produce 3M p : 2 n

D (Bn(F ) f; f )

We present now the more general fuzzy Korovkin type result. Theorem 10.41. Let (Ln )n2N be a sequence of fuzzy positive linear operators from CF (K) into itself, where K is a convex compact subset of the real normed vector space (V; k k). It has the property that there exists a ~ n )n2N of positive linear operators from C(K) into itself with sequence (L (r) ~ n (f (r) ), respectively, for all r 2 [0; 1], 8f 2 CF (K), 8n 2 N. (Ln f ) = L Furthermore assume that u ~n1 ! L 1 and

~ n (k (L

xk2 ))(x)

1

! 0; as n ! 1:

Then D (Ln f; f ) ! 0 as n ! +1, 8f 2 CF (K). Inequality (10.14) gives above convergence quantitatively and with rates. (F )

Proof. By Theorem 10.38 and that ! 1 (f; ) ! 0 as

! 0.

Example 10.42. Let K convex compact (V; k k)-Banach space. Let x 2 K and fx1 ; : : : ; xn g (n 2 N) be a …nite set of distinct points in K. The fuzzy second Shepard metric interpolation operator is de…ned by (f 2 CF (K)) Sn2 (f ; x) :=

:=

S 2 fx1 ; : : : ; xn g(f; x) 8 n Q > kx xj k2 > X n > > j=1;j6=i > > f (xi ) P ; < n n Q i=1 kx xk k2 > `=1 k=1;k6=` > > > > > : f (x);

if x 62 fx1 ; : : : ; xn g; (10.20) if x 2 fx1 ; : : : ; xn g:

Obviously Sn2 is a fuzzy positive linear operator. Notice Sn2 (f ; xi ) = f (xi ), all i = 1; : : : ; n. Actually we have that Sn2 (f ) 2 CF (K). It also ful…lls (Sn2 (f ))(x)

(r)

= S~n2 (f

(r)

)(x);

8r 2 [0; 1];

where S~n2 is the corresponding real valued positive linear operator, see [68], [95], [4]. We have S~n2 (1)(x) = 1 for all x 2 K. Furthermore it holds ! 1=2 n X 1=2 2 2 1=2 2 ~ (Sn (kt xk ))(x) =n kx xi k : (10.21) i=1

146

10. Fuzzy Korovkin Theory and Inequalities

We then apply (10.13) of Theorem 10.38: Let x 2 K then (F )

D((Sn2 f )(x); f (x))

2! 1

0

n X

@f ; n1=2

i=1

kx

xi k

2

!

1=2

1

A:

(10.22)

Consequently we get …nally:

(F ) 2! 1

D (Sn2 f; f )

0

n X

@f ; n1=2

i=1

kx

xi k

2

!

1

1=2

1

A;

8n 2 N: (10.23)

We give Theorem 10.43. Let K be a convex subset of the real normed vector space (V; k k). Let x0 2 K be …xed and a completed Borel …nite measure on K B (K), i.e. f is fuzzy continuous and with (K) = m > 0. Let also f 2 CF bounded on K. Then D

Z

f (t) (dt); f (x0 )

K (F )

+ !1

m+

Z

f;

p

K

jm

1jD(f (x0 ); o~) !

(10.24)

1=2

kt

x0 k2 d (t)

minfm + 1;

mg:

If K = R or R+ we obtain D

Z

!

f (t) (dt); f (x0 )

R or R+ (F )

+ !1

m+

0

@f ;

p

Z

jm

1jD(f (x0 ; o~) !1=2 1

x0 )2 d (t)

(t

R or R+

(10.25)

mg;

A minfm + 1;

B B for any f 2 CF (R) or f 2 CF (R+ ), respectively. Inequalities (10.24) and (10.25) are still valid if the related -…eld is the power set of K.

Proof. Clearly f is integrable. Again we have m

f (x0 ) =

Z

K

f (x0 ) d :

10.3 Main Results

147

We then observe D

Z

f (t) (dt); f (x0 ) Z D f (t) (dt); f (x0 ) m + D(f (x0 ) m; f (x0 )) K Z Z = D f (t) (dt); f (x0 ) (dt) + D(f (x0 ) m; f (x0 )) K

K

K

(by Theorem 10.19 and Lemma 1.2) Z

D(f (t); f (x0 )) (dt)

K

+jm 1jD(f (x0 ); o~) (by Lemmas 10.28, 10.29) Z

1+

x0 k2

kt

2

K

+jm = jm m+ jm p

2

Z

K

(

)

2

kt

kt

x0 k

1

Z

K

kt

x0 kd (t);

(F )

x0 k d (t) ! 1 (f ; ) (F ) ! 1 (f ;

(

1

Z

1=2

K

kt

1jD(f (x0 ); o~) + ! 1 (f ; ) minfm + 1; m +

p

1jD(f (x0 ); o~) + 1 2

Z

setting

d (t);

d (t)

1jD(f (x0 ); o~) + min m + 1

1+

K

)

1jD(f (x0 ); o~)

m; m +

K

) min m + )

kt

x0 k2 d (t)

:=

Z

K

= jm

(Z ) min

(F ) ! 1 (f ;

1=2

kt

2

x0 k d (t)

!

(F )

x0 k2 d (t)

mg;

proving the claim. We present B Application 10.44. We consider here f 2 CF (R+ ). For t CB (R+ ) the real Szasz–Mirakjan operator is de…ned as

~ n g)(t) := e (M

nt

1 X

k=0

g

k n

(nt)k ; k!

0 and g 2

(10.26)

148

10. Fuzzy Korovkin Theory and Inequalities

and the real Baskakov-type operator is de…ned as (V~n g)(t) :=

1 X

k n

g

k=0

n+k k

tk : (1 + t)n+k

1

(10.27)

Consider here Xj real independently and identically distributed random n P variables and put Sn := Xj , n 2 N. Let E denote the expectation j=1

~ n and V~n operators are of the form E(g(sn =n)). In one operator. Then M 1 k P t case the random variable X has the distribution PX := e t k! k and in k=0

the other

PX :=

1 X

k=0

k

t 1+t

1 1+t

k;

that is the Poisson and the geometric distribution, respectively, where k denotes the Dirac measure at k. In both cases, E(X) = t, while the variance Var(X) = t and Var(X) = (t + q t2 ), respectively. Clearly theqstandard 2

deviation sn =n in Poisson case is nt and in geometric case is t+t n . Among others we will apply here Theorem 10.43, its inequality (10.25). We will see that the related measure here is the distribution Fsn =n . NatB urally, we de…ne the fuzzy Szasz–Mirakjan operator (f 2 CF (R+ )), nt

(Mn f )(t) := e

1 X

(nt)k ; k!

k n

f

k=0

(10.28)

and the fuzzy Baskakov-type operator (Vn f )(t) :=

1 X

k n

f

k=0

n+k k

1

tk ; t (1 + t)n+k

0:

~ n operator the corresponding probability measure Clearly for M 1

=e

nt

1 X (nt)k

k!

k=0

and for V~n operator the probability measure 2

=

1 X n+k k

~ nf (M

(r)

)(t) =

Z

R+

f

(r)

is (10.30)

is

tk (1 + t)n+k

1

k=0

B Hence for f 2 CF (R+ ) we have

k=n ;

(10.29)

1 (dt)

k=n :

(10.31)

(10.32)

10.3 Main Results

and (V~n f

(r)

)(t) =

Z

f

(r)

2 (dt);

R+

8r 2 [0; 1];

149

(10.33)

respectively, and these real integrals exist, i.e. for every r 2 [0; 1] the func(r) tions f are integrable with respect to i , i = 1; 2. Then it follows from [67] and the Lebesgue-dominated convergence theorem that the parametrized representation Z Z (r) (r) f d i; f+ d i j 0 r 1 A

A

is a fuzzy real number for each A R+ , for i = 1; 2. Thus by De…nition R 10.18 we get that R+ f d i , i = 1; 2 exist as fuzzy real numbers. From Theorem 10.45 next we obtain r ! ~ n g)(t)j jg(t)j + 2! 1 g; t ; (10.34) j(M n j(V~n g)(t)j

jg(t)j + 2! 1

g;

r

t + t2 n

!

;

(10.35)

8g 2 CB (R+ ), t > 0, where ! 1 is the real modulus of continuity. I.e. ~ n g)(t), (V~n g)(t) as in…nite series converge, 8n 2 N. Hence easily we see (M ~ n , V~n are positive linear operators. Here for f 2 C B (R+ ) we get that that M F (r) f are equicontinuous, respectively, and uniformly bounded in r 2 [0; 1] over R+ . We notice by Proposition 10.3 that r ! t (r) (r) (r) ~ n f )(t)j j(M jf (t)j + 2! 1 f ; n r ! t (F ) =: M1 (f ) D (f; o~) + 2! 1 f; n and (r) j(V~n f )(t)j

jf

(r)

(t)j + 2! 1

D (f; o~) +

f

(F ) 2! 1

(r)

;

f;

r

r

t + t2 n t + t2 n

!

!

=: M2 (f ); 8r 2 [0; 1]:

I.e. ~ n f (r) )(t)j j(M (r) j(V~n f )(t)j

M1 (f ); M2 (f );

(10.36)

150

10. Fuzzy Korovkin Theory and Inequalities

where the constants M1 (f ), M2 (f ) 0, 8r 2 [0; 1]. ~ n f (r) )(t), (V~n f (r) )(t) converge 8r 2 [0; 1], for any So as in…nite series (M n 2 N. For convenience call the weights wkn (t) := e kn (t)

n+k k

:=

tk > 0; (1 + t)n+k

1

so that ~ n g)(t) = (M

1 X

g

k=0

and

1 X

(V~n g)(t) =

k n

g

k=0

k n

kn (t);

nt (nt) k!

k

> 0 and

t > 0; k; n 2 N;

wkn (t);

8g 2 CB (R+ ):

B (R+ ) the following by basic properties of fuzzy We notice for f 2 CF numbers, see [67]: (0)

(r)

(f (x))

(f (x))

(1)

(1)

(f (x))

(f (x))+

(r)

(0)

(f (x))+

(f (x))+ ; 8x 2 R+ :

We get that (r)

j(f (x))

(0)

(1)

max j(f (x)) j; j(f (x)) j 1 (1) (0) = j(f (x)) j + j(f (x)) j 2 =: A0;1 (x) 2 CB (R+ ): j

(0)

j(f (x))

j

(1)

j(f (x))

j

Also it holds (0)

(r)

(1)

max j(f (x))+ j; j(f (x))+ j 1 (0) (1) = j(f (x))+ j + j(f (x))+ j 2 =: A0;1 + (x) 2 CB (R+ ):

j(f (x))+ j

(0)

j(f (x))+ j

(1)

j(f (x))+ j

I.e. we have obtained that (r)

0

j(f (x))

0

j(f (x))+ j

(r)

j

A0;1 (x); A0;1 + (x);

8r 2 [0; 1]; 8x 2 R+ :

(10.37)

Consequently by (10.36) we have 0 0

~ n (A0;1 ))(t) (M (V~n (A0;1 ))(t)j

M1 (A0;1 ) < 1;

M2 (A0;1 ) < 1;

(10.38)

~ n (A0;1 ))(t), (V~n (A0;1 ))(t) converge as series, 8n 2 N. respectively, i.e. (M ~ n , V~n are positive linear operators we have Since M 0

~ n (f (r) ))(t)j j(M

~ n (jf (r) j))(t) (M

~ n (A0;1 )(t) M

10.3 Main Results

and j(V~n (f

0

(r)

(V~n (jf

))(t)j

(r)

respectively in and 8r 2 [0; 1]. In detail one has

(0;1) V~n (A (t);

j))(t)

(r)

f

k n

f

k n

wkn (t)

A0;1

k n

wkn (t);

kn (t)

A0;1

k n

kn (t);

(r)

151

(10.39)

(10.40)

8r 2 [0; 1], k 2 N, n 2 N …xed, t > 0 …xed, respectively in . ~ n (jf (r) j)(t), V~n (jf (r) j)(t) Thus by Weierstrass M -test we obtain that M as series converge uniformly in r 2 [0; 1], respectively in , 8n 2 N. And easily we get by Cauchy criterion for series uniform convergence that ~ n (f (r) )(t); M

(r) V~n (f )(t)

as series converge uniformly in r 2 [0; 1], respectively in , 8n 2 N. We then notice that ! ) ( Z Z Z (r) (r) f+ d 1 j r 2 [0; 1] f d 1; fd 1= RF 3 =

= e

= e

R+

R+

R+

(

e nt

nt

nt

1 X

f

k (nt) ;e n k!

(r)

k=0 ( 1 X

(

k

f

(r)

k=0

1 X (nt)k

k=0

k!

k

k (nt) ; n k! f

nt

1 X

k=0 1 X

(r)

f+

k=0

k (r) k ; f+ n n

(r)

! ) k (nt)k j r 2 [0; 1] n k! ! ) k (nt)k j r 2 [0; 1] n k! )

(r) f+

j r 2 [0; 1]

(by using Remark 10.1(2) and earlier comments) 1 X (nt)k (r) k (r) k f ; f+ j r 2 [0; 1] = e nt k! n n = e

nt

k=0 1 X

k=0

(nt)k k!

I.e. we proved that

f

k n

= (Mn (f ))(t):

Z

fd

1

= (Mn f )(t):

(10.41)

fd

2

= (Vn f )(t):

(10.42)

R+

Similarly we can prove that Z

R+

152

10. Fuzzy Korovkin Theory and Inequalities

So we have that (Mn f )(t);

(Vn f )(t) 2 RF ;

8t 2 R+ :

We next observe for t > 0 that D (Mn (f ))(t); o~

nt

= e

1 X

D

k=0

nt

= e

m X

lim D

m!1 nt

e

lim

m!1

! (nt)k ; o~ k!

k f n

k=0

m X (nt)k

k=0

k!

! (nt)k ; o~ k!

k f n

k ; o~ n

D f

(here D(f (x); o~) M; M m X (nt)k nt = M: Me lim m!1 k!

0)

k=0

Hence D (Mn f; o~) M . I.e. Mn f (t) is fuzzy bounded over R+ . Similarly we see that for t > 0 that D (Vn f )(t); o~

= D

1 X

k=0

=

lim D

m!1

lim

m!1

=

k f n

lim

m!1

M

m X

k=0

m X

k=0 m X

k=0 1 X

k=0

n+k k k f n

! tk ; o~ (1 + t)n+k

n+k k

k n

D f

1

n+k k

1 1

! tk ; o~ (1 + t)n+k

tk ; o~ (1 + t)n+k

tk k D f ; o~ (1 + t)n+k n ! 1 tk = M: (1 + t)n+k

n+k k

1

n+k k

I.e. D (Vn f; o~) M , that is Vn f is fuzzy bounded. Using Remark 10.1(1) and Lemma 1.3(iv) we get easily that Mn , Vn are fuzzy linear operators. Next we prove their fuzzy positivity. (r) (r) B Indeed we have: let f; g 2 CF (R+ ) then g , f 2 CB (R+ ), 8r 2 [0; 1]. (r) (r) (r) (r) So assume that f % g i¤ f g and f+ g+ , 8r 2 [0; 1]. Then f and (r)

f+

k n

(r)

k n (r)

g+

g k n

(r)

;

k n 8r 2 [0; 1];

10.3 Main Results

and f and

(nt)k k!

k n

(r)

f+

(nt)k k!

k n

(r)

g

(nt)k ; k!

k n

(r)

g+

(nt)k ; k!

k n

(r)

153

8r 2 [0; 1]:

Consequently it holds e

nt

1 X

f

(nt)k k!

k n

(r)

k=0

e

nt

1 X

g

k n

(r)

k=0

(nt)k ; k!

8r 2 [0; 1];

respectively. Therefore ~ n f (r) (t) M

~ n g (r) (t); M

8r 2 [0; 1];

respectively in . Next we use Remark 10.1(1). We notice that [(Mn f )(t)]r

=

(r)

(Mn f ) 2

= 4e

= =

1 X

k=0

1 X

nt

e "

nt

(r)

(t); (Mn f )+ (t)

k=0

e

nt

1 X

f

k f n (r)

k=0 (r)

~ n (f M

(nt)k k!

k f n

(nt)k k! k

k (nt) ;e n k!

~ n (f (r) (t) : )(t); M +

!(r)

;

!(r) 3 +

nt

5

1 X

(r) f+

k=0

k (nt)k n k!

#

(r)

(r)

~ n (f ), 8r 2 [0; 1], respectively, (that is ful…lling the = M I.e. (Mn f ) (r) basic condition of Assumption 10.20), 8n 2 N. Therefore (Mn (f )) (r) (Mn g) , respectively, i¤ Mn f % Mn g. We have proved Mn ’s positivity. The Vn ’s positivity follows similarly. We need to state Theorem 10.45. Let f 2 CB (R+ ). Then r ! ~ n f )(t) f (t)j 2! 1 f; t ; j(M n and j(V~n f )(t)

f (t)j

2! 1

f;

r

t + t2 n

!

8t > 0; n 2 N;

;

8t > 0; n 2 N;

(10.43)

(10.44)

154

10. Fuzzy Korovkin Theory and Inequalities

where ! 1 is the real basic …rst modulus of continuity. Proof. Let Z

be a probability measure on R+ . We notice that

fd

f (t)

=

Z

(f

f (t)) d

R+

R+

Z

(by Corollary 7.1.1, p. 209 of [4])

R+

jf (x)

! 1 (f ; )

f (t)j d (x)

Z

jx

tj

d (x) ( > 0)

R+

! 1 (f ; )

Z

1+

jx

tj

d (x)

R+

1

= ! 1 (f ; ) 1 +

Z

R+

jx

!

!

tjd (x)

(by Cauchy–Schwarz inequality) 0 !1=2 1 Z 1 A ! 1 (f ; ) @1 + (x t)2 d (x) R+

0

@by choosing

= So we have proved that Z

fd

f (t)

R+

0

2! 1 @f ;

Z

0

2! 1 @f

(x

R+

Z

R+

(x

:=

Z

(x

R+

!1=2 1

t)2 d (x)

!1=2 1 A t)2 d (x)

A:

!1=2 1 A: t)2 d (x)

(10.45)

Using now the content of Application 10.20 and setting as = FSn =n in ~ n f )(t), (V~n f )(t) are of the form (10.45) we get that the operator values (M Z Sn = f dFSn =n : E f n R+ R

1=2

is the standard deviation (x t)2 d (x) q q 2 in Poisson case is nt and in geometric case is t+t n . In this case

R+

Sn =n ,

which

Using now all of the above and (10.25) of Theorem 10.43 we obtain

10.3 Main Results

155

B Theorem 10.46. Operators Mn , Vn are well-de…ned on CF (R+ ), n 2 N B and they are fuzzy positive linear operators there. Also for each f 2 CF (R+ ) we have that Mn f , Vn f are uniformly in n fuzzy bounded. Furthermore it holds (r)

~ (f (r) ); = M

(r)

(r) = V~n (f );

(Mn f )

(Vn f )

(10.46)

B 8r 2 [0; 1], respectively, any f 2 CF (R+ ). It holds r ! t (F ) ; D((Mn f )(t); f (t)) 2! 1 f; n

(10.47)

and D((Vn f )(t); f (t))

(F ) 2! 1

f;

r

t + t2 n

!

B ; 8t > 0; n 2 N; f 2 CF (R+ ):

(10.48) Application 10.47. Let a real normed vector space (V; k k), let x0 2 V and a sequence ( n )n2N of positive …nite measures on V with n (V ) = mn > 0, n 2 N, where mn , > 0. Let f : V ! RF be fuzzy bounded function i.e. D(f (x); o~) M , M 0, 8x 2 V . Thus D(f (x + x0 ); o~) M , 8x 2 V , so that f ( + x0 ) is integrable by Theorem 10.17 and De…nition 10.18. We de…ne the fuzzy convolution operator Z (Ln f )(x0 ) = f (t + x0 ) n (dt); (10.49) V

for any x0 2 V , 8n 2 N. See that (Ln f )(x0 ) 2 RF and Ln is a fuzzy linear operator by Theorem 10.19. By the representation o n (r) (r) f (t + x0 ) = (f (t + x0 ); f+ (t + x0 )) j r 2 [0; 1] ; 8t 2 V

we get that: Let f; g : V ! RF fuzzy bounded functions such that f % g (r) (r) (r) (r) (r) (r) i¤ f g and f+ g+ , 8r 2 [0; 1] i¤ f (x) g (x), 8x 2 V , (r) (r) 8r 2 [0; 1], respectively, then f (t + x0 ) g (t + x0 ), 8t 2 V , 8r 2 [0; 1], respectively, then Z Z (r) (r) f (t + x0 )d (t) g (t + x0 )d (t); 8r 2 [0; 1]; V

V

respectively. Hence by De…nition 10.18 we get that Z

V

(r)

f (t + x0 )d (t)

Z

V

(r)

g(t + x0 )d (t))

;

8r 2 [0; 1];

156

10. Fuzzy Korovkin Theory and Inequalities (r)

(r)

respectively, i¤ ((Ln f )(x0 )) ((Ln g)(x0 )) , 8r 2 [0; 1], respectively, i¤ (Ln f )(x0 ) % (Ln g)(x0 ), any x0 2 V i¤ (Ln f ) % (Ln g), n 2 N. That is (Ln )n2N is a sequence of fuzzy positive operators. As before we obtain D (Ln f )(x0 ); f (x0 ) Z = D f (t + x0 ) n (dt); f (x0 ) V Z D f (t + x0 ) n (dt); f (x0 ) mn + D(f (x0 ) mn ; f (x0 )) V Z Z = D f (t + x0 ) n (dt); f (x0 ) n (dt) + D(f (x0 ) mn ; f (x0 )) V V Z D(f (t + x0 ); f (x0 )) n (dt) + jmn 1jD(f (x0 ); o~) ( > 0) ZV ktk (F ) 1jD(f (x0 ); o~) ! 1 (f ; ) n (dt) + jmn V Z ktk (F ) jmn 1jM + ! 1 (f ; ) 1+ n (dt) V Z 1 (F ) = jmn 1jM + ! 1 (f ; ) mn + ktk n (dt) V ! Z 1=2 p 1 (F ) 2 jmn 1jM + ! 1 (f ; ) mn + ktk n (dt) mn V

choosing

:=

Z

V

= M jmn

1j +

(F ) !1

f;

1=2

ktk Z

V

2

n (dt)

!

1=2

ktk

2

n (dt)

!

(mn +

p

mn ):

Finally we have established the following fuzzy convolution result. Theorem 10.48. Let (V; k k) real normed vector space and ( n )n2N positive …nite measures on V : n (V ) = mn > 0, n 2 N, where mn , > 0. Let f : V ! RF : D (f; o~) M , M 0. De…ne Z (Ln f )(x) = f (t + x) n (dt); 8x 2 V: (10.50) V

Then (Ln )n2N is a well-de…ned sequence of fuzzy positive linear operators. It holds ! Z 1=2 p (F ) D (Ln f; f ) M jmn 1j+! 1 f; ktk2 n (dt) (mn + mn ); n 2 N: V

(10.51)

10.3 Main Results

157

If mn = 1 then D (Ln f; f )

(F ) 2! 1

f;

Z

V

1=2

ktk

2

n (dt)

!

:

(10.52)

11 HIGHER ORDER FUZZY KOROVKIN THEORY USING INEQUALITIES

Here is studied with rates the fuzzy uniform and Lp , p 1, convergence of a sequence of fuzzy positive linear operators to the fuzzy unit operator acting on spaces of fuzzy di¤erentiable functions. This is done quantitatively via fuzzy Korovkin type inequalities involving the fuzzy modulus of continuity of a fuzzy derivative of the engaged function. From there we deduce general fuzzy Korovkin type theorems with high rate of convergence. The surprising fact is that basic real positive linear operator simple assumptions enforce here the fuzzy convergences. At the end we give applications. The results are univariate and multivariate. The assumptions are minimal and natural ful…lled by almost all example— fuzzy positive linear operators. This chapter follows [20].

11.1 Introduction Motivation for this chapter are [4], [18], [24], [32], [78], [96]. In fact this is continuation of [18], [24]. Here …rst we translate the necessary measure theory approximation results from [4] into the language of real positive linear operators, then by combining the facts, e.g. use of Proposition 11.2, we transfer results at the fuzzy level. G.A. Anastassiou: Fuzzy Mathematics: Approximation The., STUDFUZZ 251, pp. 159–190. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

160

11. Higher Order Fuzzy Korovkin Theory Using Inequalities

Applications are on univariate and multivariate Bernstein operators. At the beginning we provide all necessary fuzzy terminology, de…nitions and theorems we use here. In that background section we prove some results that they stand by themselves, such as in positivity. The basic ingredient to establish the results is the bridge between real operators to fuzzy ones. It is the natural realization condition: Assumption 11.21, see (11.17). This is ful…lled by almost all example positive operators, in fact by all summation and integration operators: real and fuzzy. The concept of fuzzy positivity we use is the natural analog of the real positivity, the same thing with linearity. We need De…nition 11.1. Let U (M; d) a metric space and let f : U ! RF . We de…ne the (…rst) fuzzy modulus of continuity of f by (F )

! 1 (f ; ) := for 0 <

diameter(U ). If

sup

D(f (x); f (y))

x;y2U d(x;y)

> diam(U ) then we de…ne

(F )

(F )

! 1 (f ; ) := ! 1 (f ; diam(U )): Proposition 11.2 (see [24]). Let U (M; d) metric space and f : U ! RF . Assume that (r) (F ) (r) ! 1 (f; ); ! 1 (f ; ); ! 1 (f+ ; ) are …nite for any > 0. Here ! 1 is the usual real modulus of continuity, i.e. for g : U ! R we de…ne ! 1 (g; ) :=

sup x;y2U d(x;y)

jg(x)

g(y)j;

etc. Then (F )

! 1 (f ; ) = sup max ! 1 (f r2[0;1]

(r)

(r)

; ); ! 1 (f+ ; ) :

We need De…nition 11.3. Let U open or compact (M; d) metric space and f : U ! RF . We say that f is fuzzy continuous at x0 2 U i¤ whenever xn ! x0 , then D(f (xn ); f (x0 )) ! 0. If f is continuous for every x0 2 U , we then call f a fuzzy continuous real number valued function. We denote the related space by CF (U ). Similarly one de…nes CF ([a; b]), [a; b] R, etc. De…nition 11.4. Let f : K ! RF , K open or compact (M; d) metric space. We call f a fuzzy uniformly continuous real number valued function, i¤ 8" > 0, 9 > 0: whenever d(x; y) , x; y 2 K, implies that U D(f (x); f (y)) ". We denote the related space by CF (K).

11.1 Introduction

161

De…nition 11.5. Let f : U ! RF , U (M; d) metric space. If D(f (x); o~) M , 8x 2 U , M 0, we call f a fuzzy bounded real number valued function. In particular if f 2 CF ([a; b]), [a; b] R, then f is a fuzzy bounded (F ) function, also ! 1 (f ; ) < 1 for any 0 < b a, etc. Also notice that U CF (K) = CF (K), for K compact (V; k k) real normed vector space. We use Proposition 11.6 (see [24]) Let K (V; k k) a real normed vector space and (F ) ! 1 (f ; ) = sup D(f (x); f (y)); > 0; x;y2K kx yk

the fuzzy modulus of continuity for f : K ! RF . Then U (1) If f 2 CF (K), K open convex or compact convex (F ) ! 1 (f ; ) < 1, 8 > 0.

(V; k k), then

(2) Assume that K is open convex or compact convex (V; k k), then (F ) U ! 1 (f ; ) is continuous on R+ in for f 2 CF (K). (3) Assume that K is convex, then (F )

(F )

! 1 (f; t1 + t2 )

(F )

! 1 (f; t1 ) + ! 1 (f; t2 );

t1 ; t 2

0;

that is the subadditivity property is true. Also it holds (F )

! 1 (f; n ) and

(F )

! 1 (f; where n 2 N,

(F )

d e! 1 (f; )

)

> 0,

(F )

n! 1 (f; ); (F )

( + 1)! 1 (f; );

> 0, d e is the ceiling of the number. (F )

(4) Clearly in general ! 1 (f ; ) (F ) ! 1 (f ; 0) = 0. (5) Let K be open or compact U i¤ f 2 CF (K).

0 and is increasing in (F )

> 0 and

(V; k k). Then ! 1 (f ; ) ! 0 as

#0

(6) It holds (F )

! 1 (f for

g; )

(F )

(F )

! 1 (f ; ) + ! 1 (g; );

> 0, any f; g : K ! RF , K

(V; k k) is arbitrary.

We also need Lemma 11.7 (see [24]) Let K be a compact subset of the real normed vector space (V; k k) and f 2 CF (K). Then f is a fuzzy bounded function. We use

162

11. Higher Order Fuzzy Korovkin Theory Using Inequalities

Proposition 11.8 (see [24]). Let U open or compact (M; d) metric space, (r) f 2 CF (U ). Then f are equicontinuous with respect to r 2 [0; 1] over U , respectively in . For the reverse we have Proposition 11.9 (see [24]). Let f to r 2 [0; 1] on U -open or compact , then f 2 CF (U ).

(r)

(r)

, f+ be equicontinuous with respect (M; d)-metric space, respectively in

We mention

De…nition 11.10 (see [53]). Let x; y 2 RF . If there exists a z 2 RF , such that x = y + z, then, we call z the H-di¤erence of x and y, denoted by z := x y. De…nition 11.11 (see [53]). Let T := [x0 ; x0 + ] R, with > 0. A function f : T ! RF is di¤erentiable at x 2 T , if there exists a f 0 (x) 2 RF , such that the limits lim

h!0+

f (x + h) h

f (x)

;

lim

f (x)

h!0+

f (x h

h)

;

exist and are equal to f 0 (x). We call f 0 the derivative of f at x. If f is di¤erentiable at any x 2 T , we call f di¤erentiable and it has derivative over T , the function f 0 . Here is assumed that f (x+h) f (x); f (x) f (x h) exist for small h: Similarly we de…ne higher order fuzzy derivatives. Regarding functions of several variables one can de…ne the same way partial derivatives in the fuzzy sense. n (Q) Let Q be a compact convex subset of Rk , k > 1 and n 2 N. By CF we mean all the functions from Q into RF that are n-times continuously di¤erentiable in the fuzzy sense. We use Theorem 11.12 (see [71]). Let f : [a; b] Let t 2 [a; b], 0 r 1. Clearly (r)

[f (t)]r = (f (t)) (r)

Then (f (t))

R ! RF be fuzzy di¤ erentiable. (r)

; (f (t))+

R:

(11.1)

are di¤ erentiable and (r) 0(r) 0 )+ )

[f 0r = ((f (t))

;

(11.2)

i.e. (r)

(f 0 ) We make

= (f

(r) 0

) ; for any r 2 [0; 1]:

(11.3)

11.1 Introduction

163

n Remark 11.13. 1) Let f 2 CF ([a; b]). Then by Theorem 11.12 and Propo(r) n sition 11.8 we obtain f 2 C ([a; b]) and (r) (i)

[f (i) (t)]r = ((f (t))

(r)

) ; ((f (t))+ )(i) ;

(11.4)

for i = 0; 1; 2; : : : ; n, and, in particular, we have that (r)

(f (i) )

for any r 2 [0; 1]. n 2) Let f 2 CF (Q), denote f := k P i = 1; : : : ; k and 0 < j j := i

= (f @ f @x ,

(r) (i)

) ;

(11.5)

where

:= (

1; : : : ;

k ),

i

2 Z+ ,

n, n > 1. Then we get by Theorem

i=1

11.12 that

(f for any r 2 [0; 1] and any We also make

(r)

(r)

) = (f )

:j j

n. Here f

;

(r)

(11.6) 2 C n (Q). (r)

(r)

Remark 11.14 (see [15], Remark 3). Let r 2 [0; 1], xi , yi 1; : : : ; m 2 N. Assume (r)

(r)

sup max(xi ; yi ) 2 R;

2 R, i =

i = 1; : : : ; m:

r2[0;1]

Then sup max r2[0;1]

m X

(r) xi ;

m X

(r) yi

i=1

i=1

!

m X

(r)

(r)

sup max(xi ; yi ):

(11.7)

i=1 r2[0;1]

We use (r)

Lemma 11.15. Let r 2 [0; 1], 2 I, I a …nite index set, and x Assume A := max sup max x(r) ; y (r) 2 R

;y

(r)

2 R.

2I r2[0;1]

and B := sup max max x(r) ; max y (r) 2I

r2[0;1]

2I

2 R:

(11.8)

It holds A = B: Proof. 1) First we prove B x(r) ; y (r)

(11.9)

A. We see that A;

8r 2 [0; 1] and 8 2 I:

Then A1 := max x(r) 2I

A;

A2 := max y (r) 2I

A

164

11. Higher Order Fuzzy Korovkin Theory Using Inequalities

and max(A1 ; A2 )

A:

Thus sup max(A1 ; A2 )

A:

r2[0;1]

2) We last prove A

B. We notice

x(r)

max x(r) ;

8r 2 [0; 1]; 8 2 I

2I

and y (r)

max y (r) ; 2I

8r 2 [0; 1];

8 2 I:

Then maxfx(r) ; y (r) g

max max x(r) ; max y (r) ; 2I

8r 2 [0; 1]; 8 2 I:

2I

Furthermore sup maxfx(r) ; y (r) g

r2[0;1]

sup max max x(r) ; max y (r) 2I

r2[0;1]

2I

= B; 8 2 I;

and …nally max sup maxfx(r) ; y (r) g 2I r2[0;1]

B:

We mention De…nition 11.16. Let L : CF (U ) ,! CF (U ), where U is open or compact (M; d) metric space, such that L(c1 f + c2 g) = c1 L(f ) + c2 L(g);

8c1 ; c2 2 R:

(11.10)

We call L a fuzzy linear operator. We give the following example of a fuzzy linear operator, etc. De…nition 11.17. Let f : [0; 1] ! RF be a fuzzy real function. The fuzzy algebraic polynomial de…ned by (F )

BN (f )(x) =

N X

k=0

N k x (1 k

x)N

k

f

k N

;

8x 2 [0; 1]; N 2 N; (11.11)

will be called the fuzzy Bernstein operator. We do have Theorem 11.18 (see p. 642, [66], S. Gal). If f 2 CF ([0; 1]), then (F )

D BN (f )(x); f (x)

3 (F ) 1 !1 f; p 2 N

;

N 2 N; 8x 2 [0; 1]; (11.12)

11.1 Introduction

165

i.e., (F )

lim D BN (f ); f = 0;

N !+1

(11.13)

(F )

that is BN f !n!+1 f , fuzzy uniform convergence. We also need De…nition 11.19. Let f; g : U ! RF , U (M; d) metric space. We denote (r) (r) (r) (r) f % g, i¤ f (x) % g(x), 8x 2 U , i¤ f+ (x) g+ (x) and f (x) g (x), (r) (r) (r) (r) 8x 2 U , 8r 2 [0; 1], i¤ f+ g+ and f g , 8r 2 [0; 1]. We give De…nition 11.20. Let L : CF (U ) ,! CF (U ) be a fuzzy linear operator, U open or compact (M; d) metric space. We say that L is positive, i¤ whenever f; g 2 CF (U ) are such that f % g then L(f ) % L(g), i¤ (r)

(r)

(L(f ))+ and

(r)

(L(f ))

(L(g))+ (r)

(L(g))

;

(11.14)

8r 2 [0; 1]:

(11.15)

Here we denote (r)

[L(f )]r = (L(f ))

(r)

; (L(f ))+ ;

8r 2 [0; 1]:

(11.16)

An example of a fuzzy positive linear operator is the fuzzy Bernstein operator on the domain [0; 1], etc. We will use this type of supposition. Assumption 11.21. Let L be a fuzzy positive linear operator from CF (K), K compact (M; d) metric space, into itself. Here we assume that there ~ from C(K) into itself with the property exists a positive linear operator L (r)

(Lf )

~ (r) ); = L(f

(11.17)

respectively, for all r 2 [0; 1], 8f 2 CF (K). As an example again we mention the fuzzy Bernstein operator and the real Bernstein operator ful…lling the above assumption on [0; 1]. etc. We will use Theorem 11.22. Let (X; O) be a linearly ordered vector space, (O denotes the order) and G a majorant subspace of X (i.e. for any x in X there exist z and y in G such that zOxOy). If f : G ! R is a positive linear map, then there exists (a not necessarily unique) F : X ! R positive linear map, such that F (x) = f (x), 8x 2 G. Above Theorem 11.22 is a special case of the famous Kantorovich Extension Theorem, due to L.V. Kantorovich [73]. For a proof see [1], Theorem 2.8, p. 26. See also [54], p. 72, Proposition 1.

166

11. Higher Order Fuzzy Korovkin Theory Using Inequalities

We need ~ : C n (Q) ! C(Q) be a positive linear operator, n 2 N, Lemma 11.23. Let L k Q compact R ,k 1. Then there exists a unique …nite Borel measure x , x 2 Q, such that Z ~ ))(x) = (L(f f (t)d x (t); 8f 2 C n (Q): Q

~ Notice (L(1))(x) = x (Q) < 1. ~ ))(x) is a positive linear functional on C n (Q). Also since Proof. Clearly (L( 1 2 C n (Q), C n (Q) is a majorant subspace of C(Q). Thus, by Theorem ~ 11.22 there exists a positive linear functional M : C(Q) ! R extending L, ~ = M n . Consequently, by Riesz Representation theorem there i.e. L C (Q) exists a unique …nite Borel measure x for this M , such that Z M (f ) = f (t)d x (t); 8f 2 C(Q): Q

Therefore ~ )(x) = L(f

Z

f (t)d

x (t);

Q

8f 2 C n (Q):

~ )(x). Let us assume that there exists We prove uniqueness x regarding L( another …nite Borel measure such that Z Z f (t)d x (t) = f (t)d (t); 8f 2 C n (Q): Q

Q

In particular, we have Z

Q

p(t)d

x (t) =

Z

p(t)d (t);

Q

for all polynomials p. Since by the Stone–Weierstrass approximation theorem the polynomials are uniformly dense in C(Q), it follows for any f 2 C(Q) that there exists a sequence fpn gn2N of polynomials that converges uniformly to f on Q. In particular fpn g is uniformly bounded. Now from Z Z pn (t)d x (t) = pn (t)d (t) Q

Q

and the Lebesgue Dominated Convergence Theorem, by taking the limits in the last equation, we …nd Z Z f (t)d x (t) = f (t)d (t): Q

Q

Hence, by the uniqueness of the measure in the Riesz representation theorem, we get indeed that x = .

11.2 Univariate Results

167

We give Remark 11.24. We recall from [4], p. 210–211 the function n (x)

:=

Z

jxj

0

t (jxj t)n 1 dt; h (n 1)!

(11.18)

x 2 R, n 2 N, where d e is the ceiling of the number. We have Z xn 1 l m Z jxj Z x1 xn dxn dx1 ; (x) = n h 0 0 0 and

Also it holds

0 1 1 @X (jxj n (x) = n! j=0

n (x) =

jh)n+ A ;

x 2 R:

jxjn+1 jxjn hjxjn 1 + + (n + 1)!h 2n! 8(n 1)!

n (x)

and

1

Z

(11.19)

(11.20)

;

(11.21)

x 2 R+ ;

n 2 N:

(11.22)

x 2 R;

n 2 N:

(11.23)

x n 1 (t)dt;

0

And from [4], p. 217 we have n (x)

jxjn n!

1+

jxj (n + 1)h

;

11.2 Univariate Results We need to mention Theorem 11.25 (by Corollary 7.2.2, p. 219–220 in [4] and Geometric Moment theory [13]). Consider the positive linear operator ~ : C([a; b]) ! C([a; b]): L

(11.24)

Let ck (x) :=

~ L((t

dn (x) := c(x) :=

1=n ~ L(jt xjn ; x) ; max(x a; b x) (c(x)

x)k ; x); k = 0; 1; : : : ; n; n 2 N; (11.25) (b

a)=2):

168

11. Higher Order Fuzzy Korovkin Theory Using Inequalities

Let g 2 C n ([a; b]) such that ! 1 (g (n) ; h) Then ~ x) jL(g;

g(x)j

w, where w; h > 0, 0 < h n X jg (k) (x)j

jg(x)j jc0 (x)

1j +

+w

dn (x) c(x)

k!

k=1

n (c(x))

b

a.

jck (x)j

n

;

8x 2 [a; b]: (11.26)

Inequality (11.26) is sharp. It is attained in a certain sense by w n ((t x)+ ) and a measure ~ x supported by fx; bg when x a b x, also attained by w n ((x t)+ ) and a measure ~ x supported by fx; ag when x a b x : dn (x) n n (x) n in each case with masses c0 (x) and dc(x) , respectively. c(x) Note 11.26. Assuming ! 1 (g (n) ; h) > 0 for h > 0 and all as in the context of Theorem 11.25. We prefer to write ~ x) jL(g;

g(x)j

jg(x)j jc0 (x) +

n (c(x))

1j + dn (x) c(x)

n X jg (k) (x)j

k!

k=1 n

jck (x)j

! 1 (g (n) ; h);

(11.27)

8x 2 [a; b]: We need also Theorem 11.27 (By Theorem 7.3.5, p. 231–232 in [4] and Lemma 11.23). Consider the positive linear operator ~ : C n ([a; b]) ! C([a; b]); L

n 2 N:

Assume that ~ x) > 0; L(1; and ~ jt L

xjn+1 ; x > 0;

x 2 [a; b];

Consider also > 0. Consider g 2 C n ([a; b]), n for any > 0. Then ~ x) jL(g;

g(x)j

~ x) jg(x)j jL(1;

1j +

k!

~ L((t

x)k ; x) +

~ x) 1=(n+1) L(1; ~ L(jt n! ! 1 ~ L(jt xjn+1 ; x) (n+1) ; ~ x) L(1;

1 n 2 + + 8 2 (n + 1) ! 1 g (n) ;

1, with ! 1 (g (n) ; ) > 0

n X jg (k) (x)j

k=1

(11.28)

xjn+1 ; x)

n (n+1)

(11.29)

11.2 Univariate Results

169

x 2 [a; b]; n 2 N: We will use also Theorem 11.28 (by Theorem 7.2.2, p. 216–217, in [4] and Lemma 11.23). Consider the positive linear operator ~ : C n ([a; b]) ! C([a; b]); L

n 2 N:

Assume that ~ x) > 0; L(1;

(11.30)

and ~ L(jt

xjn+1 ; x) > 0;

Consider g 2 C n ([a; b]), n ~ x) jL(g;

1, with ! 1 (g (n) ; ) > 0 for any

~ x) jg(x)j jL(1;

g(x)j

+ +

x 2 [a; b]:

~ L(jt

1j +

n+1

xj

; x) n!

n X jg (k) (x)j

k=1 n=(n+1)

k!

> 0. Then

~ jL((t

x)k ; x)j 1

~ (n+1) (L(1)(x))

1 1 ~ ! 1 g (n) ; (L(jt xjn+1 ; x)) n+1 ; (n + 1) x 2 [a; b]; n 2 N: (11.31)

~ ~ Remark 11.29. 1) If kL(1)k 1 = 0, then L = 0 the trivial operator, ~ 6 0, and as a result therefore without loss of generality we may assume L ~ we have kL(1)k > 0. 1 ~ 2) By using Hölder’s inequality and Riesz Representation theorem, for L as in (11.24), we easily derive that ~ jL((t

x)k ; x)j

~ x) L(1;

1

k n

~ x) > 0, 0 < k under the assumption L(1; ~ kL((t

1

k

k n

xjn ; x)

;

(11.32)

n. And it holds k

~ 6 0: n; L (11.33) ~ : C n ([a; b]) ! C([a; b]) positive linear Similarly, by Lemma 11.23, for L operator, it holds ~ jL((t

x)k ; x)k1

x)k ; x)j

n ~ ~ kL(1)k 1 kL(jt

~ L(jt

~ x) L(1;

1

k (n+1)

1

k (n+1)

~ L(jt

n xjn ; x)k1 ; 0
xjn+1 ; x)

k (n+1)

; 0 < k < n + 1: (11.34)

And also it holds ~ kL((t

x)k ; x)k1

~ kL(1)k 1

~ kL(jt

k

(n+1) xjn+1 ; x)k1 ; 0 < k < n + 1: (11.35)

170

11. Higher Order Fuzzy Korovkin Theory Using Inequalities

~ x) = 0, x 2 [a; b], then easily we get that L(g; ~ x) = 0, 8g 2 3) If L(1; C n ([a; b]). So that inequalities (11.26), (11.27) and (11.31) hold trivially. 4) If dn (x) = 0, x 2 [a; b], then we get that ck (x) = 0, k = 0; 1; : : : ; n, since the associated measure x is concentrated at fxg only, etc., and inequalities (11.26) and (11.27) hold again, in fact as equalities. ~ ~ 5) Similarly, if L(jt xjn+1 ; x) = 0, x 2 [a; b], then we get again L((t x)k ; x) = 0, k = 1; : : : ; n, since the associated measure x is concentrated at fxg only, etc., and inequalities (11.29) and (11.31) hold again, in fact as equalities. 6) If ! 1 (g (n) ; h) = 0 for some h > 0 then g (n) is the constant function, furthermore inequalities (11.26), (11.27), (11.29) and (11.31) are again valid. We give the …rst main fuzzy result. Theorem 11.30. Consider the fuzzy positive linear operator n L : CF ([a; b]) ! CF ([a; b]); n 2 N;

(11.36)

with the property (r)

(Lf )

~ (r) ); = L(f

(11.37)

n ~ is a positive linear ([a; b]). Here L respectively, for all r 2 [0; 1], 8f 2 CF operator such that ~ : C([a; b]) ! C([a; b]): L (11.38)

Let ck (x) := dn (x) := c(x) :=

~ (t L

x)k ; x ;

k = 0; 1; : : : ; n; 1=n

~ L(jt xjn ; x) ; max(x a; b x):

(11.39)

(F )

n Let f 2 CF ([a; b]) such that ! 1 (f (n) ; h) > 0 for any h > 0. Then 1)

D (Lf )(x); f (x)

jc0 (x)

1jD(f (x); o~) +

n X jck (x)j

k=1

dn (x) + n (c(x)) c(x)

n

k!

D(f (k) (x); o~)

(F ) ! 1 (f (n) ; h);

(11.40)

x 2 [a; b]; and 2) D (Lf; f )

~ kL1 +k

1k1 D (f; o~) + n (c(x))k1

n X ~ kL((t

x)k ; x)k1 D (f (k) ; o~) k!

k=1

~ L(jt xjn ; x) c(x)n

(F )

! 1 (f (n) ; h): (11.41) 1

11.2 Univariate Results

171

Proof. We have the following (r)

D((Lf )(x); f (x)) = sup max j(Lf )

(x)

f

(r)

(x)j;

r2[0;1]

(r)

(r)

j(Lf )+ (x)

f+ (x)j

~ sup max j(L(f

=

(r)

))(x)

f

(r)

r2[0;1]

~ (r) ))(x) (x)j; j(L(f +

(r)

f+ (x)j

(by Remark 11.13 (11.1) and (11.27)) ( n (r) X j(f (k) ) (x)j (r) sup max jf (x)j jc0 (x) 1j + jck (x)j k! r2[0;1] k=1

n

dn (x) (r) (r) + n (c(x)) ! 1 (f (n) ) ; h ; jf+ (x)j jc0 (x) 1j c(x) ) n (r) n X j(f (k) )+ (x)j dn (x) (n) (r) + jck (x)j + n (c(x)) ! 1 (f )+ ; h k! c(x) k=1

(11:7)

jc0 (x) +

r2[0;1]

n X jck (x)j

k=1

+

1j sup max jf

k!

n (c(x))

(r)

(r)

(x)j; jf+ (x)j (r)

sup max j(f (k) )

r2[0;1]

dn (x) c(x)

(r)

(x)j; j(f (k) )+ (x)j

n (r)

sup max ! 1 ((f (n) ) r2[0;1]

(r)

; h); ! 1 ((f (n) )+ ; h)

(by Proposition 11.2) = jc0 (x) 1jD(f (x); o~) n X jck (x)j dn (x) D(f (k) (x); o~) + n (c(x)) + k! c(x)

n (F )

! 1 (f (n) ; h):

k=1

That is proving (11.40). We proceed with the next main result. Theorem 11.31. Consider the fuzzy positive linear operator n L : CF ([a; b]) ! CF ([a; b]);

n 2 N;

with the property (r)

(Lf )

~ (r) ); = L(f

n ~ is a positive linear respectively, for all r 2 [0; 1], 8f 2 CF ([a; b]). Here L n ~ x) > operator from C ([a; b]) into C([a; b]). Additionally assume that L(1; (F ) (n) n 0, x 2 [a; b] and consider > 0. Let f 2 CF ([a; b]) such that ! 1 (f ; h) > 0 for any h > 0. Then

172

11. Higher Order Fuzzy Korovkin Theory Using Inequalities

1) D((Lf )(x); f (x))

~ x) 1jD(f (x); o~) jL(1; n X ~ jL((t x)k ; x)j + D(f (k) (x); o~) k! k=1

+

n 2 1 + + 8 2 (n + 1) ~ x) L(1; n!

(F ) !1

f

(11.42)

1 (n+1)

(n)

~ L(jt ~ L(jt

;

xjn+1 ; x)

xjn+1 ; x) ~ x) L(1;

n (n+1)

1 (n+1)

!

;

n 2 N; x 2 [a; b]: ~ x) > 0, 8x 2 [a; b], it holds 2) And by assuming L(1; D (Lf; f )

~ kL1

1k1 D (f; o~) +

n X ~ kL((t

k=1

x)k ; x)k1 D (f (k) ; o~) k! 1

(n+1) ~ x)k1 1 n 2 kL(1; ~ + + kL(jt + 8 2 (n + 1) n! ! 1 ~ L(jt xjn+1 ; x) (n+1) (F ) (n) !1 f ; : ~ x) L(1; 1

n

(n+1) xjn+1 ; x)k1

(11.43)

Proof. Here we are using (11.29), see also Remark 11.29(11.5). The proof is similar to the proof of Theorem 11.30 and is omitted. We present Theorem 11.32. Consider the fuzzy positive linear operator n L : CF ([a; b]) ! CF ([a; b]);

n 2 N;

(11.44)

with the property (r)

(Lf )

~ = L(f

(r)

);

(11.45)

n ~ is a positive linear respectively, for all r 2 [0; 1], 8f 2 CF ([a; b]). Here L operator such that ~ : C n ([a; b]) ! C([a; b]): L (11.46) (F )

n Let f 2 CF ([a; b]) such that ! 1 (f (n) ; h) > 0 for any h > 0. Then

11.2 Univariate Results

173

1) ~ x) 1jD(f (x); o~) jL(1; n X ~ j(L(( x)k ))(x)j + D(f (k) ; (x); o~) k!

D((Lf )(x); f (x))

k=1

~ L(j

+

(F )

!1

n

1 xjn+1 )(x) (n+1) 1 ~ x)) (n+1) + L(1; n! (n + 1) 1 ~ (11.47) f (n) ; (L(j xjn+1 )(x)) (n+1) ;

8x 2 [a; b]: And also it holds 2) ~ kL1

D (Lf; f )

+

n X ~ k(L((

1k1 D (f; o~) + ~ (L(j

n+1

xj

k=1

))(x)

n! (F ) !1

f

(n)

~ ; (L(j

x)k ))(x)k1 D (f (k) ; o~) k!

n (n+1)

~ L(1)

1 n+1

xj

1 (n+1)

1 n+1

+

1 (n + 1)

:

))(x)k1

1

(11.48)

Proof. Here we use (11.31) and see also Remark 11.29 (11.3) & (11.5). We do have (r)

D((Lf )(x); f (x)) = sup max j(Lf )

(x)

f

(r)

r2[0;1]

~ (r) )(x) sup max jL(f

=

r2[0;1]

(

sup max jf

r2[0;1]

+

~ ((L(j

(r)

n+1

xj

(r)

n X

~ x) (x)j jL(1;

))(x))

n (n+1)

~ ((L(j

k!

k!

k=1

1

~ x)) (n+1) + (L(1;

~ j(L((

~ j(L((

n

(r)

1j sup max jf

1 1 ~ x)) (n+1) (L(1; + (n + 1) ) 1

xjn+1 ))(x)) (n+1) (r)

(r)

(x)j; jf+ (x)j

x)k ))(x)j

1 (n + 1)

x)k ))(x)j

xjn+1 ))(x)) (n+1) n!

r2[0;1]

f+ (x)j

n (r) X j(f (k) ) (x)j

1

(r) ~ ! 1 (f (n) )+ ; ((L(j

~ x) jL(1;

1j +

(r)

f+ (x)j

(r)

~ (r) )(x) (x)j; jL(f +

~ x) xjn+1 ))(x)) (n+1) ; jf+ (x)j jL(1;

~ ; ((L(j

(r) j(f (k) )+ (x)j

k=1

+

(r)

n! ! 1 (f (n) )

+

f

(r)

(x)j; j(Lf )+ (x)

1j

174

11. Higher Order Fuzzy Korovkin Theory Using Inequalities

+

n X ~ j(L((

k=1

+

~ ((L(j

x)k ))(x)j (r) (r) sup max j(f (k) ) (x)j; j(f (k) )+ (x)j k! r2[0;1] n

xjn+1 ))(x)) (n+1) n! (r)

sup max ! 1 (f (n) )

1

~ x)) (n+1) + (L(1;

~ ; ((L(j

1 (n + 1) 1

xjn+1 ))(x)) (n+1) ;

r2[0;1] 1

(r) ~ ! 1 (f (n) )+ ; ((L(j

~ x) = jL(1; +

~ (L(j

xjn+1 ))(x)) (n+1) n X ~ j(L(( x)k ))(x)j D(f (k) (x); o~) 1jD(f (x); o~) + k! k=1

n+1

xj

)(x))

n (n+1)

n! (F )

!1

1

~ x)) (n+1) + (L(1;

1 (n + 1)

1

~ f (n) ; ((L(j

xjn+1 ))(x)) (n+1) :

(F )

Note 11.33. If ! 1 (f (n) ; h) = 0 for some h > 0, then by Proposition (r) (r) (r) 11.2 we get ! 1 ((f (n) ) ; h), ! 1 ((f (n) )+ ; h) = 0, 8r 2 [0; 1], i.e. (f )(n) = (r) (f (n) ) are constant real valued functions, 8r 2 [0; 1]. Consequently (see Remark 11.29(11.6) and repeat proofs) inequalities (11.40), (11.41), (11.42), (11.43), (11.47) and (11.48) are again valid. We give the following fuzzy Korovkin type theorem ([78]). Theorem 11.34. Consider the sequence of fuzzy positive linear operators n LN : CF ([a; b]) ! CF ([a; b]);

n 2 N; 8N 2 N;

(11.49)

with the property (r)

(LN f )

~ N (f =L

(r)

);

(11.50)

n ~ N gN 2N is respectively, for all r 2 [0; 1], 8f 2 CF ([a; b]), 8N 2 N. Here fL a sequence of positive linear operators such that

~ N : C n ([a; b]) ! C([a; b]): L ~ N (1)k1 Assume that kL , 8N 2 N, for some > 0. Furthermore asu ~ ~ N (j sume that LN 1 ! 1 and k(L xjn+1 ))(x)k1 ! 0, as N ! 1. Then n D (LN f; f ) ! 0 as N ! 1, 8f 2 CF ([a; b]). I.e. LN ! I, as N ! 1, fuzzy and uniformly, where I is the fuzzy unit operator.

11.3 Multidimensional Results

175

Proof. From (11.48) we get ~ N 1 1k1 D (f; o~) kL n X ~ N (( x)k ))(x)k1 k(L + D (f (k) ; o~) k!

D (LN f; f )

(11.51)

k=1

+

n

~ N (j k(L

(F ) !1

f

(n+1) 1 xjn+1 ))(x)k1 1 ~ N (1)) (n+1) (L + n! (n + 1)

(n)

~ N (j ; k(L

n+1

xj

1 (n+1)

))(x)k1

;

1

8N 2 N:

Also by (11.35) we have ~ N ((t kL

x)k ; x)k1

1

~ N (1)k1 kL

k (n+1)

~ N (jt kL

k

(n+1) xjn+1 ; x)k1 ; (11.52)

any 0 < k < n + 1, 8N 2 N. Now by using (11.52), (11.51) and the assumptions of the theorem we conclude that D (LN f; f ) ! 0, as N ! 1. Comment. Inequality (11.51), proof of Theorem 11.34, gives the convergence of LN ! I, quantitatively, and at higher rate, re‡ecting the higher order fuzzy di¤erentiability of f .

11.3 Multidimensional Results We need Theorem 11.35 (By Theorem 7.4.1, p. 236 of [4] and the Riesz Representation Theorem). Take Q := fx 2 Rk : kxk`1 1g, k 1, x 2 Q. Let ~ : C(Q) ! C(Q) positive linear operator with L(1; ~ x) = 1 and f 2 C n (Q), L n 2 N. Here f = @@x f , where := ( 1 ; : : : ; k ), i 2 Z+ , i = 1; : : : ; k, and k P 0 < j j := n. Assume for h > 0 we have w := max ! 1 (f ; h) > 0, i j j=n

i=1

where ! 1 is the usual modulus of continuity with respect to k k`1 relative to Q. Then !! k n Y X X jf (x)j ~ i ~ L (zi xi ) ;x jL(f; x) f (x)j 1! k! i=1 j=1 j j=j

~ + (L(kz

xk`1 ; x))

n (1 + kxk`1 ) w: (1 + kxk`1 )

(11.53)

Proof. We add the following, in order to transfer here Theorem 7.4.1, p. 236 of [4]. Let gz (t) := f (x + t(z x)), x; z 2 Q, all 0 t 1. For

176

11. Higher Order Fuzzy Korovkin Theory Using Inequalities

j = 1; : : : ; n we have

gz(j) (t)

"

=

k X

(zi

i=1

j

@ xi ) @xi

#

f (x + t(z

x))

with gz(j) (0)

=

"

k X

(zi

i=1

@ xi ) @xi

j

#

f (x);

8x; z 2 Q:

More precisely, we get (j) X gz (0) = j!

j j=j

Qk

i=1 (zi Qk i=1

xi ) i!

i

f (x):

(11.54)

We will use Theorem 11.36 (By Theorem 7.4.2, p. 237 of [4]). Let be a measure of mass m > 0 on Q Rk , k 1 compact and convex. Assume 1 (n + 1)

1 m

Z

Q

kx

x0 kn+1 `1

1 (n+1)

(dx)

=: h > 0;

where x0 is a …xed point of Q. Also, let f 2 C n (Q) : maxj w, where w > 0. Then Z

Q

fd

f (x0 )

jm

(11.55)

j=n

! 1 (f ; h)

Z n X 1 1j jf (x0 )j + gx(j) (0) (dx) j! Q j=1

+ mwhn

(n + 1)n 1 3 (n + 1)n + ; (11.56) 2 n! 8(n 1)!

where gx (t) := f (x0 + t(x x0 )), t 0. Above inequality (11.56) is trivially true if h = 0, or if w = 0 with h > 0. Translating last Theorem 11.36 into the terminology of positive linear operators and by expanding we have ~ be a Theorem 11.37. Let Q Rk , k 1 compact and convex. Let L ~ positive linear operator from C(Q) into C(Q). Assume L(1; x) > 0, 8x 2 Q.

11.3 Multidimensional Results

177

Consider f 2 C n (Q), n 2 N. Then

~ )(x) f (x)j jL(1)(x) ~ jL(f 1j jf (x)j ( n k X X jf (x)j Y ~ + L (zi xi ) k Q j=1 j j=j i=1 i!

)

(x)

i

i=1

~ + L(1)(x)

1 (n+1)

~ L(kz

xkn+1 `1 )(x)

3 + 2n! 8(n

n (n+1)

~ (L(kz xkn+1 1 `1 ))(x) f ; ~ (n + 1) L(1)(x)

max ! 1

j j=n

1 (n+1)

1 1)!(n + 1) ! ;

(11.57)

8x 2 Q: We have ~ x) = Corollary 11.38 (to Theorem 11.37). All as in Theorem 11.37 with L(1; 1, 8x 2 Q. Then ) ( k n Y X X jf (x)j i ~ ~ (zi xi ) (x) L jL(f )(x) f (x)j k Q i=1 j=1 j j=j ! i i=1

~ + L(kz

xkn+1 `1 )(x)

max ! 1 f ;

j j=n

n (n+1)

3 + 2n! 8(n

1 ~ (L(kz (n + 1)

1 1)!(n + 1)

xkn+1 `1 ))(x)

1 (n+1)

(11.58)

;

8x 2 Q; 8f 2 C n (Q): We further give Theorem 11.39. Let Q be a compact and convex subset of Rk , k 1, and let x0 := (x01 ; : : : ; x0k ) 2 Q be …xed and let be a measure on Q of mass m 0. Consider f 2 C n (Q), n 2 N, and suppose that each nth order partial derivative f := @@x f , where := ( 1 ; : : : ; k ), i 2 Z+ , i = 1; : : : ; k and n P j j := i = n, has relative to Q and the k k`1 , a modulus of continuity i=1

! 1 (f ; h)

w. Here we take Z h := kz Q

x0 kn+1 `1 d (z)

1 (n+1)

:

(11.59)

Then Z

Q

fd

f (x0 )

jm +

Z n X 1 gx(j) (0) (dx) 1j jf (x0 )j + j! Q j=1

whn n!

1

m (n+1) +

1 (n + 1)

;

178

11. Higher Order Fuzzy Korovkin Theory Using Inequalities

where gx (t) := f (x0 + t(x

x0 )), t

0, x 2 Q.

Proof. Here we have f (z1 ; : : : ; zk ) = gz (1) =

n (j) X gz (0)

j!

j=0

+ Rn (z; 0);

where Rn (z; 0) :=

Z

1

0

Z

Z

t1

tn

1

(gz(n) (tn )

gz(n) (0))dtn

dt1 ; z 2 Q:

0

0

In [4], p. 236 we got that (see 7.4.4 there) jRn (z; 0)j

w

n (kz

x0 k`1 ); 8z 2 Q:

But by (11.23) we have w

n (kz

x0 k`1 )

w

Z

Q

jRn (z; 0)j d (z) w n!

Z

Q

w

Z

Q

x0 kn`1 d

kz

x0 kn`1 kz x0 k`1 1+ n! (n + 1)h

w kz n!

= Hence

kz

n kz

kz x0 kn+1 `1 : (n + 1)h

x0 kn`1 +

x0 k`1 )d (z)

1 (z) + (n + 1)h

Z

Q

kz

x0 kn+1 `1 d (z)

(not to have a trivial case we take (Q) = m > 0) " n Z (n+1) 1 w n+1 (n+1) m kz x0 k`1 d (z) n! Q # Z 1 1 n+1 + kz x0 k`1 d (z) h (n + 1) Q (we choose h :=

Z

Q

x0 kn+1 `1 d

kz

1 (n+1)

(z)

> 0;

the case h = 0 is trivial and not discussed here). =

whn n!

That is, we got that Z jRn (z; 0)jd (z) Q

1

m (n+1) +

1 (n + 1)

whn n!

:

1

m (n+1) +

1 (n + 1)

:

11.3 Multidimensional Results

179

The validity of (11.59) is now clear. By using Riesz Representation theorem, Theorem 11.39 and expanding we obtain Theorem 11.40. Let Q be a compact and convex subset of Rk , k 1. Let ~ be a positive linear operator from C(Q) into itself. Consider f 2 C n (Q), L n 2 N. Then ~ )(x) f (x)j jL(1) e (x) 1j jf (x)j jL(f ( n k X X jf (x)j Y ~ + L (zi xi ) k Q j=1 j j=j i=1 ! i

)

(x)

i

i=1

+

~ (L(kz

n (n+1)

xkn+1 `1 ))(x) n!

~ (L(1))(x)

~ max ! 1 f ; (L(kz

1 (n+1)

xkn+1 `1 ))(x)

j j=n

1 (n+1)

+

;

1 (n + 1)

8x 2 Q: (11.60)

Next we give the fuzzy multidimensional related results. Theorem 11.41. Let Q := fx 2 Rk : kxk`1 fuzzy positive linear operator n L : CF (Q) ! C(Q);

1g, k

1. Consider the

n 2 N;

(11.61)

with the property (r)

(Lf )

~ (r) ); = L(f

(11.62)

n ~ is a positive linear respectively, for all r 2 [0; 1], 8f 2 CF (Q). Here L operator such that ~ : C(Q) ! C(Q); L (11.63)

~ x) = 1. Also f = with L(1; := (

1; : : : ;

k ),

i

@ f @x

is the fuzzy partial derivative of f , where k P 2 Z+ , i = 1; : : : ; k, and 0 < j j := n. i i=1

(F )

(F )

Assume for h > 0 we have that w(F ) := max ! 1 (f ; h) > 0, where ! 1 j j=n

is the fuzzy modulus of continuity with respect to k k`1 relative to Q. Then 1)

D((Lf )(x); f (x)) n X j=1

(

X

j j=j

~ + (L(kz

~ ( L

k Q

(zi

xi ) i ) (x)

i=1

xk`1 ))(x)

k Q

D(f (x); o~) i!

)

i=1 n (1 + kxk`1 ) (F ) w ; 8x 2 Q: (1 + kxk`1 )

(11.64)

180

11. Higher Order Fuzzy Korovkin Theory Using Inequalities

2) D (Lf; f )

n X j=1

~ + (L(kz

(

~ ( Q (zi L k

X

xi ) i ) (x)

i=1

k Q

j j=j

1

D (f ; o~)

i!

)

i=1

xk`1 ))(x)

n (1 + kxk`1 ) (1 + kxk`1 )

1

w(F ) :

(11.65)

1

Proof. We have the following (r)

D((Lf )(x); f (x)) = sup max j(Lf )

f

(r)

(x)j;

r2[0;1]

(r)

(r)

j(Lf )+

(11:62)

=

f+ (x)j

~ (r) ) sup max jL(f

f

(r)

r2[0;1]

~ (r) ) (x)j; jL(f +

(r)

f+ (x)j

(by Remark 11.13 (11.2), (11.6) and (11.53)) ( n k X X j(f )(r) (x)j Y ~ sup max (zi xi ) L k Q r2[0;1] j=1 j j=j i=1 ! i

i

i=1

(r) n (1 + kxk`1 ) max ! 1 ((f ) ; h); (1 + kxk`1 ) j j=n

~ + (L(kz

xk`1 )(x))

n X X

j(f )+ (x)j k Q

j=1 j j=j

(r)

k Y

~ L

(zi

xi )

(x)

i

i=1

i

i=1

) (1 + kxk ) ` (r) 1 ~ + (L(kz xk`1 )(x)) n max ! 1 ((f )+ ; h) (1 + kxk`1 ) j j=n # " k n Y X D(f (x); o~) (11:7) X i ~ (zi xi ) (x) L k Q i=1 j=1 j j=j i! i=1

~ + (L(kz

xk`1 )(x))

(

n (1

+ kxk`1 ) sup (1 + kxk`1 ) r2[0;1] (r)

max max ! 1 ((f ) j j=n

=

(by Lemma 11.15) " X D(f (x); o~) k Q j=1 j j=j i!

n X

i=1

~ L

; h); max ! 1 ((f j j=n

k Y

i=1

(zi

xi )

i

)

(r) )+ ; h)

(x)

#

(x)

11.3 Multidimensional Results

~ + (L(kz

xk`1 )(x)) (r)

181

n (1 + kxk`1 ) max sup (1 + kxk`1 ) j j=n r2[0;1] (r)

max ! 1 ((f )

; h); ! 1 ((f )+ ; h) " n (by Proposition 11.2) X X D(f (x); o~) = k Q j=1 j j=j i! i=1

~ L

k Y

(zi

xi )

i

(x)

i=1

~ + (L(kz

xk`1 (x))

#

(F ) n (1 + kxk`1 ) max ! (f ; h): (1 + kxk`1 ) j j=n 1

Inequality (11.64) is established. The next result follows. Theorem 11.42. Let Q Rk , k the fuzzy positive linear operator

1 be a compact convex subset. Consider

n L : CF (Q) ! C(Q);

n 2 N;

with the property (r)

(Lf )

~ (r) ); = L(f

n ~ is a positive linear (Q). Here L respectively, for all r 2 [0; 1], 8f 2 CF ~ ~ operator such that L : C(Q) ! C(Q), with L(1; x) > 0, 8x 2 Q. Consider n f 2 CF (Q). Then 1)

D((Lf )(x); f (x))

+

" n X X

j=1 j j=j

~ + L(1)(x)

i=1

1 (n+1)

(F )

max ! 1

j j=n

8x 2 Q: Also it holds

~ jL(1)(x) 1jD(f (x); o~) k # ~ Q (zi xi ) i (x) L i=1 D(f (x); o~) k Q i! ~ L(kz

f ;

xkn+1 `1 )(x)

n (n+1)

3 + 2n! 8(n

~ (L(kz xkn+1 1 `1 ))(x) ~ (n + 1) L(1)(x)

1 (n+1)

1 1)!(n + 1) ! ;

(11.66)

182

11. Higher Order Fuzzy Korovkin Theory Using Inequalities

2) ~ kL(1)

D (Lf; f )

+

1k1 D (f; o~) k " ~ Q (zi L n X X i=1 j=1 j j=j

xi ) k Q

(x)

i

1

D (f ; o~)

i!

#

i=1

1 (n+1)

~ (L(kz

~ + kL(1)k 1

(F )

f ;

j j=n

1

1 1)!(n + 1)

3 + 2n! 8(n

max ! 1

n (n+1)

xkn+1 `1 ))(x)

(11.67) ! 1

~ (L(kz xkn+1 1 `1 ))(x) ~ (n + 1) L(1)(x)

(n+1)

:

1

Proof. We observe the following: (r)

D((Lf )(x); f (x)) = sup max j(Lf )

(x)

f

(r)

r2[0;1]

(r)

(x)j; j(Lf )+ (x)

(r) f+ (x)j

~ (r) )(x) sup max jL(f

=

r2[0;1] (11:57)

(

~ sup max jL(1)(x)

r2[0;1]

+

n X X

(r)

j=1 j j=j

j(f ) k Q

(x)j

f

(r)

~ (r) )(x) (x)j; jL(f +

1j jf ~ L

(r)

k Y

(r)

f+ (x)j

(x)j

(zi

xi )

i

(x)

i=1

i!

i=1

1 1)!(n + 1) ! 1 n+1 ~ L(kz xk`1 )(x) (n+1) 1 (r) ~ ; jL(1)(x) max ! 1 (f ) ; ~ (n + 1) j j=n L(1)(x) " # n k X X j(f )(r) Y (r) + (x)j ~ jf+ (x)j + L (zi xi ) i (x) k Q j=1 j j=j i=1 ! i ~ +(L(1)(x))

1 (n+1)

~ L(kz

xkn+1 `1 )(x)

n (n+1)

3 + 2n! 8(n

i=1

~ + (L(1)(x))

1 (n+1)

max ! 1 (f

j j=n

~ L(kz

(r) )+ ;

xkn+1 `1 )(x)

n (n+1)

3 + 2n! 8(n

~ L(kz xkn+1 1 `1 )(x) ~ (n + 1) L(1)(x)

1 (n+1)

1 1)!(n + 1) )

1j

11.3 Multidimensional Results

~ jL(1)(x)

1jD(f (x); o~) + #

k ~ Q (zi L

X

k Q

i

(x)

i!

i=1

xkn+1 `1 )(x)

(

(r)

sup max max ! 1 (f ) r2[0;1]

xi )

i=1

1 ~ ~ (n+1) L(kz + (L(1)(x))

1 1)!(n + 1)

8(n

"

j=1 j j=j

D(f (x); o~) +

n X

183

j j=n

1 ~ (n+1) L(kz xkn+1 1 `1 )(x) ; ~ (n + 1) L(1)(x) ~ L(kz xkn+1 1 (r) `1 )(x) max ! 1 (f )+ ; ~ (n + 1) j j=n L(1)(x)

n (n+1)

3 2n!

;

1 (n+1)

)

(by Lemma 11.15 and Prop. 11.2) ~ = jL(1)(x) 1jD(f (x); o~) k Q ~ L (zi xi ) i (x) n X X i=1 D(f (x); o~) + k Q j=1 j j=j ! i i=1

~ + (L(1)(x)) max

j j=n

1 (n+1)

(F ) !1

~ L(kz

xkn+1 `1 )(x)

n (n+1)

3 + 2n! 8(n

~ (L(kz xkn+1 1 `1 ))(x) f ; ~ (n + 1) L(1)(x)

1 1)!(n + 1)

1 (n+1)

:

Inequality (11.66) is proved. Using Theorem 11.40 and working similarly as in the proof of Theorem 11.42 we obtain the very important Theorem 11.43. Let Q Rk , k the fuzzy positive linear operator

1 be a compact convex subset. Consider

n L : CF (Q) ! C(Q);

n 2 N;

with the property (r)

(Lf )

~ (r) ); = L(f

n ~ is a positive linear respectively, for all r 2 [0; 1], 8f 2 CF (Q). Here L n operator from C(Q) into itself. Consider f 2 CF (Q). Then

184

11. Higher Order Fuzzy Korovkin Theory Using Inequalities

1) D((Lf )(x); f (x))

+

n X j=1

+

(

X

j j=j

~ jL(1)(x) 1jD(f (x); o~) k ) ~ Q (zi xi ) i (x) L i=1 D(f (x); o~) k Q ! i i=1 n

~ (L(kz

(n+1) 1 xkn+1 1 `1 )(x)) (n+1) ~ (L(1))(x) + n! (n + 1) 1 (F ) ~ (n+1) ; max ! 1 f ; ((L(kz (11.68) xkn+1 `1 ))(x))

j j=n

8x 2 Q: And 2) ~ kL1

D (Lf; f )

+

1k1 D (f; o~) k " ~ Q (zi L n X X i=1 j=1 j j=j

+

~ (L(kz

xi ) k Q

max

j j=n

(x)

1

D (f ; o~)

i!

#

i=1 n

(n+1) xkn+1 `1 ))(x) 1

n!

(F ) !1

i

~ f ; (L(kz

1

~ (n+1) + (L(1)) 1

n+1 xkn+1 `1 ))(x) 1

:

1 (n + 1)

1

(11.69)

Next we give a fuzzy multivariate Korovkin type result. Theorem 11.44. Let Q Rk , k 1 be a compact convex subset. Consider the sequence of fuzzy positive linear operators n LN : CF (Q) ! C(Q);

n

1; 8N 2 N

with the property (r)

(LN f )

~ N (f (r) ); =L

n ~ N is a sequence of respectively, for all r 2 [0; 1], 8f 2 CF (Q). Here L ~ N (1)k positive linear operators from C(Q) into itself, 8N 2 N. Assume kL u ~ , > 0, 8N 2 N, and LN (1) ! 1,

~ N (kz k(L

xkn+1 ))(x)k1 ! 0; as N ! 1:

n Then D (LN f; f ) ! 0, as N ! 1, 8f 2 CF (Q) at higher rate. I.e. LN ! I, as N ! 1, fuzzy and uniformly.

11.4 Lp -estimates, p

185

1

Proof. We use (11.69) and the following. By Hölder’s inequality and Riesz Representation Theorem we obtain that ~ N (kz (L

xkj`1 ))(x)

j n+1

1 1

~ N (kz (L

xkn+1 `1 ))(x)

j (n+1)

1

for j = 1; : : : ; n. Therefore we get xkj`1 ))(x)

~ N (kz (L as N ! 1. Notice that xkj`1 ))(x)

~ N (kz (L

=

X

j j=j

1

j! k Q

! 0;

1 k Y

~N L

i=1

i!

i=1

j

n;

jzi

xi j

i

!

(x):

Hence, since in the last equality all parts are nonnegative, we have j! k Q

~N L

k Y

i=1

i!

i=1

jzi

xi j

i

~ N (kz L

(x)

xkj`1 ) (x):

Consequently we …nd that

~N L

k Y

(zi

xi )

i

i=1

(x)

i=1

1

Thus ~N L

k Y

(zi

xi )

i=1

as N ! 1, for all

k Q

j!

i

i!

~ N (kz (L

(x) 1

xkj`1 ))(x)

1

:

! 0;

: j j = j, j = 1; : : : ; n. The claim is now established.

11.4 Lp -estimates, p

1

From [24] we have Theorem 11.45. Let K be a convex and compact subset of the real normed vector space (V; k k). Let L be a fuzzy positive linear operator from CF (K) ~ from into itself with the property that there exists positive linear operator L (r) (r) ~ C(K) into itself with (Lf ) = L(f ), respectively for all r 2 [0; 1], 8f 2 CF (K). Then ~ D(L(f )(x); f (x)) kL(1)(x) 1jjD(f (x); o~) (F ) ~ ~ + ! 1 f; ((L(k xk2 ))(x))1=2 min (L(1)(x) q ~ ~ + L(1)(x)); (L(1)(x) + 1) ; 8x 2 K:

(11.70)

186

11. Higher Order Fuzzy Korovkin Theory Using Inequalities

Furthermore we get ~ D (f; o~)kL1 1k1 q (F ) ~ ~ ~ ~ + min kL(1) + L(1)k 1 ; kL(1) + 1k1 ! 1 (f; k(L(k

D (Lf; f )

(11.71) xk2 ))(x)k1=2 1 );

~ and k k1 stands for the sup-norm over K. In particular, if L(1) = 1 then (11.71) reduces to (F )

D (Lf; f )

2! 1

~ f; k(L(k

xk2 ))(x)k1=2 1 :

(11.72)

We give Theorem 11.46. All as in the assumptions of Theorem 11.45, plus (K; A; ) is a Borel measure space with (K) < 1, p 1. Then Z

1=p

Dp (L(f )(x); f (x))d (x)

K

~ kL1

Z

1=p

Dp (f (x); o~)d (x) K q ~ ~ ~ + min kL(1) + L(1)k 1 ; kL(1) + 1k1 1k1

(F )

!1

~ f; k(L(k

Proof. Let f; g 2 CF (K) and let xn x0 2 K. We have D(f (xn ); g(xn ))

1=p xk2 ))(x)k1=2 : 1 ( (K))

(11.73)

kk

! x0 , n ! 1, where fxn gn2N ,

D(f (xn ); f (x0 )) + D(f (x0 ); g(x0 )) + D(g(x0 ); g(xn ))

and D(f (x0 ); g(x0 ))

D(f (x0 ); f (xn ) + D(f (xn ); g(xn )) + D(g(xn ); g(x0 )):

Letting n ! +1, from the continuity of f and g we …nd lim D(f (xn ); g(xn )) = D(f (x0 ); g(x0 )):

n!1

Therefore the function F (x) = D(f (x); g(x)), x 2 K is a continuous real valued function. Thus D(L(f )(x); f (x)) is continuous, hence Borel measurable. Finally using (11.70) and by integrating we obtain (11.73). From now on the measure of integration will be the Lebesgue measure , and k kp , p 1 will be the Lp -norm. We present

11.4 Lp -estimates, p

187

1

~ x) > 0, 8x 2 [a; b]. All the rest as in TheoTheorem 11.47. Assume L(1; rem 11.31. Then ~ kD((Lf )(x); f (x))kp kL1 1k1 kD(f (x); o~)kp n k X ~ kL((t x) ; x)k1 + kD(f (k) (x); o~)kp k!

(11.74)

k=1

+ (b

~ (L(jt

1

~ (n+1) n 2 1 (kL(1)k 1) + + 8 2 (n + 1) n!

a)1=p

n+1

xj

; x))

n (n+1)

1

(F ) !1

f

(n)

;

~ L(jt

1 (n+1)

xjn+1 ; x) ~ x) L(1;

1

!

n 2 N: We also have Theorem 11.48. Assume all as in Theorem 11.32. Then ~ kD((Lf )(x); f (x))kp kL1 1k1 kD(f (x); o~)kp n k X k(L(( ~ x) ))(x)k1 kD(f (k) (x); o~)kp + k! k=1

+

~ k(L(j

(F ) !1

n

(n+1) 1 xjn+1 )(x))k1 1 ~ (n+1) + (L(1)) n! (n + 1) 1 (n+1)

~ f (n) ; k(L(j

xjn+1 )(x))k1

(b

1

a)1=p : (11.75)

Furthermore we list the following multivariate fuzzy Lp results, p

1.

Theorem 11.49. Assume all as in Theorem 11.41. Then kD((Lf )(x); f (x))kp n X j=1

(

X

j j=j

~ + k(L(kz We continue with

~ L

(11.76) k Q

(zi

xi )

i

i=1 k Q

(x)

1

i!

kD(f (x); o~)kp )

)

i=1

xk`1 ))(x)k1

n (1

+ kxk`1 ) (1 + kxk`1 )

w(F ) ( (Q))1=p : 1

;

188

11. Higher Order Fuzzy Korovkin Theory Using Inequalities

Theorem 11.50. Assume all as in Theorem 11.42. Then ~ kL(1) 1k1 kD(f (x); o~)kp k # ~ Q (zi xi ) i (x) L 1 i=1 kD(f (x); o~)k k Q ! i

kD((Lf )(x); f (x))kp +

" n X X

j=1 j j=j

(11.77)

i=1

~ + (kL(1)k 1)

1 (n+1)

(F )

max ! 1

~ (L(kz

f ;

j j=n

xkn+1 `1 )(x)) ~ (L(kz

1 (n + 1)

3 + 2n! 8(n

n (n+1)

1

1 (n+1)

xkn+1 `1 ))(x) ~ L(1)(x)

1

1 1)!(n + 1) ! (Q)1=p :

We …nish the main results with Theorem 11.51. Assume all as in Theorem 11.43. Then ~ kL(1) 1k1 kD(f (x); o~)kp (11.78) k ) ~ Q (zi xi ) i (x) L 1 i=1 kD(f (x); o~)kp k Q ! i

kD((Lf )(x); f (x))kp +

n X j=1

+

(

X

j j=j

i=1

n

~ k(L(kz

(n+1) xkn+1 `1 )(x))k1

n!

max

j j=n

(F ) !1

1

~ (n+1) + (L(1))

~ f ; k((L(kz

1 (n + 1) 1 (n+1)

xkn+1 `1 ))(x))k1

1

(Q)1=p :

11.5 Applications 1 ([0; 1]), and the real Bernstein operators Let f 2 CF

(BN (g))(x) :=

N X

k=0

g

k N

N k x (1 k

x)N

k

; 8x 2 [0; 1]; 8g 2 C 1 ([0; 1]):

We have BN 1 = 1, (BN (id))(x) = x, also (BN ( Furthermore we get (BN ((

x)2 ))(x) =

x(1 x) N

x))(x) = 0, 8x 2 [0; 1]. 1 ; 4N

11.5 Applications

189

with equality at x = 12 . By (11.11) and (11.17) we obtain ! r r x(1 x) 3 x(1 x) (F ) 0 (F ) D(BN (f )(x); f (x)) !1 f; 2 N N 3 1 ) p ! (F ; (11.79) f 0; p 1 4 N 2 N 1 8N 2 N; 8x 2 [0; 1]; 8f 2 CF ([0; 1]): (F )

1 Clearly then lim D (BN (f ); f ) = 0, fuzzy and uniformly, 8f 2 CF ([0; 1]), N !+1

at higher speed than (11.13). 2) Inequality (11.68) for n = 1 becomes ~ jL(1)(x)

D((Lf )(x); f (x))

1jD(f (x); o~) (11.80) ! ~ i xi ))(x)jD @f (x); o~ j(L(z + @xi i=1 q q 1 ~ ~ xk2`1 ))(x)) (L(1))(x) + + ((L(kz 2 q @f (F ) ~ xk2`1 ))(x)) ; 8x 2 Q: max ! 1 ; ((L(kz @xi i=f1;:::;kg k X

~ ~ i If L(1)(x) = 1, (L(z (11.68) reduces to D((Lf )(x); f (x))

xi ))(x) = 0, 8x 2 Q, all i = 1; : : : ; k, n = 1, then 3 2

q

~ ((L(kz

xk2`1 ))(x)) q @f (F ) ~ max ! 1 ; ((L(kz @xi i2f1;:::;kg

(11.81) xk2`1 ))(x)) ;

8x 2 Q: Let g 2 C([0; 1])2 , the two-dimensional Bernstein polynomials of g are de…ned by (Bm;n (g))(t1 ; t2 ) :=

m X n X

k=0 `=0

g

k ` ; m n

m k

n k t (1 ` 1

t1 )m

k ` t2 (1

t2 )n

`

;

(11.82) for all t := (t1 ; t2 ) 2 [0; 1]2 , all (m; n) 2 N2 . It is known that Bm;n (g) ! g uniformly on [0; 1]2 . Clearly (Bm;n (1))(t1 t2 ) = 1, 8(t1 ; t2 ) 2 [0; 1]2 , 8(m; n) 2 N2 . Using Schwarz’s inequality we get ! r r q t1 (1 t1 ) t2 (1 t2 ) 2 ((Bm;n (k tk`1 ))(t)) + m n 1 1 1 p +p ; 2 m n

(11.83)

190

11. Higher Order Fuzzy Korovkin Theory Using Inequalities

8(m; n) 2 N2 ; 8t 2 [0; 1]2 : We have easily that (Bm;n (zi

ti ))(t1 ; t2 ) = 0;

i = 1; 2:

(11.84)

Next we de…ne the fuzzy two-dimensional Bernstein operators as follows (F )

(Bm;n (f ))(t1 ; t2 ) :=

m n X X

f

k=1 `=0

m k

k ` ; m n

n k t (1 ` 1

t1 )m

k ` t2 (1

t2 )n

`

;

(11.85)

8(t1 ; t2 ) 2 [0; 1]2 ; 8(m; n) 2 N2 ; 8f 2 CF ([0; 1]2 ): We observe as valid the following (F )

(r)

(Bm;n (f ))

= Bm;n (f

(r)

);

(11.86)

respectively, for all r 2 [0; 1], 8f 2 CF ([0; 1]2 ). Finally, by (11.81) and (11.83) we derive that (F )

D (Bm;n (f ))(t1 ; t2 ); f (t1 ; t2 ) ! r r t1 (1 t1 ) t2 (1 t2 ) 3 + 2 m n ( ) r r @f t1 (1 t1 ) t2 (1 t2 ) (F ) + max ! 1 ; @xi m n i2f1;2g ( 1 @f 1 1 3 1 1 (F ) p +p p +p max ! 1 ; ; 4 @x1 2 m m n n ) @f 1 1 1 (F ) p +p ; (11.87) !1 ; @x2 2 m n 1 8(m; n) 2 N2 ; 8(t1 ; t2 ) 2 [0; 1]2 ; 8f 2 CF ([0; 1]2 ):

Clearly then

(F )

lim D (Bm;n (f ); f ) = 0, fuzzy and uniformly,

m;n!1 2

1 8f 2 CF ([0; 1] ), at a higher rate. One can give many similar other applications of the produced theorems.

12 FUZZY WAVELET LIKE OPERATORS

The basic wavelet type operators Ak , Bk , Ck , Dk , k 2 Z were studied extensively in the real case, e.g., see [9]. Here they are extended to the fuzzy setting and are de…ned similarly via a real valued scaling function. Their pointwise and uniform convergence with rates to the fuzzy unit operator I is presented. The produced Jackson type inequalities involve the fuzzy …rst modulus of continuity and usually are proved to be sharp, in fact attained. Furthermore all fuzzy wavelet like operators Ak , Bk , Ck , Dk preserve monotonicity in the fuzzy sense. Here we do not suppose any kind of orthogonality condition on the scaling function ', and the operators act on fuzzy valued continuous functions over R. This chapter follows [14].

12.1 Background We use the following De…nition 12.1 ([11]). Let f : R ! RF be a fuzzy real number valued function. We de…ne the (…rst) fuzzy modulus of continuity of f by (F )

! 1 (f; ) :=

sup

D(f (x); f (y));

> 0:

x;y2R jx yj

G.A. Anastassiou: Fuzzy Mathematics: Approximation The., STUDFUZZ 251, pp. 191–207. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

192

12. Fuzzy Wavelet Like Operators

Denote by C(R; RF ) the space of fuzzy continuous functions and by Cb (R; RF ) the space of bounded fuzzy continuous functions on R with respect to the metric D: De…nition 12.2 ([11]). Let f : R ! RF . We call f a uniformly continuous fuzzy real number valued function, i¤ for any " > 0 there exists > 0: whenever jx yj ; x; y 2 R, implies that D(f (x); f (y)) ". We denote U it as f 2 CF (R). (F )

U Proposition 12.3 ([11]). Let f 2 CF (R). Then ! 1 (f; ) < +1, any > 0.

Proposition 12.4 ([11]). It holds (F )

(i) ! 1 (f; ) is nonnegative and nondecreasing in RF . (ii) lim

(F )

#0

> 0, any f : R !

(F )

U ! 1 (f; ) = ! 1 (f; 0) = 0, i¤ f 2 CF (R).

(F )

(iii) ! 1 (f; RF .

1

+

(F )

(iv) ! 1 (f; n ) (F )

2)

(F )

! 1 (f; (F )

n! 1 (f; ),

1)

(F )

+ ! 1 (f;

2 ),

1; 2

> 0, any f : R !

> 0, n 2 N, any f : R ! RF .

(F )

(F )

(v) ! 1 (f; ) d e! 1 (f; ) ( + 1)! 1 (f; ), > 0, d e is the ceiling of the number, any f : R ! RF . (F )

(vi) ! 1 (f

g; )

(F )

(F )

! 1 (f; ) + ! 1 (g; ),

> 0, where

> 0, any f; g : R ! RF .

(F )

U (R). (vii) ! 1 (f; ) is continuous on R+ , for f 2 CF

12.2 Results We present the …rst main result. Theorem 12.5. Let f 2 C(R; RF ) and the scaling function '(x) a real valued bounded function with supp '(x) [ a; a], 0 < a < +1, '(x) 0, 1 P such that '(x j) 1 on R. For k 2 Z, x 2 R put j= 1

(Bk f )(x) :=

1 X

f

j= 1

j 2k

'(2k x

j);

(12.1)

which is a fuzzy wavelet like operator. Then D(Bk f )(x); f (x))

(F )

!1

f;

a ; 2k

(12.2)

12.2 Results

193

and D (Bk f; f )

(F )

!1

f;

a ; 2k

(12.3)

(F )

U all x 2 R, and k 2 Z. If f 2 CF (R), then as k ! +1 we get ! 1 0 and lim Bk f = f , pointwise and uniformly with rates.

f; 2ak !

k!+1

Proof. Notice that X

(Bk f )(x) =

f

j 2k x j2[a;a]

j 2k

'(2k x

j):

We would like to estimate

D((Bk f )(x); f (x))

0

B = DB @

0

B = DB @

f

j 2k

'(2k x

j); f (x)

X

f

j 2k

'(2k x

j);

'(2k x

j);

j 2k x j2[ a;a]

j 2k x j2[ a;a]

f (x)

1 X

'(2k x

j= 1

0

B = DB @

X

X

f

j 2k x j2[ a;a]

X

f (x)

X

'(2k x

1

j)A j 2k

'(2k x

j 2k x j2[ a;a]

j 2k x j2[ a;a]

j)D f

1

C j)C A

j 2k

; f (x)

1

C 1C A

194

12. Fuzzy Wavelet Like Operators

X

(F )

'(2k x

j)! 1

f;

j 2k x j2[ a;a]

=

B B @

X

C (F ) a f; k j)C A !1 2

'(2k x

j 2k x j2[ a;a] (F )

1 !1

f;

x

h a ai j 2 ; 2k 2k 2k 1

here x 0

j 2k

a : 2k

It follows the next important result. Theorem 12.6. Let f 2 Cb (R; RF ) and the scaling function '(x) a real valued function with supp '(x) [ a; a], 0 < a < +1, ' is continuous on 1 R1 P [ a; a], '(x) 0, such that '(x j) = 1 on R ( then 1 '(x)dx = j= 1

1). De…ne

2k=2 '(2k t j); for k; j 2 Z; Z j+a 2k f (t) 'kj (t)dt; (F R)

'kj (t) := hf; 'kj i :=

t 2 R;

(12.4) (12.5)

j a 2k

and set (Ak f )(x) :=

1 X

hf; 'kj i

'kj (x);

j= 1

x 2 R;

(12.6)

which a fuzzy wavelet like operator. Then (F )

D((Ak f )(x); f (x))

!1

f;

and

a ; 2k 1 (F )

D ((Ak f ); f )

!1

f;

x 2 R; k 2 Z; a 2k 1

:

(12.7)

(12.8)

U (R) and bounded, then again we get Ak ! unit operator I with If f 2 CF rates as k ! +1.

Proof. Since ' is compactly supported we have 'kj (t) 6= 0 i¤

a

2k t

j

a; i¤

j

a 2k

t

j+a : 2k

Also it holds that (Ak f )(x) :=

X

j 2k x j2[ a;a]

hf; 'kj i

'kj (x);

k 2 Z:

12.2 Results

195

We would like to estimate 0

B = DB @

D((Ak f )(x); f (x))

0

B = DB @

X

hf; 'kj i

X

hf; 'kj i

j 2k x j2[ a;a]

j 2k x j2[ a;a]

X

f (x)

B = DB @

X

j 2k x j2[ a;a]

C 'kj (x); f (x)C A 'kj (x);

'(2k x

j 2k x j2[ a;a]

0

1

hf; 'kj i

1

C j)C A 'kj (x); 1

X

f (x)

X

'kj (x)D(hf; 'kj i; 2

2

k=2

j 2k x j2[ a;a]

j 2k x j2[ a;a]

C 'kj (x)C A

k=2

f (x)) =: ( ):

Next we estimate separately k=2

D(hf; 'kj i; 2

= D (F R) k=2

= D 2

f (x)) Z

j+a 2k

f (t)

j a 2k

(F R)

Z

j+a 2k j a 2k

k=2

'kj (t)dt; 2 f (t)

k

'(2 t

!

f (x)

j)dt; 2

k=2

!

f (x)

(in Fuzzy-Riemann integral we can have linear change of variables) Z j+a du u k=2 '(u j) k ; 2 k=2 f (x) = D 2 (F R) f k 2 2 j a Z j+a u = D 2 k=2 (F R) f k '(u j)du; 2 k=2 f (x) 2 j a Z j+a u k=2 = 2 D (F R) f k '(u j)du; f (x) 1 =: ( ): 2 j a

196

12. Fuzzy Wavelet Like Operators

Notice that have

R1

'(u

1

j)du = 1, j 2 Z and by compact support of ' we

Z

j+a

'(u

j)du = 1:

j a

Hence

(

k=2

)=2 Z

f (x)

D (F R)

j+a

k=2

(F R)

Z

j+a

u 2k

f

j a

'(u

D (F R)

Z

j+a

f

j a

j+a

f (x)

'(u

j)du;

j)du

j a

= 2

Z

'(u

u 2k

'(u

j)du;

j)du

j a

(Lemma 1, [11]) and 2

k=2

'(u

= 2

k=2

Z

k=2

D f

u 2k

'(u

j)) du j+a

'(u

j)D f

'(u

j)! 1

j a j+a

2

j+a

j a

(Lemma 12.7 next) f (x)

Z

Z

(F )

u ; f (x) du 2k f;

j a

u 2k

x

du

a u a x notice that 2k 1 2k 2k 1 Z j+a a (F ) 2 k=2 '(u j)du ! 1 f; k 1 2 j a a k=2 (F ) 2 !1 f; k 1 : 2

I.e. we prove that

D(hf; 'kj i; 2

k=2

f (x))

2

k=2

(F )

!1

f;

a : 2k 1

j);

12.2 Results

197

Going back to ( ) we get ( )

X

'kj (x)2

j 2k x j2[ a;a]

0

B = B @ 0

= @

X

j 2k x j2[ a;a] 1 X

'(2k x

j= 1

'(2k x 1

k=2

(F )

!1

f;

a 2k 1

1

C (F ) a f; k 1 j)C A !1 2

(F ) j)A ! 1 f;

a 2k 1

(F )

= 1 !1

f;

a ; x 2 R: 2k 1

Here we use Lemma 12.7. Let f : R ! RF fuzzy continuous and bounded, i.e. 9M1 > 0 : D(f (x); o~) M1 , 8x 2 R. Let also g : J R ! R+ continuous and bounded, i.e. 9M2 > 0 : g(x) M2 , 8x 2 J, where J is an interval. Then f (x) g(x) is fuzzy continuous function 8x 2 J. Proof. Let xn ; x0 2 J, n = 1; 2; : : :, such that xn ! x0 . Thus D(f (xn ); f (x0 )) ! 0, as n ! +1 and jg(xn ) g(x0 )j ! 0. We need to establish that n

:= D(f (xn )

g(xn ); f (x0 )

g(x0 )) ! 0;

as n ! +1. We have 2

= D(2 (f (xn ) g(xn )); 2 (f (x0 ) g(x0 )) (notice for u 2 RF that u u = 2 u) D(f (xn ) g(xn ) f (xn ) g(xn ) f (x0 ) g(xn ) f (xn ) g(x0 ); f (x0 ) g(xn ) f (xn ) g(x0 ) f (x0 ) g(x0 ) f (x0 ) g(x0 )) D(f (xn ) g(xn ); f (x0 ) g(xn )) + D(f (xn ) g(xn ); f (xn ) g(x0 )) + D(f (x0 ) g(xn ); f (x0 ) g(x0 )) + D(f (xn ) g(x0 ); f (x0 ) g(x0 )) (by Lemma 2.2, [31]) g(xn )D(f (xn ); f (x0 )) +jg(xn ) g(x0 )jD(f (xn ); o~) + jg(xn ) g(x0 )jD(f (x0 ); o~) + g(x0 )D(f (xn ); f (x0 )) 2M2 D(f (xn ); f (x0 )) + 2M1 jg(xn ) g(x0 )j ! 0; as n ! +1: n

We proceed with the following related result.

198

12. Fuzzy Wavelet Like Operators

Theorem 12.8. All assumptions here are as in Theorem 12.5. De…ne for k 2 Z, x 2 R the fuzzy wavelet like operator 1 X

(Ck f )(x) :=

2k

(F R)

Z

k

2

f

0

j= 1

!

j t+ k 2

'(2k x

dt

j): (12.9)

Then (F )

D((Ck f )(x); f (x))

!1

f;

a+1 2k

;

(12.10)

and (F )

D ((Ck f ); f )

!1

f;

a+1 2k

;

all k 2 Z; x 2 R:

(12.11)

U (R) then as k ! +1 we get Ck ! I with rates. When f 2 CF

Proof. We need to estimate D((Ck f )(x); f (x)) 0 B = DB @

X

k

2

2

k

(F R)

2k

(F R)

2k

D

X

2

f

j t+ k 2

k

2

!

k

(f (x) Z

2

k

f

0

Z

2

k

(f (x)

0

2k

Z

(F R)

j 2k x j2[ a;a]

(F R)

Z

dt

!

dt

!

j 2k x j2[ a;a]

'(2k x

1)dt

!

!

'(2k x

j 2k

dt;

!

k

Z

0

C j)C A

j t+ k 2

2

dt

j)

k

f

t+

j);

1

'(2k x

'(2 x

j)D (F R)

'(2k x

1)dt

0

j 2k x j2[ a;a]

2

f

j t+ k 2

0

j 2k x j2[ a;a]

X

k

k

1

X

X

2

0

j 2k x j2[ a;a]

'(2k x j); f (x) 0 B = DB @

(F R)

Z

j);

12.2 Results

(F R)

Z

!

k

2

f (x)dt

0

(by Lemma 1, [11])

199

X

2k

'(2k x

j)

j 2k x j2[ a;a]

Z

2

k

D f

t+

0

(here 0

( )

t

2k

1 2k

j 2k

and x

X

j 2k

a , 2k

thus t +

(F )

'(2k x

; f (x)) dt =: ( )

j)! 1

f;

j 2k x j2[ a;a]

j 2k

a+1 ). 2k

x

a+1 2k

2

k

(F )

= !1

Hence

f;

a+1 2k

:

Next we give the corresponding result for the last fuzzy wavelet type operator we are dealing with here. Theorem 12.9. All assumptions here are as in Theorem 12.5. De…ne for k 2 Z, x 2 R the fuzzy wavelet like operator (Dk f )(x) :=

1 X

kj (f )

'(2k x

j);

(12.12)

j= 1

where

kj (f )

:=

n X

wr~

j r~ + k k 2 2 n

f

r~=0

; n 2 N; wr~

0;

n X

wr~ = 1: (12.13)

r~=0

Then D((Dk f )(x); f (x))

(F )

!1

f;

a+1 2k

;

(12.14)

and D (Dk f; f )

(F )

!1

f;

a+1 2k

;

all k 2 Z; x 2 R:

U When f 2 CF (R) then as k ! +1 we get Dk ! I with rates.

(12.15)

200

12. Fuzzy Wavelet Like Operators

Proof. We need to upper bound

D((Dk f )(x); f (x)) 0 B = DB @

X

j 2k x j2[ a;a]

n X

wr~

f (x)

'(2k x

X

'(2k x

j)D

j 2k x j2[ a;a]

j 2k x j2[ a;a] n X

(wr~

f

r~=0

X

'(2k x

j);

1

C j)C A

n X

wr~

f

r~=0

!

!

j r~ + k 2k 2 n

j r~ + k 2k 2 n

;

f (x))

r~=0

X

'(2k x

j 2k x j2[ a;a]

j)

n X

wr~D f

r~=0

j r~ + k 2k 2 n

a+1 r~ j + k x k 2 2 n 2k n X a+1 (F ) '(2k x j) wr~! 1 f; k 2

; f (x)

notice that X

j 2k x j2[ a;a]

(F )

= !1

r~=0

f;

a+1 2k

:

Next we prove optimality for three of the above main results.

Proposition 12.10. Inequality (12.2) is attained, that is sharp.

Proof. Take '(x) = [ 12 ; 21 ) (x), the characteristic function on Fix u 2 RF and take f (x) = q(x) u, where 8 < 0; 1; q(x) := : k+1 2 x + 1;

x x 2

2 0;

k 1

k 1

< x < 0;

1 1 2; 2

.

12.2 Results

k 2 Z …xed, x 2 R. Clearly q(x) (Bk f )(x)

1 X

=

0. We observe that j 2k

q

j= 1

0

= @ 0

= @

1 X

q

'(2k x

u

j 2k

j= 1

1 X

201

'(2k x 1

j)A

'(2k x

j=0

j) 1

j)A

u

u:

Hence

k 1

D((Bk f )( 2

); f ( 2

00 1 X 1 ) = D @@ '

k

1 2

j=0

1

j A

1

u; o~A = D(u; o~):

Furthermore we see that (F )

! 1 (f; 2 =

)=

sup

D(f (x); f (y))

x;y2R jx yj 2 k

sup x;y2R jx yj 2 k

=

k 1

D(q(x)

1

u; q(y)

u)

1

0 (by Lemma 2.2, [31]) B @

sup x;y2R jx yj 2 k

jq(x) 1

1 D(u; o~):

1

C q(y)jA D(u; o~)

I.e. we got that (F )

! 1 (f; 2

k 1

)

D(u; o~):

So that by (12.2) and the above we …nd D((Bk f )( 2

k 1

); f ( 2

k 1

(F )

)) = ! 1 (f; 2

k 1

);

proving the sharpness of (12.2). Proposition 12.11. Inequalities (12.10) and (12.14) are attained, i.e. they are sharp.

202

12. Fuzzy Wavelet Like Operators

Proof. (I) Consider as optimal elements ', q, u, and f , exactly as in the proof of Proposition 12.10. Here a = 12 . We observe that (F )

!1

f; =

a+1 2k

(F )

= !1

sup

3 2k+1

f;

D(q(x)

x;y jx yj

=

sup

D(f (x); f (y))

x;y jx yj

3 2k+1

u; q(y)

u)

3 2k+1

0 (by Lemma 2.2, [31]) B @ 0

That is

B =@

x;y jx yj

1 2k+1

(F )

!1

f;

1

a+1 2k

Call kj (f )

k

:= 2

3 2k+1

(F R)

D(u; o~):

Z

2

k

f

j 2k

t+

0

We obtain k(

1) (f )

2k

=

(F R)

2k

=

Z

Z

2

k

q t

0

2

k

q t

0

k

=

2

k

= I.e.

2

Z

Z

0

q(t)dt 1 2k

1 2k !

0

q(t)dt 1 2k+1

2

dt:

1 2k !

u dt

dt

u

u

!

u=

1 u: 4 ~, and kj (f ) = o~, all j 2) (f ) = o k( 1) (f )

Moreover k( j 0. Hence

C q(y)jA D(u; o~)

jq(x)

x;y jx yj

C q(y)jA D(u; o~) = 1 D(u; o~):

jq(x)

sup

sup

1

1 4

u:

=

+1 X

1 (Ck f )(x) = 4 '(2k x + 1) + '(2k x 4 j=0

2, and 3

j)5

kj (f )

u:

= u, all

12.2 Results

203

We easily see then that 1 2k+1

(Ck f )

= u;

1 2k+1

also f

= o~:

Therefore 1 2k+1

D (Ck f )

1 2k+1

;f

= D(u; o~):

From the above and (12.10) we conclude that 1

D (Ck f )

1

;f

2k+1

(F )

= !1

2k+1

a+1 2k

f;

;

k 2 Z;

proving the sharpness of (12.10). (II) The sharpness of (12.14) is treated similarly to (I). Notice that 0, and kj (f ) = o~, all j 2. We observe that kj (f ) = u, all j 1

' 2k

( 1)

2k+1

='

1 2

= 0:

Furthermore D (Dk f ) 0 = D@

1 2k+1 1 X

1 2k+1

;f

' 2k

kj (f )

j= 1

00 1 X = D @@ 1' j=0

1

1 2

So that by (12.14) and the above D (Dk f )

1 2k+1

j A 1 2k+1

;f

1

1

j ; o~A

2k+1 1

u; o~A = D(1 (F )

= !1

f;

u; o~) = D(u; o~):

a+1 2k

;

proving sharpness of (12.14). Remark 12.12. We notice that (Lk f )(x) = L0 (f (2

k

))(2k x);

all x 2 R; k 2 Z;

where Lk = Bk , Ak , Ck , Dk . Clearly Lk ’s are linear over R operators. In the following we present a monotonicity result for the fuzzy wavelet like operators Bk and Dk . For that we need

204

12. Fuzzy Wavelet Like Operators

De…nition 12.13. Let f : R ! RF , then f is called a nondecreasing function i¤ whenever x1 x2 , x1 ; x2 2 R, we have that f (x1 ) f (x2 ), i.e. (r) (r) (r) (r) (f (x1 )) (f (x2 )) and (f (x1 ))+ (f (x2 ))+ , 8r 2 [0; 1]. Theorem 12.14. Let f 2 C(R; RF ), and the scaling function '(x) a real valued bounded function with supp ' [ a; a], 0 < a < +1, such that (i)

1 P

'(x

j)

j= 1

1 on R,

(ii) there exists a b 2 R such that ' is nondecreasing for x nonincreasing for x b,

b and ' is

(the above imply ' 0). Let f (x) be nondecreasing fuzzy function, then (Bk f )(x), (Dk f )(x) are nondecreasing fuzzy valued functions for any k 2 Z. Remark 12.15. We give two examples of '’s as in Theorem 12.14. (i) ( 1 1; x < 12 ; 2 '(x) = 0; elsewhere. (ii)

8 < x + 1; 1 x; '(x) = : 0;

1 x 0; 0 < x 1; elsewhere.

Proof of Theorem 12.14. Let xn ; x 2 R such that xn ! x, as n ! +1, then D(f (xn ); f (x)) ! 0 by fuzzy continuity of f . But we have (r)

D(f (xn ); f (x)) = sup maxfj(f (xn ))

(r)

(f (x))

r2[0;1] (r)

(r)

j; j(f (xn ))+

(r)

(r)

(f (x))+ jg:

(f (x)) j ! 0, all 0 r 1, as n ! +1, respectively. That is, j(f (xn )) (r) Therefore (f ) 2 C(R; R), all 0 r 1, i.e. real valued continuous functions on R. Since f is fuzzy nondecreasing by De…nition 12.13 we get (r) that (f ) are nondecreasing, 8r 2 [0; 1], respectively. Then by Theorem 6.3, p. 156, [9], see also [36], we get that the corresponding real wavelet (r) type operators map to the functions (Bk (f ) )(x) that are nondecreasing on R for all r 2 [0; 1], any k 2 Z. Also by Lemma 8.2, p. 186, [9], see also [7], we get that the corresponding real wavelet type operators map to the (r) functions (Dk (f ) )(x) that are nondecreasing on R for all r 2 [0; 1], any k 2 Z. We notice for any r 2 [0; 1] that [(Bk f )(x)]r =

+1 X

j= 1

f

j 2k

r

'(2k x

j):

12.2 Results

205

That is (r)

((Bk f )(x))

= 4 =

h

"

+1 X

=

j= 1

k

2

(r) ; ((Bk f )(x))+

'(2 x +1 X

j) f

j= 1 (r)

(Bk (f )

(r)

j 2k

'(2k x

j);

(r)

x2 we get (f ) (r)

(Bk (f )

+1 X

(r)

(x1 )

(f ) (r)

)(x1 )

f

j= 1

i (r) )(x); (Bk (f )+ )(x) :

So whenever x1

f

(Bk (f )

j 2k

j 2k

(r)

; f

(r)

j 2k

+

3

(r)

j)5

'(2k x +

(x2 ), respectively, and

)(x2 );

8r 2 [0; 1]:

Therefore (Bk f )(x1 ) (Bk f )(x2 ), that is (Bk f ) is nondecreasing. Next we observe that ! +1 n r X X r~ j r + k [(Dk f )(x)] = wr~ f '(2k x j): k 2 2 n j= 1 r~=0

That is (r)

(r)

; ((Dk f )(x))+ " n X j r~ wr~ f k + k 2 2 n

((Dk f )(x)) =

+1 X

j= 1

'(2k x 2 +1 X =4

r~=0

j= 1

n X

wr~ f

(r) +

#

r~=0

n X

wr~ f

r~=0

(r)

j r~ + k 2k 2 n

'(2k x (r)

j r~ + k 2k 2 n

'(2k x +

i h (r) (r) = (Dk (f ) )(x); (Dk (f )+ )(x) :

So whenever x1

j r~ ; f k + k 2 2 n

j)

j= 1

+1 X

(r)

j); 3

j)5

x2 we get (r)

(Dk (f )

)(x1 )

(r)

(Dk (f )

)(x2 );

8r 2 [0; 1]:

Therefore (Dk f )(x1 ) (Dk f )(x2 ), so that (Dk f ) is nondecreasing. Finally we present the corresponding monotonicity results for the fuzzy wavelet like operators Ak , Ck .

#

206

12. Fuzzy Wavelet Like Operators

Theorem 12.16. Let f 2 Cb (R; RF ) and ' as in Theorem 12.14 which is continuous on [ a; a]. Let f (x) be nondecreasing fuzzy function, then (Ak f )(x) is a nondecreasing fuzzy valued function for any k 2 Z. (r)

Proof. Since f is fuzzy nondecreasing we get again that (f ) are nondecreasing, 8r 2 [0; 1], respectively. Then by Theorem 6.1, p. 149, [9], see also [36], we get that the corresponding real wavelet type operators map to the (r) functions (Ak (f ) )(x) that are nondecreasing on R for all r 2 [0; 1], any k 2 Z. Using Theorem 3.2 ([67]), for any r 2 [0; 1] we notice that "Z j+a # Z j+a [hf; 'kj i]r

2k

=

"Z

=

j a 2k j+a 2k

(r)

(f (t)

'kj (t))

(r)

(f (t))

j a 2k

'kj (t)dt;

2k

dt; Z

(f (t)

j a 2k j+a 2k

j a 2k

(r)

'kj (t))+ dt #

:

#

'kj (x)

(r) (f (t))+ 'kj (t)dt

We observe for any r 2 [0; 1] that +1 X

[(Ak f )(x)]r =

[hf; 'kj i]r 'kj (x):

j= 1

That is h i (r) (r) ((Ak f )(x)) ; ((Ak f )(x))+ "Z j+a Z +1 X 2k (r) = (f (t)) 'kj (dt)dt; j a 2k

j= 1

2

=4

Z

+1 X

j= 1

+1 X

j= 1

Z

j+a 2k j a 2k

j+a 2k j a 2k

(f (t))

j a 2k

!

'kj (t)dt 'kj (x); !

3

(r)

)(x); (Ak (f )+ )(x) :

(r)

So whenever x1 x2 we have that (f ) (x1 ) and (r) (r) (Ak (f ) )(x1 ) (Ak (f ) )(x2 ); Hence (Ak f )(x1 )

(r) (f (t))+ 'kj (t)dt

(r) (f (t))+ 'kj (t)dt 'kj (x)5

(r)

= (Ak (f )

(r)

j+a 2k

(r)

(f )

(x2 ), respectively,

8r 2 [0; 1]:

(Ak f )(x2 ), that is (Ak f ) is nondecreasing.

Theorem 12.17. Let f and ' as in Theorem 12.14. Let f (x) be nondecreasing fuzzy function, then (Ck f )(x) is a nondecreasing fuzzy valued function for any k 2 Z.

12.2 Results

207

Proof. By Lemma 8.2, p. 186, [9], see also [7], we get that the corresponding (r) real wavelet type operators map to the functions (Ck (f ) )(x) that are nondecreasing on R for all r 2 [0; 1], any k 2 Z. Using Theorem 3.2 ([67]), for any r 2 [0; 1] we notice that #r " Z 2 k +1 X j r k [(Ck f )(x)] = 2 (F R) f t + k dt '(2k x j) 2 0 j= 1 #r " k Z +1 2 (j+1) X k = 2 (F R) f (t)dt '(2k x j) j= 1 +1 X

=

j= 1

2

k

Z

2

(j+1)

(r)

(f )

k

(t)dt; 2

kj

Z

2

2

k

(j+1)

kj

(r) (f )+ (t)dt

#

j) k

2

Z

2k (r)

(j+1)

(r)

(f )

!

(t)dt '(2k x

kj

Z

k

2

(j+1)

!

(r)

(f )+ (t)dt '(2k x

kj

2

j= 1

(Ck (f )

k

2

2

j= 1 +1 X

k

2

'(2k x 2 +1 X = 4

=

kj

2

"

(r)

)(x); (Ck (f )+ )(x) :

j); 3

j)5

That is, for any r 2 [0; 1] we found (r)

((Ck f )(x)) So whenever x1

(r)

; ((Ck f )(x))+

(r)

x2 we have (f ) (r)

(Ck (f )

)(x1 )

respectively. Hence (Ck f )(x1 )

(r)

= (Ck (f ) (x1 ) (r)

(Ck (f )

(r)

(f )

)(x2 );

(r)

)(x); (Ck (f )+ )(x) : (x2 ) and 8r 2 [0; 1];

(Ck f )(x2 ), that is (Ck f ) is nondecreasing.

13 ESTIMATES TO DISTANCES BETWEEN FUZZY WAVELET LIKE OPERATORS

The basic fuzzy wavelet like operators Ak , Bk , Ck , Dk , k 2 Z were …rst introduced in [14], see also Chapter 12, where they were studied among others for their pointwise/uniform convergence with rates to the fuzzy unit operator I. Here we continue this study by estimating the fuzzy distances between these operators. We give the pointwise convergence with rates of these distances to zero. The related approximation is of higher order since we involve these higher order fuzzy derivatives of the engaged fuzzy continuous function f . The derived Jackson type inequalities involve the fuzzy (…rst) modulus of continuity. Some comparison inequalities are also given so we get better upper bounds to the distances we study. The de…ning of these operators scaling function ' is of compact support in [ a; a], a > 0 and is not assumed to be orthogonal. This chapter is based on [23].

13.1 Background This Chapter is motivated by [14], especially from the following result: Theorem 13.1 ([14]). Let f be a function from R into the fuzzy real numbers RF which is fuzzy continuous. Let '(x) be a real valued bounded G.A. Anastassiou: Fuzzy Mathematics: Approximation The., STUDFUZZ 251, pp. 209–236. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

210

13. Estimates to Distances Between Fuzzy Wavelet Like Operators

scaling function with supp '(x) [ a; a], a > 0, '(x) 1 P '(x j) 1 on R. For k 2 Z, x 2 R put

0, such that

j= 1

(Bk f )(x) := fuzzy sum

1 X

j 2k

f

j= 1

'(2k x

j)

( denotes fuzzy multiplication). Clearly Bk is a fuzzy wavelet type operator. Then the fuzzy distance D((Bk f )(x); f (x))

(F )

!1

f;

a ; 2k

any x 2 R, and k 2 Z, and sup D((Bk f )(x); f (x)) x2R

(F )

!1

f;

a : 2k

(F )

Here ! 1 stands for the fuzzy (…rst) modulus of continuity. If f is fuzzy uniformly continuous then lim Bk f = f uniformly with rates. k!+1

All fuzzy wavelet like operators Ak , Bk , Ck , Dk , k 2 Z are reintroduced here and we …nd upper bounds to their distances D((Bk f )(x), (Ck f )(x)), D((Dk f )(x), (Ck f )(x)), D((Bk f )(x), (Dk f )(x)), D((Ak f )(x), (Bk f )(x)), D((Ak f )(x), (Ck f )(x)), and D((Ak f )(x), (Dk f )(x)). Their proofs rely a lot on the found here upper bounds for D (Bk f )(x), f x 2ak , D (Ck f )(x), f x 2ak and D (Dk f )(x), f x 2ak , k 2 Z, x 2 R. The produced associated inequalities involve fuzzy (…rst) moduli of continuity of the engaged function and its fuzzy derivatives. For fuzzy uniformly continuous functions f and its likewise derivatives we obtain pointwise convergence with rates to zero of all the above mentioned distances among the stated sequences of fuzzy wavelet like operators. For these see Section 13.2. Our methods of proving here are based only on fuzzy theoretic concepts. We use the following De…nition 13.2 ([11]). Let f : R ! RF be a fuzzy real number valued function. We de…ne the (…rst) fuzzy modulus of continuity of f by (F )

! 1 (f; ) :=

sup

D(f (x); f (y));

> 0:

x;y2R jx yj

De…nition 13.3 ([11]). Let f : R ! RF . If D(f (x); o~) M1 , 8x 2 R, M1 > 0, we call f a bounded fuzzy real number valued function. De…nition 13.4 ([11]). Let f : R ! RF . We say that f is fuzzy continuous at a 2 R if whenever xn ! a, then D(f (xn ), f (a)) ! 0. If f is continuous

13.1 Background

211

for every a 2 R, then we call f a fuzzy continuous real number valued function. We denote it as f 2 C(R; RF ). De…nition 13.5 ([11]). Let f : R ! RF . We call f a fuzzy uniformly continuous real number valued function, i¤ for any " > 0 there exists > 0: whenever jx yj , x; y 2 R, implies that D(f (x); f (y)) ". We denote U it as f 2 CF (R). (F )

U Proposition 13.6 ([11]). Let f 2 CF (R). Then ! 1 (f; ) < +1, any > 0.

Denote f : R ! RF which is bounded and fuzzy continuous, as f 2 Cb (R; RF ). Proposition 13.7 ([11]). It holds (F )

(i) ! 1 (f; ) is nonnegative and nondecreasing in RF . (ii) lim

(F )

#0

> 0, any f : R !

(F )

U (R). ! 1 (f; ) = ! 1 (f; 0) = 0, i¤ f 2 CF

(F )

(iii) ! 1 (f; RF .

1

+

2)

(F )

(F )

! 1 (f; (F )

(iv) ! 1 (f; n )

n! 1 (f; ),

(F )

(F )

1)

+ ! 1 (f;

2 ),

1; 2

> 0, any f : R !

> 0, n 2 N, any f : R ! RF .

(F )

(F )

(v) ! 1 (f; ) d e! 1 (f; ) ( + 1)! 1 (f; ), > 0, d e is the ceiling of the number, any f : R ! RF . (F )

(vi) ! 1 (f

g; )

(F )

(F )

! 1 (f; ) + ! 1 (g; ),

> 0, where

> 0, any f; g : R ! RF .

(F )

U (R). (vii) ! 1 (f; ) is continuous on R+ , for f 2 CF

Lemma 13.8 ([11]). Let f 2 C(R; RF ), r 2 N. Then the following integrals Z sr 1 Z s r 2 Z sr 1 (F R) f (sr )dsr ; (F R) f (sr )dsr dsr 1 ; : : : ; a

(F R)

a

Z

a

s

Z

a

s1

Z

a

sr

2

a

Z

sr

1

f (sr )dsr dsr

1

ds1 ;

a

are fuzzy continuous functions in sr 1 , sr sr 2 ; : : : ; s a and all are real numbers.

2 ; : : : ; s,

respectively. Here sr

1,

Here we use a lot the following fuzzy Taylor’s formula. Theorem 13.9 ([11]). Let T := [x0 ; x0 + ] R, with > 0. We assume that f (i) : T ! RF are fuzzy di¤ erentiable for all i = 0; 1; : : : ; n 1, for any x 2 T (i.e., there exist in RF the H-di¤ erences f (i) (x + h) f (i) (x),

212

13. Estimates to Distances Between Fuzzy Wavelet Like Operators

f (i) (x) f (i) (x h), i = 0; 1; : : : ; n 1 for all small 0 < h < . Furthermore there exist f (i+1) (x) 2 RF such that the limits in D-distance exist and f (i+1) (x) = lim+ h!0

f (i) (x + h) h

f (i) (x)

= lim+

f (i) (x)

f (i) (x h

h!0

h)

;

for all i = 0; 1; : : : ; n 1). Also we assume that f (i) , i = 0; 1; : : : ; n are fuzzy continuous on T . Then for s a, s; a 2 T we obtain f (s)

= f (a)

f 0 (a)

(s a)n 1 (n 1)!

(s

f 00 (a)

a)

(s

a)2 2!

f (n

1)

(a)

Rn (a; s)

where Rn (a; s) := (F R)

Z

s

a

Z

Z

s1

a

sn

1

f (n) dsn dsn

1

ds1 :

a

Here Rn (a; s) is fuzzy continuous over T as a function of s. Note 13.10. This formula is invalid when s < a, as it is based on Theorem 3.6 of [53]. We denote by C N (R; RF ) the space of N -times continuously di¤erentiable in the fuzzy sense functions from R into RF , N 2 N. We also denote NU (R), N 2 N, the space of functions f : R ! RF , such that the by CF fuzzy derivatives exist up to order N and all f , f 0(N ) are fuzzy uniformly continuous. Finally we make use of Lemma 13.11 ([14]). Let f : R ! RF be fuzzy continuous and bounded function, i.e. 9M1 > 0 : D(f (x); o~) M1 , 8x 2 R. Let also g : from interval J R ! R+ continuous and bounded function. Then f (x) g(x) is a fuzzy continuous function on J.

13.2 Results We present the …rst main result. Theorem 13.12. Let f 2 C N (R; RF ), N 2 N and the scaling function '(x) a real valued bounded function with supp '(x) [ a; a], 0 < a < +1, 1 P '(x) 0, such that '(x j) 1 on R. For k 2 Z, x 2 R put j= 1

(Bk f )(x) :=

1 X

j= 1

f

j 2k

'(2k x

j);

(13.1)

13.2 Results

213

which is a fuzzy wavelet like operator. Then it holds a 2k

D (Bk f )(x); f x N X1 i=1

+

ai

(F )

2i(k 1) i! aN

(F )

2N (k 1) N ! =:

D(f (i) (x); o~) + ! 1

k

D(f (N ) (x); o~) + 3! 1 a ; 2k

f (i) ;

a 2k

f (N ) ;

a 2k

for any x 2 R; k 2 Z:

(13.2)

NU If f 2 CF (R) and as k ! +1 we obtain with rates that

lim D (Bk f )(x); f x

k!+1

a 2k

= 0:

Corollary 13.13 (to Theorem 13.12, for N = 1). It holds D Bk f )(x); f x

a 2k

a 2k 1

(F )

D(f 0 (x); o~) + 3! 1

f 0;

a 2k

;

for any x 2 R, k 2 Z. Corollary 13.14 (to Theorem 13.12). The following improvement of (13.2) holds a a a (F ) D (Bk f )(x); f x min 2! 1 ; f; k ; k k k 2 2 2 for any x 2 R, k 2 Z. Proof. From Theorem 13.1 we get D((Bk f )(x); f (x))

(F )

!1

f;

a : 2k

Then we observe a 2k a 2k

D (Bk f )(x); f x +D f (x); f x

D((Bk f )(x); f (x)) (F )

2! 1

f;

a : 2k

Proof of Theorem 13.12. Because ' is of compact support in [ a; a] we see that X j (Bk f )(x) = f '(2k x j): (13.3) k 2 j 2k x j2[ a;a]

That is for speci…c x 2 R, k 2 Z we have the j’s in (13.3) to ful…ll x

a 2k

j 2k

x+

a : 2k

214

13. Estimates to Distances Between Fuzzy Wavelet Like Operators

Using the fuzzy Taylor formula (Theorem 1, [11]) we get

f

j 2k

=

N 1 X

j+a 2k

a 2k

f (i) x

i=0

i

x

RN

i!

a j ; 2k 2k

x

; (13.4)

where

RN

a j ; 2k 2k

x

:= (F R)

Z

x

Z

sN

1

a 2k

x

Z

j 2k a 2k

s1

x

a 2k

f (N ) (sN )dsN dsN

ds1 : (13.5)

1

Then

f

j 2k

k

'(2 x

j)

N 1 X

=

f

(i)

x

i=0

RN

x

j+a 2k

a 2k

x

i

'(2k x

i!

a j ; 2k 2k

'(2k x

j)

j)

and X

j 2k

f

j 2k x j2[ a;a]

k

'(2 x

N 1 X

X

j) =

f (i) x

j i=0 2k x j2[ a;a]

j+a 2k

x

a 2k

i

i! X

j 2k x j2[ a;a]

'(2k x RN

x

j) a j ; 2k 2k

'(2k x

j):

That is we have

(Bk f )(x)

=

N 1 X i=0

X

f (i) x

j 2k x j2[ a;a]

X

j 2k x j2[ a;a]

RN

x

a 2k a j ; 2k 2k

j+a 2k

x

i

'(2k x

i! '(2k x

j):

j)

(13.6)

13.2 Results

215

Next we estimate a 2k

D (Bk f )(x); f x 0

X

B = DB @(Bk f )(x); N 1 X

= D

i=1

X

RN

X

j+a 2k

X

x

N X1 i=1

(F )

!1

+

X

ai

j 2k x j2[ a;a]

f (i) ;

a 2k X

2i(k

1) i!

'(2k x

'(2k x

j)D RN

j 2k x j2[ a;a]

k

'(2 x

'(2k x

j)D RN

j 2k x j2[ a;a]

i

'(2k x !

j); o~

i

i!

'(2k x

x i!

a j ; 2k 2k

x

j 2k x j2[ a;a]

+

a C C 2k A

f x

j+a 2k

a 2k

f (i) x

j 2k x j2[ a;a]

j 2k x j2[ a;a]

i=1

j)

j 2k x j2[ a;a]

X

N X1

'(2k x

1

a ; o~ 2k

j)D f (i) x

x

a j ; 2k 2k

; o~

j) D(f (i) (x); o~)+

x

a j ; 2k 2k

; o~ =: ( ):

Next we work on

D RN

a j ; 2k 2k

x

= D RN j+a 2k

N!

a j ; 2k 2k

x x

; o~ f (N ) x

N

;f

(N )

x

a 2k

a 2k j+a 2k

N!

x

N

!

j)

216

13. Estimates to Distances Between Fuzzy Wavelet Like Operators

D RN

a j ; 2k 2k

x

; f (N ) x j+a 2k

a 2k

+ D f (N ) x

j+a 2k

a 2k

x

x

N

N! !

N

!

; o~ :

N!

So that

D RN

a j ; 2k 2k

x

; o~

a j x ; 2k 2k ! N

D RN j+a 2k

x

N! aN

+

(F )

2N (k 1) N !

D(f (N ) (x); o~) + ! 1

; f (N ) x

a 2k (13.7)

f (N ) ;

a 2k

:

At the end we estimate

a j D RN x ; 2k 2k Z jk 2 = D (F R) Z

2

x

a 2k

= D (F R)

x

(F R)

Z

x

a 2k

j 2k a 2k

j 2k a 2k

Z

Z

a 2k

x

Z

sN

Z

a 2k

s1

x

a 2k

sN

x

1

f

j 2k a 2k

1

a 2k

x

x

(by Lemmas 1,4 of [11]) Z

sN

(N )

a 2k

Z

s1

x

a 2k

!

(sN )dsN dsN

1 dsN dsN Z

N

!

!

1

Z

(N )

a 2k

s1

x

1

f

a 2k

x

N!

sN

x

x

j+a 2k

a 2k

x

s1

s1

x

Z

Z

a 2k

x

a f (N ) x 2k Z jk

;f

(N )

1

!

ds1

ds1 ;

1

!

!

f (N ) (sN )dsN dsN x

!

! a dsN dsN 2k

!

ds1 ;

1

1

!

ds1

!

13.2 Results

Z

sN

D f

!

Z

j 2k a 2k

x

(N )

a 2k

x

Z

1

Z

s1 a 2k

x

(sN ); f sN

1

a 2k

x

(N )

x

(F )

!

a 2k

dsN dsN

f (N ) ; sN +

!1

a 2k

217

!

ds1

1

x dsN dsN

1

ds1 (F ) !1

f

(N )

j a ; k + k 2 2

x

j a + k 2k 2

x

Z

j 2k

x

!

a 2k

Z

Z

s1

x

a 2k

x N!

a 2k

sN a 2k

x

!

1

1 dsN dsN

1

ds1 (F )

f (N ) ;

= !1 aN

(F )

2N (k 1) N !

f (N ) ;

!1

j 2k

N

a : 2k 1

That is, we have found that D RN

a j ; 2k 2k

x

; f (N ) x aN

2N (k

(F )

1) N !

!1

j+a 2k

a 2k

x

N

N!

f (N ) ;

!

a : 2k 1

(13.8)

Therefore by (13.7) and (13.8) we obtain D RN

x

a j ; 2k 2k

; o~

aN

(F )

2N (k 1) N !

D(f (N ) ; (x); o~) + 3! 1

a 2k (13.9)

f (N ) ;

Using (13.9) into ( ) we get ( )

N X1 i=1

+

ai 2i(k 1) i! aN

2N (k 1) N !

(F )

D(f (i) (x); o~) + ! 1

f (i) ;

(F )

D(f (N ) (x); o~) + 3! 1

a 2k

f (N ) ;

a 2k

:

Inequality (13.2) has been established. The next related main result is given. Theorem 13.15. All assumptions are as in Theorem 13.12. De…ne for k 2 Z, x 2 R the fuzzy wavelet like operator (Dk f )(x) :=

1 X

j= 1

kj (f )

'(2k x

j);

(13.10)

:

218

13. Estimates to Distances Between Fuzzy Wavelet Like Operators

where kj (f )

:=

n X

wr~

j r~ + k 2k 2 n

f

r~=0

; n 2 N; wr~

0;

n X

wr~ = 1: (13.11)

r~=0

Then a 2k

D (Dk f )(x); f x N X1 i=1

+

a (2a + 1)i (F ) D(f (i) (x); o~) + ! 1 f (i) ; k 2ki i! 2

(2a + 1)N 2kN N ! (F )

+3! 1 =:

k

D(f (N ) (x); o~)

f (N ) ;

a ; 2k

a 2k

(F )

+ !1

f (N ) ;

1 2k

for any x 2 R; k 2 Z:

(13.12)

NU (R), as k ! 1, we obtain with rates that If f 2 CF

a 2k

lim D (Dk f )(x); f x

k!+1

= 0:

Corollary 13.16 (to Theorem 13.15, for N = 1). It holds a 2k

D (Dk f )(x); f x

a 2k 1

+

(F )

1 2k f 0;

+3! 1

(D(f 0 (x); o~) a 2k

(F )

+ !1

f 0;

1 2k

;

for any x 2 R, k 2 Z. Corollary 13.17 (to Theorem 13.15). The following improvement of (13.12) holds D (Dk f )(x); f x

a 2k

(F )

min k

!1 a 2k

f;

a+1 2k

(F )

+ !1

;

for any x 2 R, k 2 Z. Proof. From Theorem 4, inequality (14) of [14] we obtain D((Dk f )(x); f (x))

(F )

!1

f;

a+1 2k

:

f;

a 2k

;

13.2 Results

219

Then we see that a 2k

D (Dk f )(x); f x

D((Dk f )(x); f (x)) + D f (x); f x (F )

!1

f;

a+1 2k

(F )

+ !1

f;

a 2k

a : 2k

Proof of Theorem 13.15. Because ' is of compact support in [ a; a] we observe that ! n X X j r~ wr~ f (Dk f )(x) = + k '(2k x j): k 2 2 n j r~=0

2k x j2[ a;a]

(13.13) Again for speci…c x 2 R, k 2 Z we have that the j’s in (13.12) satisfy a 2k

x

j 2k

x+

a : 2k

Using the fuzzy Taylor formula (Theorem 1, [11]) we get f

j r~ + k k 2 2 n

N 1 X

=

i=0

RN

(j+a) 2k

a 2k

f (i) x

r~ 2k n

x

i

i!

r~ a j ; k + k k 2 2 2 n

x

+

;

(13.14)

where r~ a j ; + k 2k 2k 2 n Z jk + kr~ Z 2 2 n := (F R)

RN

(13.15)

x

x

!

a 2k

Z

s1

x

a 2k

sN

1

a 2k

x

!

f (N ) (sN )dsN dsN

1

ds1 :

Then kj (f )

=

n X

wr~

f

r~=0 n X

wr~

r~ j + k 2k 2 n

(j+a) 2k

+

r~=0

wr~

f (i) x

i=0

x

i

a 2k

i!

r~=0 n

X

r~ 2k n

=

N 1 X

RN

x

a j r~ ; k + k k 2 2 2 n

;

(13.16)

220

13. Estimates to Distances Between Fuzzy Wavelet Like Operators

and (Dk f )(x)

N 1 X

=

a 2k

f (i) x

i=0

X

(13.17)

'(2k x

n X

j)

j 2k x j2[ a;a] k

'(2 x

j)

j 2k x j2[ a;a]

n X

r~

x

2k n

i

i!

!

wr~

r~=0

a j r~ ; + k 2k 2k 2 n

x

+

r~=0

X

RN

wr~

(j+a) 2k

:

Next we estimate a 2k

D (Dk f )(x); f x 0

B = DB @(Dk f )(x);

= D

N 1 X

X

k

'(2 x

j)

j 2k x j2[ a;a]

'(2k x

j)

j 2k x j2[ a;a]

N X1 i=1

n X

wr~

k

'(2 x

j)

j 2k x j2[ a;a]

+

j 2k x j2[ a;a]

k

'(2 x

wr~

RN

(j+a) 2k

+

n X

wr~

j)

n X r~=0

a C C 2k A

r~ 2k n

x

i

i!

r~ a j ; k + k k 2 2 2 n

x

(j+a) 2k

+

r~ 2k n

x

!

i

i!

r~=0

D f (i) x X

n X r~=0

r~=0

X

f x

a 2k

i=1

X

j)

j 2k x j2[ a;a]

f (i) x

X

'(2k x

1

! 1

C ; o~C A

!

a ; o~ 2k

wr~D RN

x

a j r~ ; k + k k 2 2 2 n

; o~

!

13.2 Results N X1

a (2a + 1)i (F ) D(f (i) (x); o~) + ! 1 f (i) ; k 2ki i! 2

i=1

+

X

221

k

'(2 x

n X

j)

j 2k x j2[ a;a]

r~=0

wr~D RN

r~ a j ; k + k k 2 2 2 n

x

!

; o~

=: ( ):

Next we work on r~ a j ; + k ; o~ 2k 2k 2 n r~ a j ; + k D RN x 2k 2k 2 n ! N (j+a) r~ x + k k 2 2 n N!

D RN

x

+D f

(N )

(j+a) 2k

a 2k

x

a 2k

; f (N ) x

+ 2kr~n N!

x

!

N

; o~ :

So that a j r~ ; k + k ; o~ k 2 2 2 n r~ a j D RN x ; k + k k 2 2 2 n ! N (j+a) r~ + 2k n x 2k N!

D RN

x

(2a + 1)N 2kN N !

+

a 2k

; f (N ) x

(F )

D(f (N ) (x); o~) + ! 1

f (N ) ;

a 2k

: (13.18)

Next we observe, as in the proof of Theorem 13.12, that D RN

x

r~ a j ; k + k k 2 2 2 n

;f

(N )

x

j+a 2k

a 2k

r~ 2k n

+

x

N

N!

!

N

(2a + 1) 2a + 1 (F ) f (N ) ; ! 2kN N ! 1 2k

:

(13.19)

Hence the previous inequality (13.19) implies D RN

x

a j r~ ; + k 2k 2k 2 n

(2a + 1)N 2kN N !

; o~

(13.20) (F )

D(f (N ) (x); o~) + 3! 1

f (N ) ;

a 2k

(F )

+ !1

f (N ) ;

1 2k

:

222

13. Estimates to Distances Between Fuzzy Wavelet Like Operators

Using the last inequality (13.20) into ( ) we obtain N X1

( )

i=1

+

(2a + 1)i a (F ) D(f (i) (x); o~) + ! 1 f (i) ; k ki 2 i! 2

(2a + 1)N 2kN N ! (F )

+! 1

(F )

D(f (N ) (x); o~) + 3! 1

f (N ) ;

1 2k

f (N ) ;

(13.21) a 2k

:

By (13.21) the proof of the theorem is now …nished. We need also the following main result. Theorem 13.18. All assumptions are as in Theorem 13.12. De…ne for k 2 Z, x 2 R the fuzzy wavelet like operator ! Z 2 k 1 X j k 2 (F R) (Ck f )(x) := f t + k dt '(2k x j): 2 0 j= 1 (13.22)

Then a 2k

D (Ck f )(x); f x N X1 i=1

+

(13.23)

(2a + 1)i a (F ) D(f (i) (x); o~) + ! 1 f (i) ; k 2ki i! 2

(2a + 1)N 2kN N !

(F )

D(f (N ) (x); o~) + 3! 1 (F )

+ !1

=

f (N ) ; k

f (N ) ;

a 2k

1 2k

a ; 2k

all k 2 Z, x 2 R. NU (R) and as k ! +1 we obtain with rates that If f 2 CF lim D (Ck f )(x); f x

k!+1

a 2k

= 0:

Corollary 13.19 (to Theorem 13.18, for N = 1). It holds D (Ck f )(x); f x

a 2k

a 1 + k 2k 1 2 (F )

+3! 1 for any x 2 R, k 2 Z.

f 0;

(D(f 0 (x); o~) a 2k

(F )

+ !1

f 0;

1 2k

;

13.2 Results

223

Corollary 13.20 (to Theorem 13.18). The following improvement of (13.23) holds a 2k

D (Ck f )(x); f x (F )

min

!1

f;

a+1 2k

(F )

+ !1

f;

a 2k

;

a 2k

k

;

for any x 2 R, k 2 Z. Proof. From Theorem 3, inequality (10), of [14] we get (F )

D((Ck f )(x); f (x))

!1

f;

a+1 2k

:

Then we observe that a 2k

D (Ck f )(x); f x

D((Ck f )(x); f (x)) + D f (x); f x (F )

!1

f;

a+1 2k

(F )

+ !1

f;

a 2k

a : 2k

Proof of Theorem 13.18. Because ' is of compact support in [ a; a] we see that ! Z 2 k X j k (Ck f )(x) = 2 (F R) f t + k dt '(2k x j): 2 0 j 2k x j2[ a;a]

(13.24) Hence for speci…c x 2 R, k 2 Z we have the j’s in (13.24) to satisfy x

a 2k

j 2k

x+

a : 2k

Using the fuzzy Taylor formula (Theorem 1, [11]) we …nd j f t+ k 2

=

N 1 X

f

(i)

i=0

a x k 2

(j+a) 2k

t+

x

i

RN

i!

x

a j ;t + k ; 2k 2 (13.25)

where RN

x

a j ;t + k 2k 2

:=

(F R)

Z

t+

a 2k

x

Z

Z

j 2k

sN

x

x

1

f a 2k

s1

(N )

(13.26) a 2k

!

(sN )dsN dsN

1

!

ds1 :

224

13. Estimates to Distances Between Fuzzy Wavelet Like Operators

Then by Theorem 3.4, [67] we get k

2

Z

(F R)

k

2

j t+ k 2

f

0

Z

k

2 i!

N 1 X

dt =

i=0

k

2

f (i) x

(j + a) x 2k Z 2 k (F R) RN x

i

t+

0

2k

a 2k

0

dt

(13.27)

j a ;t + k k 2 2

dt:

Thus we have (Ck f )(x) =

N 1 X

a 2k

f (i) x

i=0

X

k

k

'(2 x

2 i!

j)

j 2k x j2[ a;a]

X

k

k

'(2 x

j) 2

Z

2

k

(j + a) t+ 2k

0

(F R)

Z

k

2

RN

0

j 2k x j2[ a;a]

x

i

x

dt

!

(13.28)

j a ;t + k 2k 2

!

dt :

Next we estimate a 2k

D (Ck f )(x); f x 0

B = DB @(Ck f )(x); N 1 X

= D

X

'(2k x

j)

f x

j 2k x j2[ a;a]

a 2k

f (i) x

i=1

X

k

'(2 x

2k i!

j)

j 2k x j2[ a;a]

X

'(2k x

Z

0

2

k

2

(F R)

Z

0

2

i

x

j)

k

RN

a C C 2k A

(j + a) t+ 2k

j 2k x j2[ a;a] k

1

x

a j ;t + k k 2 2

dt

!!

!

; o~

!

dt

13.2 Results N X1

X

i=1

'(2k x

j)

j 2k x j2[ a;a]

a ; o~ k 2 X '(2k x

D f (i) x +

2k i!

Z

k

2

0

(F R)

2

k

RN

0

x

i

x

dt

!

j) 2k D

j 2k x j2[ a;a]

Z

(j + a) t+ 2k

225

!!

j a ;t + k 2k 2

dt; o~

(by Lemma 1, [11]) NX1 (2a + 1)i a (F ) D(f (i) (x); o~) + ! 1 f (i) ; k ki i! 2 2 i=1 X + '(2k x j) j 2k x j2[ a;a]

2

k

Z

k

2

D RN

0

=

x

j a ;t + k 2k 2

; o~ dt

!

: ( ):

Next we work on D RN

x

D RN

j a ;t + k 2k 2 x

+ D f (N ) x

; o~

j a ;t + k 2k 2

; f (N ) x t+

a 2k

(j+a) 2k

t+

a 2k x

(j+a) 2k

x

N

N!

N

!

!

; o~ :

N!

So that D RN

x

D RN + for 0

t

a j ;t + k 2k 2 x

(2a + 1)N 2kN N ! 2

k

:

; o~

a j ;t + k 2k 2

;f

(N )

x

(F )

D(f (N ) (x); o~) + ! 1

t+

a 2k f (N ) ;

(j+a) 2k

N! a 2k

;

x

N

!

(13.29)

226

13. Estimates to Distances Between Fuzzy Wavelet Like Operators

Next, we act as in the proof of Theorem 13.12, we observe that D RN

a j ;t + k k 2 2

x

;f

(N )

t+

a 2k

x

(j+a) 2k

x

N

N!

!

N

(2a + 1) 2a + 1 (F ) ! f (N ) ; 2kN N ! 1 2k

;

(13.30)

for 0 t 2 k : Therefore we obtain D RN

a j ;t + k 2k 2

x

(2a + 1)N 2kN N ! (F )

+! 1 for 0

t

2

( )

k

; o~

(13.31) (F )

D(f (N ) (x); o~) + 3! 1

f (N ) ;

1 2k

a 2k

f (N ) ;

;

. Using the last inequality (13.31) into ( ) we get N X1 i=1

+

a (2a + 1)i (F ) D(f (i) (x); o~) + ! 1 f (i) ; k 2ki i! 2

(2a + 1)N 2kN N ! (F )

+! 1

(F )

D(f (N ) (x); o~) + 3! 1

f (N ) ;

1 2k

f (N ) ;

a 2k

:

(13.32)

By (13.32) we have the validity of the theorem. The previous results lead to the following important theorems. Theorem 13.21. All assumptions are as in Theorem 13.12. It holds D((Bk f )(x); (Dk f )(x))

k

a 2k

+

k

a ; 2k

(13.33)

NU (R), as k ! +1, we obtain with rates for any x 2 R, k 2 Z. If f 2 CF that lim D((Bk f )(x); (Dk f )(x)) = 0: k!+1

Proof. By D((Bk f )(x); (Dk f )(x))

D (Bk f )(x); f x

+ D (Dk f )(x); f x

a 2k

(by (13.2) and (13.12)) k

a 2k

+

a 2k

k

a : 2k

13.2 Results

227

Theorem 13.22. All assumptions are as in Theorem 13.12. It holds D((Dk f )(x); (Ck f )(x))

2

a ; 2k

k

(13.34)

NU for any x 2 R, k 2 Z. If f 2 CF (R), as k ! +1, we get with rates that

lim D((Dk f )(x); (Ck f )(x)) = 0:

k!+1

Proof. By D((Dk f )(x); (Ck f )(x))

a 2k

D (Dk f )(x); f x

+ D (Ck f )(x); f x

a 2k

(by (13.12) and (13.23)) k

a 2k

+

k

a 2k

=2

a : 2k

k

Theorem 13.23. All assumptions are as in Theorem 13.12. It holds D((Bk f )(x); (Ck f )(x))

k

a 2k

+

k

a ; 2k

(13.35)

NU (R), as k ! +1, we …nd with rates that all x 2 R, k 2 Z. If f 2 CF

lim D((Bk f )(x); (Ck f )(x)) = 0:

k!+1

Proof. By D((Bk f )(x); (Ck f )(x))

D (Bk f )(x); f x

+ D (Ck f )(x); f x (by (13.2) and (13.23)) k

a 2k

a 2k a 2k +

k

a : 2k

It follows another family of basic interesting related results. Theorem 13.24. Here ' is as in Theorem 13.12 and f 2 C(R; RF ). Fuzzy wavelet like operators Bk de…ned by (13.1), and Ck de…ned by (13.22). Then 1 (F ) D((Bk f )(x); (Ck f )(x)) ! 1 f; k ; (13.36) 2 U for any x 2 R, k 2 Z. If f 2 CF (R), as k ! +1, we obtain with rates that lim D((Bk f )(x); (Ck f )(x)) = 0: k!+1

228

13. Estimates to Distances Between Fuzzy Wavelet Like Operators

Proof. We have that 0

1 X

D((Bk f )(x); (Ck f )(x)) = D @

j 2k

'(2k x

f

j t+ k 2

!

j 2k

k

f

j= 1

1 X

2k

(F R)

2

k

'(2 x

j)D f

j 2k x j2[ a;a]

= 2

X

k

k

'(2 x

(F R)

2

k

f

0

;2

dt

(F R)

j)D (F R)

Z

2

k

k

X

k

'(2 x

j)

2

k

j 2k

k

f

j t+ k 2

dt;

Z

2

k

D f

0

j 2k x j2[ a;a]

t+

2

dt

2

f

j)A

!

j t+ k 2

(by Lemma 1, [11])

Z

j 2k

f

1

'(2k x

0

0

j 2k x j2[ a;a]

Z

k

0

j= 1

X

Z

j);

j 2k

;

dt X

k

'(2 x

j 2k x j2[ a;a]

j)

Z

2

k

(F )

(F )

! 1 (f; t)dt

!1

0

f;

1 2k

:

Theorem 13.25. Here ' is as in Theorem 13.12 and f 2 C(R; RF ). Fuzzy wavelet like operators Dk de…ned by (13.10) and (13.11), and Ck de…ned by (13.22). Then

D((Dk f )(x); (Ck f )(x))

(F )

!1

f;

1 2k

1

;

(13.37)

U for any x 2 R, k 2 Z. If f 2 CF (R), as k ! +1, we get with rates that

lim D((Dk f )(x); (Ck f )(x)) = 0:

k!+1

dt

!

13.2 Results

229

Proof. We have that D((Dk f )(x); (Ck f )(x)) 0 1 X = D@ '(2k x kj (f )

j);

j= 1

1 X

2k

Z

(F R)

2

k

f

0

j= 1

X

'(2 x

X

'(2k x

k

j)D

!

j t+ k 2 kj (f ); 2

(F R)

j)

j 2k x j2[ a;a]

(F R)

k

2

f

= 2

X

k

n X

'(2 x

j)

n X

wr~D (F R)

Z

(by Lemma 1, [11])

2

k

f 2k

Z

2

0

2k

j t+ k 2 X

dt

'(2k x

j)

X

'(2k x

j 2k x j2[ a;a] (F )

= !1

f;

n X

1 2k 1

:

j)

n X r~=0

wr~

dt;

wr~

r~=0

0

r~=0

r~ j + k 2k 2 n

!

j r~ j + k ; f t+ k dt 2k 2 n 2 Z 2 k n X X (F ) '(2k x j) wr~ !1 f;

j 2k x j2[ a;a]

2k

k

f

j 2k x j2[ a;a]

D f

dt

; 2k

0

r~=0

0

k

!

dt k

(F R)

2

f

j t+ k 2

!

j 2k x j2[ a;a]

Z

k

2

r~ j + k k 2 2 n

wr~D f

r~=0

j t+ k 2

0

Z

0

j 2k x j2[ a;a]

Z

j)A

'(2k x

dt

k

1

Z

0

2

k

(F )

!1

f;

r~ 2k n 1 2k

1

t

dt

dt

230

13. Estimates to Distances Between Fuzzy Wavelet Like Operators

Theorem 13.26. Here ' is as in Theorem 13.12 and f 2 C(R; RF ). Fuzzy wavelet like operators Bk de…ned by (13.1), and Dk de…ned by (13.10) and (13.11). Then (F )

D((Bk f )(x); (Dk f )(x))

!1

f;

1 2k

;

(13.38)

U for any x 2 R, k 2 Z. If f 2 CF (R), as k ! +1, we …nd with rates that

lim D((Bk f )(x); (Dk f )(x)) = 0:

k!+1

Proof. We see that D((Bk f )(x); (Dk f )(x)) 0 1 X j = D@ f 2k j= 1

'(2k x

j);

1 X

j)A

'(2k x

j= 1

X

'(2k x

j)D f

X

'(2k x

j)

X

'(2k x

j 2k x j2[ a;a]

j 2k x j2[ a;a]

j 2k

;

n X

wr~D f

n X

wr~! 1

kj (f )

j 2k

r~=0

j 2k x j2[ a;a]

kj (f )

1

j)

(F )

r~ j + k 2k 2 n

;f

r~ 2k n

f;

r~=0

(F )

!1

f;

1 2k

:

Next we present the corresponding comparison results based on the previously given theorems. Corollary 13.27 (to Theorem 13.21, and Theorem 13.26). All assumptions here are as in Theorem 13.12. It holds D((Bk f )(x); (Dk f )(x))

(F )

min ! 1

f;

1 2k

;

k

a 2k

+

a ; 2k (13.39)

k

for any x 2 R, k 2 Z. Corollary 13.28 (to Theorem 13.22 and Theorem 13.25). All assumptions here are as in Theorem 13.12. It holds D((Dk f )(x); (Ck f )(x)) for any x 2 R, k 2 Z.

(F )

min ! 1

f;

1 2k

1

;2

k

a 2k

;

(13.40)

13.2 Results

231

Corollary 13.29 (to Theorem 13.23 and Theorem 13.24). All assumptions here are as in Theorem 13.12. It holds (F )

D((Bk f )(x); (Ck f )(x))

min ! 1

f;

1 2k

;

k

a 2k

+

k

a ; 2k (13.41)

all x 2 R, k 2 Z.

Next we study similarly the Fuzzy wavelet operator Ak .

Theorem 13.30. Let f 2 Cb (R; RF ) and the scaling function '(x) a real valued function with supp '(x) [ a; a], 0 < a < +1, ' is continuous on 1 R1 P [ a; a], '(x) 0, such that '(x 1) = 1 on R (then 1 '(x)dx = j= 1

1). De…ne

2k=2 '(2k t j); for k; j 2 Z; t 2 R; Z j+a 2k f (t) 'kj (t)dt; (F R)

'kj (t) := hf; 'kj i :=

(13.42) (13.43)

j a 2k

and set (Ak f )(x) :=

1 X

hf; 'kj i

'kj (x);

j= 1

any x 2 R:

(13.44)

The fuzzy wavelet like operator (Bk f )(x) is de…ned by (13.1). Then it holds D((Ak f )(x); (Bk f )(x))

(F )

!1

f;

a ; 2k

(13.45)

U (R) and bounded, then for any x 2 R, k 2 Z. If f 2 CF

lim D((Ak f )(x); (Bk f )(x)) = 0 with rates.

k!+1

Proof. We see easily that (Ak f )(x) =

1 X

(F R)

j+a

f

j a

j= 1

Also it holds

Z

Z

u 2k

'(u

j)du

'(2k x

j): (13.46)

j+a

j a

'(u

j)du = 1:

(13.47)

232

13. Estimates to Distances Between Fuzzy Wavelet Like Operators

So we observe

D((Ak f )(x); (Bk f )(x)) 0 Z 1 X = D@ (F R)

f

j a

j= 1

'(2k x

j+a

j);

1 X

j 2k

f

j= 1

X

'(2k x

u 2k

'(u

j)du 1

'(2k x

j)D (F R)

Z

j+a

j)A

u 2k

f

j a

j 2k x j2[ a;a]

(F R)

Z

j+a

j a

(by Lemma 1, [11] and Lemma 2, [14])

f

j 2k

'(u

X

'(u

j)du;

'(u

j)

j)du k

'(2 x

j)

Z

j+a

j a

j 2k x j2[ a;a]

j u ;f du k 2 2k Z j+a a (F ) '(2k x j) '(u j)! 1 f; k du 2 j a D f

X

j 2k x j2[ a;a] (F )

= !1

f;

a : 2k

Theorem 13.31. Let ', f , Ak as in Theorem 13.30. Let fuzzy wavelet like operator Dk as in (13.10) and (13.11). Then

D((Ak f )(x); (Dk f )(x))

(F )

!1

f;

a+1 2k

;

U for any x 2 R, k 2 Z. If f 2 CF (R) and bounded then

lim D((Ak f )(x); (Dk f )(x)) = 0 with rates.

k!+1

(13.48)

13.2 Results

233

Proof. We notice that D((Ak f )(x); (Dk f )(x)) 0 1 X (by (13.46)) @ = D

(F R)

'(2 x

1 X

j);

j+a

'(2k x

kj (f )

j= 1

X

'(2k x

j)D (F R)

j)du;

n X

wr~

X

(by (13.47))

f

'(2k x

j)

j 2k x j2[ a;a]

j)du; (F R)

(by Lemma 1, [11] and Lemma 2, [14]) j+a

'(u

Z

'(2k x

j 2k x j2[ a;a] (F )

= !1

f;

a+1 2k

n X

j)du

u 2k

! wr~D (F R)

j+a

f X

Z

j+a

j a

r~ j + k 2k 2 n

'(u

'(2k x

j)

j 2k x j2[ a;a]

n X

f

u 2k

j)du

wr~

r~=0

u j r~ ;f + k du 2k 2k 2 n Z j+a n X a+1 (F ) j) wr~ '(u j)! 1 f; k 2 j a

j)D f

j a

X

f

'(u

r~=0

j a

Z

j+a

j r~ + k 2k 2 n

r~=0

'(u

Z

j)A

j a

j 2k x j2[ a;a]

'(u

u 2k 1

f

j a

j= 1

k

Z

du

r~=0

:

Theorem 13.32. Let ', f , Ak as in Theorem 13.30. Let fuzzy wavelet like operator Ck as in (13.22). Then D((Ak f )(x); (Ck f )(x))

(F )

!1

f;

a+1 2k

;

U for any x 2 R, k 2 Z. If f 2 CF (R) and bounded, then

lim D((Ak f )(x); (Ck f )(x)) = 0 with rates:

k!+1

(13.49)

234

13. Estimates to Distances Between Fuzzy Wavelet Like Operators

Proof. We observe that D((Ak f )(x); (Ck f )(x))

0

1

X (by (13.46) and (13.22)) = D@ '(2k x

j= 1

'(2k x

j)D (F R)

1 X

2k

(F R)

Z

2

(F R)

Z

"

j+a

'(u

j)

(by Lemma 1, [4]

Z

j+a

Z

2k

(F R)

Z

2

dt

k

f

2

k

0

X

(by Lemma 1, [11])

2

0

X

k

k

(F ) !1

2

k

'(2k x

u f; k 2

k

! ! j t+ k dt du 2 Z j+a j) '(u j)2k

u D f k ;f 2

j a

j 2k x j2[ a;a] 2

Z

j 2k

t

!

!

dt du

j)A

dt du

!

j)

j a

j 2k x j2[ a;a]

Z

#

f

j t+ k 2

!

dt du

! ! j f t + k dt du 2 0 Z j+a '(2k x j) '(u j)2k

u f k dt; (F R) 2

1

j)du;

j a

j 2k x j2[ a;a]

D (F R)

'(2k x

j 2k x j2[ a;a]

j+a

Z

'(u

j t+ k 2

Z 2 u '(u j)D f k ; 2k (F R) 2 j a 0 Z j+a X '(2k x j) '(u j)2k

=

!

j)du

!

u 2k

0

'(2k x

dt

'(u

u 2k

f

k

X

and Lemma 2, [14])

j 2k

t+

j a

j a

0

f

j '(u j)du; 2 (F R) f t+ k 2 0 Z j+a X '(2k x j)D (F R) f

j 2k x j2[ a;a]

u 2k

f

k

2

j a

k

=

j+a

0

j 2k x j2[ a;a]

Z

Z

j a

j= 1

Z

j);

X

(F R)

!

13.2 Results

X

'(2k x

j)

k

2

a+1 f; k 2

(F ) !1

0

(F )

= !1

a+1 2k

f;

j+a

j)2k

'(u

j a

j 2k x j2[ a;a]

Z

Z

235

!

!

dt du

:

Example 13.33. The following scaling function ' ful…lls the assumptions of the presented theorems 8 1 x 0; < x + 1; 1 x; 0 < x 1; '(x) = : 0; elsewhere. Note 13.34. Ak , Bk , Ck , Dk are linear operators over R.

Remark 13.35. On Theorems 13.24, 13.25, 13.26, 13.30, 13.31 and 13.32. It is enough to comment Theorem 13.24, similar conclusions can be derived from the rest of them. Assume that f 2 C(R; RF ) ful…lls the following Lipschitz condition D(f (x); f (y))

M jx

yj ;

0<

1; M > 0:

(13.50)

Then clearly it holds (F )

!1

f;

1 2k

M ; 2k

k 2 Z:

(13.51)

M ; 2k

(13.52)

So that from (13.36) we obtain D((Bk f )(x); (Ck f )(x)) and k 2 Z, x 2 R. And consequently D ((Bk f ); (Ck f ))

M ; 2k

(13.53)

for any k 2 Z. Finally we get the global error estimate sup f

Etc.

f 2C(R;RF ) :

as in (13.50)

D ((Bk f ); (Ck f ))

M : 2k

(13.54)

236

13. Estimates to Distances Between Fuzzy Wavelet Like Operators

Next we give two independent but related and useful results. Proposition 13.36. Let ', f be both even functions as in Theorem 13.24. Let (Bk f ) de…ned by (13.1). Then (Bk f )(x) is an even function. Proof. We observe for x 2 R that (Bk f )( x)

1 X

=

=

j 2k

f

j= 1 1

X

j 2k

f

j= 1 1 X

=

=

'( 2k x

j := j=1 1

X

'(2k x + j) (a …nite sum)

j 2k

f

'(2k x

j 2k

f

j = 1

j)

'(2k x

j )

j ) = (Bk f )(x):

Proposition 13.37. Let ', f be both even functions as in Theorem 13.30. Let (Ak f ) de…ned by (13.43) and (13.44). Then (Ak f )(x) is an even function. Proof. We observe for x 2 R that (Ak f )( x)

(13:46)

=

1 X

(F R)

j= 1

=

1 X

(F R)

Z

j+a

f

j a

u 2k

'(u

j)du

'( 2k x j) (a …nite sum) j+a

f

j a

j= 1

Z

u 2k

'( u + j)du

'(2k x + j) (linear change of variables is valid in (F R)-integrals) ! Z j +a 1 X w = (F R) f k '(w j )dw 2 j a j := j=1 '(2k x

j )

(13:46)

=

(Ak f )(x):

Note. In this chapter we use only fuzzy methods in our proofs. However in Chapter 16 we use real and fuzzy analysis together in our proofs. Consequently, we get some improvements of some of our results here, namely improvements for Theorems 13.21, 13.22 and Theorem 13.23, and for Corollaries 13.27, 13.28 and Corollary 13.29. See there Theorem 16.16

14 FUZZY APPROXIMATION BY FUZZY CONVOLUTION OPERATORS

Here we study four sequences of naturally arising fuzzy integral operators of convolution type that are integral analogs of known fuzzy wavelet like operators, de…ned via a scaling function. Their fuzzy convergence with rates to the fuzzy unit operator is established through fuzzy inequalities involving the fuzzy modulus of continuity. Also their high order fuzzy approximation is given similarly by involving the fuzzy modulus of continuity of the N th order (N 1) H-fuzzy derivative of the engaged fuzzy number valued function. The fuzzy global smoothness preservation property of these operators is presented too. This chapter relies on [15].

14.1 Introduction Here C(R) is the space of continuous functions from R into R, and C N (R), N 1 the space of N -times continuously di¤erentiable functions on R. We denote by CB (R) the space of continuous and bounded functions from R into R. For f : R ! R we de…ne its …rst modulus of continuity by ! 1 (f; ) :=

sup x;y2R : jx yj

jf (x)

f (y)j;

> 0:

G.A. Anastassiou: Fuzzy Mathematics: Approximation The., STUDFUZZ 251, pp. 237–261. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

238

14. Fuzzy Approximation by Fuzzy Convolution Operators

Let CU (R) be the space of uniformly continuous functions from R into R. For f 2 CU (R) we have ! 1 (f; ) < +1, > 0, see [30], p. 298 and [34]. Also it is a well-known fact, see [56], p. 40, that lim ! 1 (f; ) = ! 1 (f; 0) = 0; #0

i¤ f 2 CU (R):

Let ' be a real valued function of compact support [ a; a], a > 0, ' 0, ' is Lebesgue measurable and such that Z 1 '(x u) du = 1; 8x 2 R; (14.1) 1

equivalently

Z

1

'(u) du = 1:

(14.2)

1

Examples (i) '(x) :=

1 1 2;2

(x) =

8 < :

the characteristic function; (ii) 8 < 1 x; 1 + x; '(x) := : 0;

1 1 ; ; 2 2 elsewhere,

1;

x2

0;

0

x 1 1 x 0 elsewhere;

the hat function. Here we follow [30], pp. 287–293, and [35]. We introduce the following positive linear operators for k 2 Z and f 2 CU (R). (i) Case of Ak operators: here additionally we assume that ' is an even continuous function. De…ne Z 1 (Ak f )(x) := rkf (u)'(2k x u) du; 1

where rkf (u) := 2k is continuous in u. (ii) (Bk f )(x) :=

Z

(iii) (Lk f )(x) :=

Z

1

u) dt

(14.3)

1

1

Z

f (t)'(2k t

f

1 1 1

u '(2k x 2k

cfk (u)'(2k x

u) du:

u) du;

(14.4)

(14.5)

14.1 Introduction

239

where cfk (u)

:= 2

k

Z

2

2

k

(u+1) k

f (t) dt = 2 ku

k f )(x)

2

k

f t+

0

is continuous in u. (iv) (

Z

:=

Z

1 1

f k k (u)'(2 x

u dt 2k

u) du;

(14.6)

where f k (u)

:=

n X

wj f

j=0

j u + k 2k 2 n

;

n 2 N …xed; wj

0;

n X

wj = 1;

j=0

is continuous in u. We have proved in [30], p. 289, p. 295, respectively, and [35] that Ak , Bk , Lk , k are shift invariant operators, and map continuous probabilistic distribution functions into continuous probabilistic distribution functions. More importantly we proved the following theorems [30], pp. 290–293 and [35]. The operators Ak , Bk , Lk , k , k 2 Z converge to the unit operator I with rates as given by next Theorem 14.1. For k 2 Z, a ; 2k 1 a f; k ; 2 a+1 ; f; k 2

j(Ak f )(x)

f (x)j

! 1 f;

(14.7)

j(Bk f )(x)

f (x)j

!1

(14.8)

j(Lk f )(x)

f (x)j

!1

(14.9)

and j(

k f )(x)

The operators Ak , Bk , Lk , Preservation.

f (x)j k

! 1 f;

:

(14.10)

ful…ll the property of Global Smoothness

Theorem 14.2. For all f 2 CU (R) and any ! 1 (Ak f; ) ! 1 (Lk f; )

a+1 2k

> 0 we have

! 1 (f; ); ! 1 (Bk f; ) ! 1 (f; ); ! 1 (f; ); and ! 1 ( k f; ) ! 1 (f; ):

Inequalities (14.11) are sharp, attained by f (x) = x 2 CU (R).

(14.11)

In [12] we continued the above study by establishing the high order of approximation of operators Ak , Bk , Lk , k to the unit operator I.

240

14. Fuzzy Approximation by Fuzzy Convolution Operators

Theorem 14.3. Let f 2 C N (R), N j(Ak f )(x)

f (x)j

1. Then one has N X jf (i) (x)j

i!

i=1

ai 2i(k

(14.12)

1)

aN a ! 1 f (N ) ; k 1 ; 2 N !2N (k 1) N X jf (i) (x)j ai +

j(Bk f )(x)

f (x)j

i!

i=1

2ki

a aN ! 1 f (N ) ; k ; kN N !2 2 N (i) X jf (x)j (a + 1)i i! 2ki i=1

(14.13)

+

j(Lk f )(x) j( k f )(x)

f (x)j f (x)j

+

(a + 1)N a+1 ! 1 f (N ) ; k kN N !2 2

(14.14) ;

for any k 2 Z, and any x 2 R. If f (N ) is uniformly continuous function, then as k ! +1 we obtain that (Lk f )(x) ! f (x); for Lk := Ak , Bk , Lk , k , pointwise with rates. In this chapter we generalize the above mentioned motivating results to the Fuzzy Theory setting. We study the same operators Ak , Bk , Lk , k , k 2 Z, when they act on fuzzy real number valued functions. So according to the context one can understand clearly, when these operators are applied to real valued functions, and when they are applied to fuzzy real valued functions. The proofs of the presented results rely a lot on this introduction. For that we also need.

14.2 Background We mention Lemma 14.4. Let g : [a; b] ! R+ has existing ordinary Riemann integral Rb Rb g(x)dx. Let u 2 RF be …xed. Then (F R) a (u g(x))dx exists in RF , a and Z b Z b u g(x) dx = (F R) (u g(x)) dx: (14.15) a

Rb

a

Proof. Since a g(x)dx exists we have that for any " > 0 there exists > 0 such that for any division P = f[v; w]; g of [a; b] with norms (P ) < it

14.2 Background

holds :=

X

(w

Z

v)g( )

Rb

D

X

(w

a

b

g(x) dx < ":

a

P

Notice that u get

241

g(x) dx 2 RF . Hence by Lemmas 2.1, 2.2(iii) of [17] we

v)g( )

Z

u; u

b

g(x) dx

a

P

!

X

= D u

u

(w

P

Z

b

!

v)g( ) ;

!

g(x) dx

a

D(u; o~)

"D(u; o~):

That is proving the claim. De…nition 14.5. Let f : R ! RF be such that (i) (F R)

Rt a

f (x) dx exists for every real number t (F R)

Z

1

f (x) dx = lim (F R) t!+1

a

a, a 2 R, then

Z

t

f (x) dx

a

provided the limit exists in RF in the D-metric. (ii) If (F R)

Rb t

f (x) dx exists for every real number t (F R)

Z

b

f (x) dx = lim (F R) 1

t! 1

Z

b, b 2 R then

b

f (x) dx

t

provided the limit exists in RF in the D-metric. The improper inteR1 Rb grals (F R) a f (x) dx and (F R) 1 f (x) dx are called convergent if the corresponding limit exists in RF and divergent if the limit does not exist in RF .

Ra R1 (iii) If both (F R) a f (x) dx and (F R) 1 f (x) dx are convergent for some a 2 R, then we de…ne the improper fuzzy-Riemann integral over R as follows: Z 1 Z a Z +1 (F R) f (x) dx = (F R) f (x) dx (F R) f (x) dx: 1

In this case we say that (F R) say it diverges.

1

R1

1

a

f (x) dx converges. Otherwise we

242

14. Fuzzy Approximation by Fuzzy Convolution Operators

Remark 14.6. (i) Let f : R ! RF be fuzzy continuous and of compact support [a; b] R (See Theorem 14.10 next), then (F R)

Z

1

>

X

D

f (x) dx = (F R)

1

(ii) Given that I := (F R) 2.4 ([17]) we have "

Z

b

f (x) dx:

Rb a

f (x) dx, a

(v

u)

f ( ); I

P

X

= D ( 1)

(v

b, exists in RF from De…nition !

!

u)

f ( ) ; ( 1)

P

X

= D

( 1)

((v

u)

f ( )); ( 1)

P

X

= D

(u

v)

f ( ); ( 1)

I

P

That last motivates us to de…ne Z a (F R) f (x) dx := ( 1)

(F R)

b

Also (F R) Similarly given that (F R) (F R)

Z

1

(14.16)

a

R1

1

Z

Z

!

I

I

!

!

:

b

f (x) dx:

(14.17)

a

a

f (x) dx := o~:

(14.18)

a

f (x) dx exists in RF , one can de…ne

f (x) dx := ( 1)

(F R)

1

Z

1

f (x) dx:

(14.19)

1

Linear change of variable is possible and valid in (F R)-integrals. Theorem 14.7. Let x := '(t) := mt + , m > 0, 2 R, t 2 [ ; ]. R Call a := m + , b := m + . Then (F R) f ('(t)) dt exists in RF i¤ Rb (F R) a f (x) dx exists in RF , and (F R)

Z

a

b

f (x) dx = m

(F R)

Z

f ('(t)) dt:

(14.20)

Proof. Here x 2 [a; b] and ' is an (1 1) and onto map from [ ; ] to R [a; b]. Also f : [a; b] ! RF and f ' : [ ; ] ! RF . If (F R) f ('(t)) dt

14.2 Background

243

1 exists in RF and equals m I, by De…nition 2.4 ([17]) for every " > 0 there exists > 0 such that for any division P = f[u; v]; g of [ ; ] with norm (P ) < , we have ! X 1 D (v u) f ('( )); I < ": m P

Equivalently we …nd X

D

(mv

mu)

f ('( )); I

P

!

< m" =: "0

and equivalently we have X

D

((mv + )

(mu + ))

f ('( )); I

P

Denoting v 0 := mv + , u0 := mu + , X

D

(v 0

u0 )

P0

0

:= m + !

f ( 0 ); I

!

< "0 :

we have equivalently,

< "0 ;

for P 0 = f[u0 ; v 0 ]; 0 g any division of [a; b] with norm (P 0 ) < m =: 0 . By Rb De…nition 2.4 ([17]) we get equivalently that (F R) a f (x) dx exists in RF and equals I. The counterpart of the last result comes next. Theorem 14.8. Let x := '(t) := mt + , m < 0, 2 R, t 2 [ ; ]. R Call a := m + , b := m + . Then (F R) f ('(t)) dt exists in RF i¤ Rb (F R) a f (x) dx exists in RF , and (F R)

Z

b

a

f (x) dx = jmj

(F R)

Z

f ('(t)) dt:

(14.21)

Proof. Similar to Theorem 14.7. Next we present a result regarding linear change of variables for improper (F R)-integrals. Theorem 14.9. …xed reals and t 2 R. R 1 Let x := '(t) := mt + , m, Rare 1 Then (F R) 1 f ('(t)) dt exists in RF i¤ (F R) 1 f (x) dx exists in RF , and Z 1 Z 1 (F R)

1

f (x) dx = jmj

Proof. Clearly ' is (1

(F R)

f ('(t)) dt:

1

1) and onto from R ! R.

(14.22)

244

14. Fuzzy Approximation by Fuzzy Convolution Operators

(i) HereRm > 0. 1 If (F R) 1 f ('(t)) dt exists in RF then 9 2 R such that (cf. De…nition 14.5(iii)) (F R)

Z

1

f ('(t)) dt = (F R)

1

where the improper integrals Z f ('(t)) dt; (F R)

Z

f ('(t)) dt

(F R)

1

(F R)

1

Z

1

f ('(t)) dt

R That is there exist (F R) f ('(t)) dt for any for any , and in the D-metric we have Z Z (F R) f ('(t)) dt = lim (F R) and (F R)

Z

1

Z

f ('(t)) dt = lim (F R) !1

1

f ('(t)) dt;

exist in RF : R

and (F R)

f ('(t)) dt

f ('(t)) dt;

! 1

1

Z

f ('(t)) dt:

Hence by Theorem 14.7 we obtain (F R)

Z

f ('(t)) dt

=

1

=

1 m

lim (F R)

! 1

1 m

(F R)

Z

Z

m +

f (x) dx

m +

m +

f (x) dx; 1

and (F R)

Z

1

f ('(t)) dt

= =

1 m

lim (F R)

!+1

1 m

(F R)

Z

Z

m +

f (x) dx

m +

1

f (x) dx:

m +

Consequently we obtain m

(F R)

Z

1

f ('(t)) dt

=

(F R)

1

=

(F R)

(ii) The case of m < 0 is similar. We need

Z Z

m +

f (x) dx 1 1 1

(F R)

Z

1

m +

f (x) dx:

f (x) dx

14.2 Background

245

Theorem 14.10 ([67]). Let f : [a; b] ! RF be fuzzy continuous with respect Rb to metric D. Then (F R) a f (x) dx exists and belongs to RF , furthermore it holds #r "Z # " Z Z b

b

(F R)

=

f (x) dx

(r)

(f )

b

(x) dx;

a

a

(r)

(f )+ (x) dx ;

a

8r 2 [0; 1]:

(14.23)

Clearly f

(r)

: [a; b] ! R are continuous functions.

Denote by C(R; RF ) the space of fuzzy continuous functions and by Cb (R; RF ) the space of bounded fuzzy continuous functions on R with respect to metric D. We use also the following De…nition 14.11 ([11]). Let f : R ! RF be a fuzzy real number valued function. We de…ne the (…rst) fuzzy modulus of continuity of f by (F )

! 1 (f; ) :=

D(f (x); f (y));

sup

> 0:

(14.24)

x;y2R jx yj

De…nition 14.12 ([11]). Let f : R ! RF . We call f a uniformly continuous fuzzy real number valued function, i¤ for any " > 0 there exists > 0: whenever jx yj ; x; y 2 R, implies that D(f (x); f (y)) ". We denote U (R). it as f 2 CF We denote by CbU (R; RF ) the space of uniformly continuous functions from R ! RF that are bounded. (F )

U (R). Then ! 1 (f; ) < +1, any Proposition 14.13 ([11]). Let f 2 CF > 0.

Proposition 14.14 ([11]). It holds (F )

(i) ! 1 (f; ) is nonnegative and nondecreasing in RF . (ii) lim

(F )

#0

> 0, any f : R !

(F )

U ! 1 (f; ) = ! 1 (f; 0) = 0, i¤ f 2 CF (R).

(F )

(iii) ! 1 (f; RF .

1

+

(F )

(iv) ! 1 (f; n ) (F )

2)

(F )

! 1 (f; (F )

n! 1 (f; ),

1)

(F )

+ ! 1 (f;

2 ),

1; 2

> 0, any f : R !

> 0, n 2 N, any f : R ! RF .

(F )

(F )

(v) ! 1 (f; ) d e! 1 (f; ) ( + 1)! 1 (f; ), > 0, d e is the ceiling of the number, any f : R ! RF . (F )

(vi) ! 1 (f

g; )

(F )

(F )

! 1 (f; ) + ! 1 (g; ),

> 0, where

> 0, any f; g : R ! RF .

246

14. Fuzzy Approximation by Fuzzy Convolution Operators (F )

U (vii) ! 1 (f; ) is continuous on R+ , for f 2 CF (R).

It follows a very important and useful representation result for moduli of continuity. Proposition 14.15. Let f : R ! RF be a fuzzy real number valued func(F ) (r) (r) tion. Assume that ! 1 (f; ), ! 1 (f ; ), ! 1 (f+ ; ) are …nite for > 0. Then it holds (F )

! 1 (f; ) = sup max ! 1 (f

(r)

r2[0;1]

(r)

; ); ! 1 (f+ ; ) :

(14.25)

Formula (14.25) is valid also when f and the moduli of continuity are de…ned over [a; b] R. Proof. Let x; y 2 R : jx D(f (x); f (y))

yj

. We have from [17], Section 2 that (r)

(r)

=

(f (y))

sup max j(f (x))

r2[0;1]

sup max ! 1 (f

(r)

r2[0;1]

Thus

(F )

! 1 (f; )

(r)

j; j(f (x))+

(r)

; ); ! 1 (f+ ; ) :

sup max ! 1 (f

(r)

r2[0;1]

(r)

; ); ! 1 (f+ ; ) :

Next we observe for any r 2 [0; 1] and any x; y 2 R : jx (F )

! 1 (f; )

(r)

D(f (x); f (y))

(r)

(f (y))+ j

(r)

j(f (x))

(f (y))

yj (r)

j; j(f (x))+

that (r)

(f (y))+ j:

Therefore ! 1 (f

(r)

(r)

(F )

; ); ! 1 (f+ ; )

! 1 (f; );

8r 2 [0; 1]:

Clearly it holds sup max ! 1 (f r2[0;1]

(r)

(r)

; ); ! 1 (f+ ; )

(F )

! 1 (f; ):

We have established formula (14.25). Remark 14.16. (i) Let f : [a; b] ! RF be fuzzy continuous. From [17], Section 2 we have D(f (t); o~) = sup max jf r2[0;1]

(r)

(r)

(t)j; jf+ (t)j ;

8t 2 [a; b]:

(14.26)

Since f is fuzzy bounded we have that 9M > 0 such that D(f (t); o~) M , 8t 2 [a; b]. Since D(f (t); o~) is continuous in t 2 [a; b] for some t 2 [a; b] we have that D(f (t ); o~) = sup D(f (t); o~): t2[a;b]

14.2 Background

247

Notice that by the principle of iterated suprema we get sup D(f (t); o~)

=

t2[a;b]

sup max jf

(r)

(t)j; jf+ (t)j

sup max sup jf

(r)

(t)j; jf+ (t)j :

sup

t2[a;b] r2[0;1]

=

r2[0;1]

t2[a;b]

(r)

(r)

Clearly we …nd that sup D(f (t); o~) = sup max kf

t2[a;b]

(r)

r2[0;1]

(r)

k1 ; kf+ k1 :

(14.27)

We observe here easily that 8r 2 [0; 1], 8t 2 [a; b] we derive jf

(r)

(r)

D(f (t ); o~)

(t)j; jf+ (t)j

M;

so we can apply above the principle of iterated suprema. (r) (ii) Let f : R ! RF be fuzzy continuous, then clearly f are continuous U functions from R into R, 8r 2 [0; 1]. If f 2 CF (R); equivalently (by Proposi(F ) (F ) tion 14.14(ii)) we have lim #0 ! 1 (f; ) = ! 1 (f; 0) = 0. Thus by formula (r) (r) (14.25) we get lim #0 ! 1 (f ; ) = ! 1 (f ; 0) = 0, 8r 2 [0; 1], equivalently (r) (r) U (R) then f 2 CU (R), 8r 2 [0; 1]. f 2 CU (R). I.e. If f 2 CF We need the following Lemma 14.17 ([14]). Let f : R ! RF fuzzy continuous and bounded. Let g: J R ! R+ continuous and bounded, where J is an interval. Then f (x) g(x) is fuzzy continuous function 8x 2 J. (r) (r) Remark 14.18. Here r 2 [0; 1], xi , yi 2 R, i = 1; : : : ; m 2 N. Suppose that (r) (r) sup max(xi ; yi ) 2 R; for i = 1; : : : ; m: (14.28) r2[0;1]

We see that (r)

(r)

xi

(r)

yi

(r)

(r)

(r)

(r)

(r)

max(xi ; yi )

sup max(xi ; yi );

(r)

sup max(xi ; yi ):

(r)

max(xi ; yi )

r r

Hence m X

i=1 m X i=1

(r)

xi

(r)

yi

m X

i=1 m X i=1

(r)

(r)

(r)

(r)

sup max(xi ; yi ); r

sup max(xi ; yi ): r

248

14. Fuzzy Approximation by Fuzzy Convolution Operators

Consequently m X

max

(r) xi ;

i=1

m X

(r) yi

i=1

!

m X i=1

(r)

(r)

sup max(xi ; yi ): r

We conclude that m X

sup max r2[0;1]

i=1

(r) xi ;

m X

(r) yi

i=1

!

m X

(r)

(r)

sup max(xi ; yi ):

(14.29)

i=1 r2[0;1]

Inequality (14.29) is used in the proofs of the main results here. We denote by C N (R; RF ), N 1, the space of all N -times continuously fuzzy di¤erentiable functions from R into RF . We need the following Theorem 14.19 ([71]). Let f : [a; b] Let t 2 [a; b], 0 r 1. Clearly

R ! RF be H-fuzzy di¤ erentiable.

(r)

[f (t)]r = (f (t)) (r)

Then (f (t))

(r)

; (f (t))+

R:

(14.30)

are di¤ erentiable and (r) 0

[f 0r = ((f (t)) I.e. (f 0 )(r) = (f

(r)

) ; ((f (t))+ )0 :

(r) 0

);

8r 2 [0; 1]:

(14.31) (14.32)

Remark 14.20. Let f 2 C N (R; RF ), N 1. Then by Theorem 14.19 we obtain (r) (r) [f (i) (t)]r = ((f (t)) )(i) ; ((f (t))+ )(i) ; for i = 0; 1; 2; : : : ; N and in particular we have that (f

(i) (r)

)

= (f

(r) (i)

) ;

(14.33)

for any r 2 [0; 1].

14.3 Main results Let here f 2 CbU (R; BF ), and let ' 2 CB (R) of compact support a > 0, ' 0 and be such that Z 1 '(x u) du = 1; 8x 2 R; 1

[ a; a],

(14.34)

14.3 Main results

equivalently

Z

1

'(u) du = 1:

249

(14.35)

1

(See Example (ii).) (F ) By Proposition 14.13 we have that ! 1 (f; ) < +1, for any > 0. Let k 2 Z, x 2 R. We introduce the following fuzzy convolution type operators, (i) Z 1 u f k '(2k x u) du; (14.36) (Bk f )(x) := (F R) 2 1 (ii) (Lk f )(x) := (F R) where cfk (u)

:= 2

k

(F R)

Z

2

k

2

Z

1 1

cfk (u)

'(2k x

(u+1) k

f (t) dt = 2 ku

Z

2

(14.37)

k

f t+

0

(equality true by Theorems 14.7, 14.10), (iii) Z 1 f ( k f )(x) := (F R) k (u)

'(2k x

u) du;

u dt 2k (14.38)

(14.39)

1

where

f k (u)

(F R)

u) du;

:=

n X

wj

f

j=0

j u + k k 2 2 n

; n 2 N …xed; wj

0;

m X

wj = 1;

j=0

(iv) here additionally we assume that ' is even and de…ne Z 1 (Ak f )(x) := (F R) rkf (u) '(2k x u) du;

(14.40)

(14.41)

1

where rkf (u) := 2k

(F R)

Z

1

'(2k t

f (t)

u) dt:

(14.42)

1

We emphasize again that the above de…ned fuzzy operators Ak , Bk , Lk , are the fuzzy analogs of the real convolution operators Ak , Bk , Lk , k discussed in Section 14.1. Introduction, de…ned in an analogous way. We present our results k

Theorem 14.21. It holds D (Bk f; f ) and

(F )

! 1 (Bk f; )

(F )

!1 (F )

f;

! 1 (f; );

a ; 2k 8 > 0;

(14.43)

(14.44)

250

14. Fuzzy Approximation by Fuzzy Convolution Operators

that is Bk ful…lling the property of Fuzzy Global Smoothness preservation. Proof of inequality(14.43). We notice (by (14.15) & (14.34)) = D (F R)

D((Bk f )(x); f (x)) '(2k x

u) du; (F R)

Z

(by (2.9) of [17]) 2k x+a

2k x

a

=

D f

u 2k

Z

2k x+a

'(2k x

Z

'(2k x

f (x)

f

'(2k x

u) du

'(2k x

u); f (x)

u)D f

a

2k x+a

a

u 2k

a

2k x+a

2k x

2k x+a

2k x

2k x

Z

Z

u) du

u ; f (x) du 2k

u x du k 2 a u a a x notice 2k 2k 2k Z 2k x+a a (F ) '(2k x u)! 1 f; k du 2 2k x a ! Z 2k x+a a a (F ) (F ) = f; k ; '(2k x u) du ! 1 f; k = ! 1 2 2 k 2 x a (F )

'(2k x

u)! 1

2k x

by

R 2k x+a 2k x a

'(2k x

f;

u) du = 1. That is proving D((Bk f )(x); f (x))

(F )

!1

f;

a ; 2k

hence (14.43) is true. Proof of inequality (14.44). Let x; y 2 R: jx that D((Bk f )(x); (Bk f )(y)) Z 1 = D (F R) f '(2k y

u 2k

1

u) du

= D (F R) (F R)

Z

Z

1

1

f

x

1

f

1

(call (u) :=

y

k

'(2 x

( u + 2k x) 2k ( u + 2k y) 2k

u + 2k x; (u) :=

yj

. Then we observe

u) du; (F R)

Z

1

f

1

u 2k

'( u + 2k x) du; '( u + 2k y) du u + 2k y)

14.3 Main results

251

Z

1 (u) = D (F R) f x '( (u)) du; k 2 1 Z 1 (u) f y (F R) '( (u)) du k 2 1 Z 1 w (14:22) '(w) dw; = D (F R) f x k 2 1 Z 1 w f y (F R) '(w) dw 2k 1 Z a w = D (F R) f x '(w) dw; 2k a Z a w '(w) dw f y (F R) 2k a (by (2.9) of [17]) Z a w w D f x '(w); f y '(w) dw k k 2 2 a Z a w w = '(w)D f x ;f y dw k k 2 2 a Z a Z a (F ) (F ) '(w)! 1 (f; jx yj) dw '(w)! 1 (f; ) dw a a Z a (14:35) (F ) (F ) (F ) = '(w) dw ! 1 (f; ) = 1 ! 1 (f; ) = ! 1 (f; ): a

We have proved that (F )

D((Bk f )(x); (Bk f )(y))

! 1 (f; )

which clearly implies (14.44). We continue with Theorem 14.22. It holds D ((Lk f ); f )

(F )

!1

f;

a+1 2k

;

(14.45)

and (F )

! 1 (Lk f; )

(F )

! 1 (f; );

8 > 0;

(14.46)

that is Lk ful…lling the property of Fuzzy Global Smoothness preservation.

252

14. Fuzzy Approximation by Fuzzy Convolution Operators

Proof. First we prove that cfk (u) is fuzzy continuous in u. We notice that ! Z 2 k Z 2 k um u D (F R) f t + k dt; (F R) f t + k dt 2 2 0 0 Z 2 k um u (by (2.9) of [17]) D f t + k ;f t + k dt 2 2 0 Z 2 k jum uj (F ) dt !1 f; 2k 0 jum uj (F ) = 2 k !1 ! 0; as um ! u; f; 2k where um ; u 2 R. That is proving fuzzy continuity of cfk (u). Also we see by (2.9) of [17] that ! Z 2 k Z 2 k u u D (F R) f t + k dt; o~ D f t + k ; o~ dt 2 2 0 0 k

M2

< +1;

where M > 0 such that D(f (x); o~) M , 8x 2 R. That is proving that cfk (u) is also fuzzy bounded. By Lemma 14.17 now we have that cfk (u) '(2k x u) 2 C(R; RF ) as a function of u. Again we get that (Lk f )(x) = (F R)

Z

a+2k x

cfk (u)

a+2k x

'(2k x

u)du;

(14.47)

and by [17], Section 2 (r)

D((Lk f )(x); f (x)) = sup max j(Lk f )

(x) f

(r)

(r)

(r)

(x)j; j(Lk f )+ (x) f+ (x)j :

r2[0;1]

(14.48) We see that [cfk (u)]r

=

2

(14:23)

"

=

=

"

k

k

2

(F R)

Z

2

0

Z

2

k

f

(r)

0

(f

ck

(r)

)

k

u f t + k dt 2

#r

u t + k dt; 2k 2

Z

(r) (f+ )

(u); ck

2

0

k

(r) f+

u t + k dt 2

#

(u) :

That is proving (cfk (u))

(r)

(f

= ck

(r)

)

(u);

8u 2 R:

(14.49)

14.3 Main results

Thus [(Lk f )(x)]r

"

=

(F R)

"Z

(14:23)

=

Z

Z

=

a+2k x

a+2k x

a+2k x

(cfk (u))

(f

a+2k x

a+2k x

(Lk (f

[(Lk f )(x)]r = [(Lk (f

'(2k x

(r)

ck

)

u) du

#r

u) du; #

u) du

(u)'(2k x

u) du;

(r)

(f ) ck + (u)'(2k x

(r)

That is

(r)

'(2k x

(r) (cfk (u))+ '(2k x

a+2k x

a+2k x

cfk (u)

a+2k x

a+2k x

a+2k x

"Z

=

Z

253

u) du

#

(r)

))(x); (Lk (f+ ))(x) :

(r)

(r)

))(x); (Lk (f+ ))(x)];

which gives (r)

((Lk f )(x))

= (Lk (f

(r)

))(x);

8r 2 [0; 1]:

(14.50)

Therefore D((Lk f )(x); f (x)) = sup max j(Lk (f

(r)

))(x)

f

(r)

(x)j;

r2[0;1] (r)

(14:9)

(r)

j(Lk (f+ ))(x)

f+ (x)j

sup max ! 1 f

(r)

;

r2[0;1] (r)

! 1 f+ ;

a+1 2k

(14:25)

=

a+1 2k

(F )

!1

f;

;

a+1 2k

:

We have established that D((Lk f )(x); f (x))

(F )

!1

f;

a+1 2k

:

(14.51)

From [17], Section 2 we get (14.45). Next we see that (F )

! 1 (Lk f; )

(14:25)

=

(r)

sup max ! 1 ((Lk f ) r2[0;1]

=

sup max ! 1 (Lk (f r2[0;1]

(14:11)

sup max ! 1 (f r2[0;1]

(r)

(r)

(r)

; ); ! 1 ((Lk f )+ ; ) (r)

); ); ! 1 (Lk (f+ ); ) (r)

; ); ! 1 (f+ ; )

(14:25)

=

(F )

! 1 (f; );

254

14. Fuzzy Approximation by Fuzzy Convolution Operators

proving (14.46). Next we give Theorem 14.23. It holds D ((

(F )

k f ); f )

!1

f;

a+1 2k

;

(14.52)

and (F )

!1 ( that is

k

k f;

(F )

)

! 1 (f; );

8 > 0;

(14.53)

ful…lling the property of Fuzzy Global Smoothness preservation. f k (u)

Proof. Here 0

D@

n X

wj

is fuzzy continuous in u. We observe that j um + k k 2 2 n

f

j=0

n X

(by (2.1) of [17])

;

n X

(F )

wj ! 1

f;

j=0

f

j=0

um j + k k 2 2 n

wj D f

j=0

n X

wj

1 u j A + k ; 2k 2 n

jum uj 2k

(F )

= !1

f;

u j + k k 2 2 n

;f

jum uj 2k

! 0;

as um ! u, where fum g, u 2 R. That is proving fuzzy continuity of Next we observe 0 1 n X u j D( fk (u); o~) = D @ wj f + k ; o~A k 2 2 n j=0 n X

wj D f

j=0

j u + k 2k 2 n

n X

; o~

f k (u).

wj M = M < +1;

j=0

where M > 0 such that D(f (x); o~) M , 8x 2 R. That is fk (u) is fuzzy bounded. By Lemma 14.17 we get again that fk (u) '(2k x u) is fuzzy continuous in u. We have from [17], Section 2, that D((

k f )(x); f (x))

= sup max (

(r) (x) kf )

r2[0;1]

f

(r)

(x)j; j(

(r) k f )+ (x)

(r)

f+ (x)j ; (14.54)

where (

k f )(x)

= (F R)

Z

a+2k x a+2k x

f k (u)

'(2k x

u) du:

(14.55)

14.3 Main results

255

We easily see 8r 2 [0; 1] that [

f r k (u)]

n X

=

wj f

j u + k 2k 2 n

(r)

j=0

(f k

=

(r)

)

(r)

; f+

j u + k 2k 2 n

(r)

(f+ ) (u) k

(u);

:

Therefore we obtain [(

r k f )(x)]

"

=

(F R)

"Z

(14:23)

=

Z

Z

=

a+2k x

a+2k x

( a+2k x

(r) f k k (u))+ '(2 x

a+2k x

(f k

a+2k x

a+2k x

)

u) du

u) du; u) du

(u)'(2k x

(f+ ) (u)'(2k x k

u) du

(r) k (f+ ))(x)

))(x); (

#

u) du;

(r)

a+2k x (r)

(r)

#r

'(2k x

(r) f '(2k x k (u))

( a+2k x

k (f

f k (u)

a+2k x

a+2k x

"Z

=

Z

#

:

That is, we have proved that ((

(r) k (f ))(x))

=(

k (f

(r)

))(x);

8r 2 [0; 1]:

(14.56)

Consequently we derive D((

k f )(x); f (x))

= sup max j(

k (f

(r)

))(x)

f

(r)

(x)j;

r2[0;1]

(r)

(r) k (f )+ ))(x)

j(

f+ (x)j

(14:10)

sup max ! 1 f

(r)

;

r2[0;1] (r)

! 1 f+ ;

a+1 2k

(14:25)

=

a+1 2k

(F )

!1

f;

;

a+1 2k

:

I.e. we get D((

k f )(x); f (x))

hence establishing (14.52).

(F )

!1

f;

a+1 2k

;

(14.57)

256

14. Fuzzy Approximation by Fuzzy Convolution Operators

Next we see that (F )

!1 (

k f;

)

=

(r) ; kf )

sup max ! 1 ((

); ! 1 ((

r2[0;1]

=

sup max ! 1 ((

k (f

(r)

(r) k f )+ ;

)); ); ! 1 ((

r2[0;1] (14:11)

sup max ! 1 (f

(r)

r2[0;1]

)

(r) k (f+ ));

(r)

)

(F )

; ); ! 1 (f+ ; ) = ! 1 (f; ):

That is proving (14.53). It follows Theorem 14.24. It holds (F )

D ((Ak f ); f )

!1

f;

a 2k 1

;

(14.58)

and (F )

(F )

! 1 (Ak f; )

! 1 (f; );

8 > 0;

(14.59)

that is Ak ful…lling the property of Fuzzy Global Smoothness preservation. Proof. Here we observe the following, '(2k x i¤ 2k x

a

u

u) 6= 0 i¤ a

2k x

u

a

2k x + a:

Let us …x u 2 [2k x a; 2k x + a]. Also we have '(2k t u) 6= 0 i¤ a 2k t u a i¤ u2ka t u+a . 2k u a u+a Let t 2 2k ; 2k , then t 2 x 2ka 1 ; x + 2ka 1 . If t > x + 2ka 1 then '(2k t u) = 0 and if t < x 2ka 1 then '(2k t u) = 0. Therefore we have rkf (u) = 2k

(F R)

Z

x+

x

a 2k 1

f (t)

'(2k t

u) dt:

(14.60)

'(2k x

u) du:

(14.61)

a 2k 1

Also we have (Ak f )(x) = (F R)

Z

a+2k x a+2k x

rkf (u)

We prove that rkf (u) is a fuzzy continuous and bounded function in u 2 [2k x a; 2k x + a]. Let um 2 [2k x a; 2k x + a] be such that um ! u, as m ! +1. Clearly '(2k t um ) ! '(2k t u), so '(2k t u) is continuous in u.

14.3 Main results

257

We notice that D(rkf (um ); rkf (u))

k

= 2 D (F R)

Z

x+

(F R)

f (t)

'(2k t

um ) dt;

k

!

a 2k 1

x

Z

a 2k 1

a 2k 1

x+

f (t)

'(2 t

u) dt

(by Lemma 14.17 f (t) '(2k t um ); f (t)

'(2k t

u)

a 2k 1

x

are fuzzy continuous functions in t, using also (2.9) of [17] to get) 2k

Z

x+

x

2k

Z

a 2k 1

'(2k t

D(f (t)

'(2k t

um ); f (t)

a 2k 1

u)) dt

(by Lemma 2.1 of [17]) x+

x

a 2k 1

a 2k 1

4aM ! 1 ('

j'(2k t

[ 3a;3a]

'(2k t

um )

; jum

u)jD(f (t); o~) dt

uj) ! 0;

as m ! +1;

where M > 0 such that D(f (x); o~) M , 8x 2 R. (Clearly here if t 2 x 2ka 1 ; x + 2ka 1 then 2k t u, 2k t um 2 [ 3a; 3a].) That is proving rkf (u) is fuzzy continuous in u 2 [2k x a; 2k x + a]. Next we see that 2k D (F R)

Z

x+

x

a 2k 1

(by (2.9) of [17])

'(2k t

f (t)

a 2k 1

k

2

Z

x+

x

= 2

k

Z

x+

x

a 2k 1

D(f (t)

a 2k 1

u) dt; o~ D(f (t)

'(2k t

u); o~) dt

a 2k 1

'(2k t

a 2k 1

(by Lemma 2.1 of [17])

!

2k

Z

u); f (t)

x+

x

a 2k 1

o~) dt

D(f (t); o~)'(2k t

u) dt

a 2k 1

4aM L < +1; where L > 0 be such that '(x) L, 8x 2 R. That is proving rkf (u) is fuzzy bounded in u 2 [2k x a; 2k x + a]. Again by Lemma 14.17 we get that rkf (u) '(2k x u) is fuzzy continuous in u 2 [2k x a; 2k x + a], and to repeat f (t) '(2k t u) is fuzzy continuous

258

14. Fuzzy Approximation by Fuzzy Convolution Operators

in t 2 x

a 2k

1

;x +

[rkf (u)]r

a 2k

. Furthermore we observe 8r 2 [0; 1] that

1

=

2

(14:23)

=

k

"Z Z

"

(F R)

x+

a 2k 1

x

a 2k 1

x+

a 2k 1

a 2k 1

(f

(r)

rk

)

a 2k 1

x+

f

#r

k

f (t)

'(2 t

(t)'(2k t

u) dt;

u) dt

a 2k 1

x

x

=

Z

(r)

(r) f+ (t)'(2k t

u) dt

#

(r)

(f+ )

(u); rk

(u) :

That is, we proved (rkf (u))

(f

(r)

(r)

= rk

)

(u):

(14.62)

Then

r

[(Ak f )(x)]

= (14:23)

=

"

(F R)

"Z

2k x+a a

2k x+a

=

"Z Z

2k x+a

(Ak (f

(f

(r)

rk

a

2k x a

=

(rkf (u))

2k x+a

2k x

rkf (u) (r)

k

'(2 x

'(2k x

(r) (rkf (u))+ '(2k x

2k x a

(14:62)

2k x+a

2k x a

2k x

Z

Z

)

(u)'(2k x

u) du; u) du

(r)

#

u) du;

(r)

(f ) rk + (u)'(2k x

u) du

#r

#

u) du

(r)

))(x); (Ak (f+ ))(x) :

We have established that (r)

((Ak f )(x))

= (Ak (f

(r)

))(x):

(14.63)

14.3 Main results

259

Next we see that D((Ak f )(x); f (x))

=

(r)

sup max j(Ak f )

(x)

f

(r)

(x)j;

r

(r)

(14:63)

=

j(Ak f )+ (x)

f+ (x)j

sup max j(Ak (f

(r)

))(x)

f

(r)

(x)j;

r

(r)

(14:25)

=

(r)

j(Ak (f+ ))(x) f+ (x)j n a a o (r) (r) sup max ! 1 f ; k+1 ; ! 1 f+ ; k 1 2 2 r a (F ) !1 f; k 1 : 2

Hence (F )

D((Ak f )(x); f (x))

!1

f;

a ; 2k 1

(14.64)

and from [17], Section 2 we get (14.58). Finally we treat (14:25)

(F )

! 1 (Ak f; )

=

(r)

sup max ! 1 ((Ak f ) r2[0;1]

(14:63)

=

sup max ! 1 (Ak (f

(r)

r2[0;1] (14:11)

sup max ! 1 (f r2[0;1]

(r)

(r)

; ); ! 1 ((Ak f )+ ; ) (r)

); ); ! 1 (Ak (f+ ); ) (r)

; ); ! 1 (f+ ; )

(14:25)

=

(F )

! 1 (f; );

so proving (14.59). In the following three theorems of high order fuzzy approximation to the fuzzy unit operator by the fuzzy operators Ak , Bk , Lk , k the scaling function ' will be as before in this Section 14.3. However now we take U (R) and f 2 for consideration f 2 C N (R; RF ), N 1, with f (N ) 2 CF (r) N Cb (R; RF ). Clearly here by Theorem 14.19, f 2 C (R) and of course (f

(r) (N ) (14:33)

)

=

(f

(N ) (r)

)

2 CU (R);

8r 2 [0; 1]:

We present Theorem 14.25. It holds D((Ak f )(x); f (x))

N X D(f (i) (x); o~) i=1

8k 2 Z, 8x 2 R.

i!

ai 2i(k

+ 1)

aN N !2N (k

(F )

1)

!1

a ; 2k 1 (14.65)

f (N ) ;

260

14. Fuzzy Approximation by Fuzzy Convolution Operators

Proof. We have that D((Ak f )(x); f (x))

=

(r)

sup max j((Ak f )(x)

f

(r)

(x)j;

r2[0;1]

(r)

(14:63)

=

(r)

j((Ak f )(x))+

f+ (x)j

sup max j(Ak (f

(r)

))(x)

f

(r)

(x)j;

r2[0;1]

(r)

(r)

j(Ak (f+ ))(x)

N (r) X j(f )(i) (x)j

(14:12)

sup max r2[0;1]

+

2i(k

i!

a N !2N (k

(r)

1)

1)

! 1 (f+ )(N ) ;

a 2k 1

N (i) X j(f )(r) (x)j

sup max r2[0;1]

+

ai 2i(k 1)

a aN (r) ! 1 (f )(N ) ; k 1 ; N (k 1) 2 N !2 N (r) (i) i X j(f+ ) (x)j a

+ =

i!

i=1

i=1 N

(14:33)

f+ (x)j

i!

i=1

ai 2i(k 1)

aN a (N ) ! 1 (f )(r) ; k 1 ; 2 N !2N (k 1) N (i) X j(f+ )(r) (x)j ai 2i(k

i!

i=1

1)

aN a (N ) ! 1 (f+ )(r) ; k 1 2 N !2N (k 1) # " N X (i) (r) (i) (r) sup max j(f ) (x)j; j(f+ ) (x)j

+ (14:29)

i=1

r2[0;1]

i

a i!2i(k

1) N

+

a N !2N (k

1)

sup max ! 1 (f r2[0;1]

a 2k 1 (by (14.25) and (14.26)) (N )

! 1 (f+ )(r) ;

=

N X D(f (i) (x); o~) i=1

+

aN N !2N (k

i!

(F )

1)

!1

ai 2i(k

1)

f (N ) ;

a : 2k 1

(N ) (r)

)

;

a 2k 1

;

14.3 Main results

261

We continue with Theorem 14.26. It holds D((Bk f )(x); f (x))

N X D(f (i) (x); o~) ai aN a (F ) + ! f (N ) ; k ; ki kN 1 i! 2 n!2 2 i=1 (14.66)

8k 2 Z, 8x 2 R. Proof. Using (14.13) and very similar to the proof of Theorem 14.25. Finally we give Theorem 14.27. It holds D((Lk f )(x); f (x)); D(( k f )(x); f (x))

N X D(f (i) (x); o~) (a + 1)i i! 2ki i=1

+

(14.67)

(a + 1)N (F ) a+1 ! f (N ) ; k N !2kN 1 2

;

8k 2 Z, 8x 2 R. Proof. Using (14.14) and very similar to the proof of Theorem 14.25. U Note 14.28. Since here f (N ) 2 CF (R), as k ! +1, we derive (F )

!1

f (N ) ;

a 2k 1

(F )

; !1

f (N ) ;

a+1 a (F ) ; !1 f (N ) ; k k 2 2

! 0;

Thus from (14.65), (14.66) and (14.67) we obtain that D((Ak f )(x); f (x)) ! 0, D((Bk f )(x); f (x)) ! 0, D((Lk f )(x); f (x)) ! 0, and D(( k f )(x); f (x)) ! 0, pointwise with rates.

15 DEGREE OF APPROXIMATION OF FUZZY NEURAL NETWORK OPERATORS, UNIVARIATE CASE

In this chapter we study the rate of convergence to the unit operator of very speci…c well described univariate Fuzzy neural network operators of Cardaliaguet–Euvrard and “Squashing” types. These Fuzzy operators arise in a very natural and common way among Fuzzy neural networks. The rates are given through Jackson type inequalities involving the Fuzzy modulus of continuity of the engaged Fuzzy valued function or its derivative in the Fuzzy sense. Also several interesting results in Fuzzy real analysis are presented to be used in the proofs of the main results. This chapter is based on [11].

15.1 Background Let f : R ! RF be a uniformly continuous Fuzzy real valued function. For each n 2 N, the neural network we deal with here has the following structure: it is a three-layer feedforward network with one hidden layer. It has one input and one output unit. The hidden layer has (2n2 +1) processing units. To each pair of connecting units (input to each processing unit) we assign the same weight n1 , 0 < < 1. The threshold values nk are one for each processing unit k. G.A. Anastassiou: Fuzzy Mathematics: Approximation The., STUDFUZZ 251, pp. 263–288. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

264

15. Approximation of Fuzzy Neural Network Operators, Univariate Case

The activation function b (or S) is the same for each processing unit. 1 The Fuzzy weights associated with the output unit are f nk , one R1 R 1I ( ) n for each processing unit k, where I = 1 b(x)dx (or I = 1 S(x)dx), “ " denotes the scalar Fuzzy multiplication. The above fully described neural networks give rise to some associated completely described Fuzzy neural network operators of Cardaliaguet– Euvrard and “Squashing" types. We study here thoroughly the Fuzzy pointwise convergence of these operators to the unit operator, we give also some Lp , p 1 analogs; see Theorems 15.17, 15.19, 15.24, 15.26 and Corollaries 15.21 and 15.22. This is done with rates through Jackson type inequalities involving Fuzzy moduli of continuity of the engaged Fuzzy functions. On the way to establish these results we produce some new results on Fuzzy Real Analysis, especially see Theorem 15.14, where we show a Fuzzy Taylor’s formula with Fuzzy integral remainder. The real ordinary theory of the above mentioned operators was presented earlier in [6] and [47]. And, of course, this chapter is motivated from there. The monumental revolutionizing work of L. Zadeh [103] is the foundation of this chapter, as well as another strong motivation. Fuzziness in Computer Science and Engineering is one of the main trends today. This way of quantitative approach over Fuzzy neural networks appeared recently in the literature. It determines the rates of convergence precisely in a natural quantitative manner through very tight inequalities using the measurement of smoothness of the engaged Fuzzy functions. As in Remark 4.4 ([31]) one can show easily that a sequence of operators of the form Ln (f )(x) :=

n X

f (xkn )

wn;k (x);

k=0

n 2 N;

P ( denotes the fuzzy summation) where f : R ! RF , xkn 2 R, wn;k (x) real valued weights, are linear over R, i.e., Ln (

f

g)(x) =

Ln (f )(x)

Ln (g)(x);

8 ; 2 R, any x 2 R; f; g : R ! RF . (Proof based on Lemma 4.1(iv) of [31].)

15.2 Basic Properties

265

15.2 Basic Properties We need the following De…nition 15.1. Let f : R ! RF be a fuzzy real number valued function. We de…ne the (…rst) fuzzy modulus of continuity of f by (F )

! 1 (f; ) :=

sup

D(f (x); f (y));

> 0:

(15.1)

x;y2R jx yj

De…nition 15.2. Let f : R ! RF . If D(f (x); o~) M , 8x 2 R, M > 0, we call f a bounded fuzzy real number valued function. De…nition 15.3. Let f : R ! RF . We say that f is continuous at a 2 R if whenever xn ! a, then D(f (xn ); f (a)) ! 0. If f is continuous for every a 2 R, then we call f a continuous fuzzy real number valued function. We denote it as f 2 CF (R). Remark 15.4. Let f be bounded from R into RF . Then we observe that (F )

! 1 (f; ) :=

sup D(f (x); f (y)) x;y2R jx

=

yj

sup

D(f (x)

o~; f (y)

o~)

x;y2R jx yj

sup (D(f (x); o~) + D(f (y); o~))

2M:

x;y2R jx yj (F )

That is, ! 1 (f; ) < +1. De…nition 15.5. Let f : R ! RF . We call f a uniformly continuous fuzzy real number valued function, i¤ for any " > 0 there exists > 0: whenever jx yj ; x; y 2 R, implies that D(f (x); f (y)) ". We denote it as U f 2 CF (R). (F )

U (R). Then ! 1 (f; ) < +1, any Proposition 15.6. Let f 2 CF

> 0.

Proof. Let "0 > 0 be arbitrary but …xed. Then there exists 0 > 0: jx yj (F ) "0 < +1. That is ! 1 (f; 0 ) "0 < +1. 0 implies D(f (x); f (y)) Let now > 0 arbitrary, x; y 2 R such that jx yj . Choose n 2 N: n 0 and set xi := x + ni (y x), 0 i n. Then D(f (x); f (y)) = D f (x)

n 1 X

f (xi );

k=1

n 1 X

f (xi )

f (y)

k=1

D(f (x); f (x1 )) + D(f (x1 ); f (x2 )) + +D(f (xn

1 ); f (y))

(F )

n! 1 (f;

0)

!

n"0 < +1;

266

15. Approximation of Fuzzy Neural Network Operators, Univariate Case (F )

1 since jxi xi+1 j = n1 jx yj i n. Therefore ! 1 (f; ) 0, 0 n n"0 < +1. B Denote f : R ! RF which is bounded and continuous, as f 2 CF (R).

Proposition 15.7. It holds (F )

(i) ! 1 (f; ) is nonnegative and nondecreasing in RF . (F )

> 0, any f : R !

(F )

U (ii) lim ! 1 (f; ) = ! 1 (f; 0) = 0, i¤ f 2 CF (R). #0

(F )

(iii) ! 1 (f; RF .

1

+

(F )

(F )

2)

! 1 (f; (F )

(iv) ! 1 (f; n )

n! 1 (f; ),

(F )

1)

(F )

+ ! 1 (f;

2 ),

1; 2

> 0, any f : R !

> 0, n 2 N, any f : R ! RF .

(F )

(F )

(v) ! 1 (f; ) d e! 1 (f; ) ( + 1)! 1 (f; ), > 0, d e is the ceiling of the number, any f : R ! RF . (F )

(vi) ! 1 (f

g; )

(F )

(F )

! 1 (f; ) + ! 1 (g; ),

> 0, where

> 0, any f; g : R ! RF .

(F )

U (vii) ! 1 (f; ) is continuous on R+ , for f 2 CF (R).

Proof. (i) is obvious. (F ) (ii) Clearly ! 1 (f; 0) = 0. (F ) (F ) ()) Let ! 1 (f; ) ! 0 as # 0. Then 8" > 0 9 > 0, ! 1 (f; ) ". I.e., U for any x; y 2 R: jx yj we get D(f (x); f (y)) ". That is, f 2 CF (R). U (() Let f 2 CF (R). Then 8" > 0 9 > 0: whenever jx yj ; x; y 2 R, (F ) it implies D(f (x); f (y)) ". I.e., 8" > 0 9 > 0: ! 1 (f; ) ". That is, (F ) ! 1 (f; ) ! 0 as # 0. (iii) Let x1 ; x2 2 R be such that jx1 x2 j 1 + 2 . Then there exists x 2 R: jx x1 j x2 j 1 and jx 2. We have D(f (x1 ); f (x2 ))

= D(f (x1 ) f (x); f (x2 ) f (x)) D(f (x1 ); f (x)) + D(f (x); f (x2 )) (F )

(F )

! 1 (f; jx1

(F ) ! 1 (f; 1 )

xj) + ! 1 (f; jx +

(F ) ! 1 (f; 2 ):

x2 j)

Therefore (iii) is true. (iv) and (v) are obvious. (vi) Notice that D(f (x)

g(x); f (y)

That is (vi) is now clear.

g(y))

D(f (x); f (y)) + D(g(x); g(y)):

15.2 Basic Properties

267

(vii) For any f : R ! RF it holds by (iii) that (F )

j! 1 (f;

1

+

2)

(F )

! 1 (f;

(F )

1 )j

! 1 (f;

(F )

U Let now f 2 CF (R), then by (ii) lim ! 1 (f;

2)

2 #0

(F ) ! 1 (f;

2 ):

= 0. That is proving the

continuity of ) on R+ . We mention the following fundamental theorem of Fuzzy calculus Theorem 15.8 ([52]). If f : [a; b] ! RF is di¤ erentiable on [a; b], then f 0 (x) is (FH)-integrable over [a; b] and Z s f 0 (x)dx; for any s t; s; t 2 [a; b]: f (s) = f (t) (F H) t

The Fuzzy–Henstock integral (F H)

R

is de…ned in [52], De…nition 2.1.

Note. In Theorem 15.8 when s < t the formula is invalid! Since fuzzy real numbers correspond to closed intervals etc. We need Corollary 15.9. Let f : [a; b] ! RF be fuzzy di¤ erentiable on [a; b], and the fuzzy derivative f 0 : [a; b] ! RF is assumed to be fuzzy continuous. Then it holds Z s f (s) = f (a)

f 0 (t)dt;

(F R)

a

for any s 2 [a; b]:

Proof. By Corollary 13.2 of [66], p. 644 we get that (F R) s b exists in RF . Clearly we have Z s Z s (F R) f 0 (t)dt = (F H) f 0 (t)dt: a

Rs a

f 0 (t)dt, a

a

By Theorem 3.6 of [52], f 0 (t) is (F H)-integrable over [a; b], and Z s Z s f (a) (F R) f 0 (t)dt = f (a) (F H) f 0 (t)dt = f (s): a

a

We need also Lemma 15.10. If f; g : [a; b] R ! RF are continuous, then the function F : [a; b] ! R+ de…ned by F (x) := D(f (x); g(x)) is continuous on [a; b] and ! Z Z b Z b b D (F R) f (u)du; (F R) g(u)du F (x)dx: a

a

a

Proof. Exactly the same as in Lemma 13.2 (ii) for 2 -periodic functions, see [66], p. 644.

268

15. Approximation of Fuzzy Neural Network Operators, Univariate Case

Lemma 15.11. Let f : [a; b] ! RF continuous, then D(f (x); o~) 8x 2 [a; b], M > 0, that is f is fuzzy bounded.

M,

Proof. Let xn ; x 2 [a; b] such that xn ! x, as n ! +1, then D(f (xn ); f (x)) ! 0, by continuity of f . But (r)

(r)

D(f (xn ); f (x)) = sup maxfj(f (xn ))

(f (x))

r2[0;1] (r)

(r)

j; j(f (xn ))+

(r)

(f (x))+ jg:

(r)

Hence j(f (xn )) (f (x)) j ! 0, all 0 r 1, as n ! +1. That is, (r) (r) (r) (f (xn )) ! (f (x)) , all 0 r 1, as n ! +1. Hence (f ) 2 C([a; b]), (r) all 0 r 1. Consequently, (f ) are bounded on [a; b], all 0 r 1. Here (r) (r) D(f (x); o~) = sup maxfj(f (x)) j; j(f (x))+ jg: r2[0;1]

From basic Fuzzy theory we get that (0)

(f (x))

(1)

(f (x))+

(f (x))

(r)

(f (x))

(1)

(r)

(f (x))+ :

;

and (f (x))+

(0)

Thus (r)

j(f (x))

(0)

j

max(j(f (x))

(1)

j; jf (x))

j);

and (0)

(r)

all 0

r

(1)

max(j(f (x))+ j; j(f (x))+ j);

j(f (x))+ j 1. Therefore

(0)

(1)

maxfj(f (x)) j; j(f (x)) M; 8x 2 [a; b];

D(f (x); o~)

jg

for some M > 0. I. e. for all 0 r 1; M [a; b] () f (x) ; f (x) 2 R : F M M Lemma 15.12. Let f : [a; b] (F R)

Z

a

maxfkf

k1 ; kf

(1)

(r)

M; 8x 2

(0)

(f (x))

k1 g

R ! RF be continuous. Then

x

f (t)dt

is a continuous function in x 2 [a; b]:

Proof. By Corollary 13.2, p. 644 of [66], see also [52], f is (FR) integrable on [a; b] and on its closed subintervals. Using Theorem 2.5 of [52] and a property of D, and without loss of generality by assuming sn s, as

15.2 Basic Properties

269

sn ! s, n ! +1, we have Z sn Z s D (F R) f (t)dt; (F R) f (t)dt a a Z s Z sn Z s = D (F R) f (t)dt (F R) f (t)dt; (F R) f (t)dt a s a Z Z sn Z sn f (t)dt; (F R) f (t)dt; o~ = D (F R) = D (F R) Z

(Lemma15:10)

o~dt

s

s

s

sn

sn

D(f (t); o~)dt

s

Z

(Lemma15:11)

sn

M dt

M (sn

s) ! 0:

s

That is D (F R)

Z

sn

f (t)dt; (F R)

a

as sn ! s, n ! +1.

Z

s

f (t)dt

a

! 0;

Lemma 15.13. Let f 2 CF (R), r 2 N. Then the following integrals Z sr 1 Z s r 2 Z sr 1 (F R) f (sr )dsr ; (F R) f (sr )dsr dsr 1 ; a a a Z s Z s1 Z sr 2 Z sr 1 ; (F R) f (sr )dsr dsr 1 a

a

a

ds1 ;

a

are continuous functions in sr 1 , sr 2 ; : : : ; s, respectively. Here sr 1 , sr 2 ; : : : ; s a and all are real numbers. Proof. By Lemma 15.12. We present the following new interesting result, which is the Fuzzy Taylor’s formula. Theorem 15.14. Let T := [x0 ; x0 + ] R, with > 0. We assume that f (i) : T ! RF are di¤ erentiable for all i = 0; 1; : : : ; n 1, for any x 2 T . (I.e., there exist in RF the H-di¤ erences f (i) (x + h) f (i) (x), f (i) (x) f (i) (x h), i = 0; 1; : : : ; n 1 for all small 0 < h < . Furthermore there exist f (i+1) (x) 2 RF such that the limits in D-distance exist and f (i+1) (x) = lim+ h!0

f (i) (x + h) h

f (i) (x)

= lim

f (i) (x)

h!0+

f (i) (x h

h)

;

for all i = 0; 1; : : : ; n 1.) Also we assume that f (i) , i = 0; 1; : : : ; n are continuous on T in the fuzzy sense. Then for s a; s; a 2 T we obtain f (s)

= f (a)

f 0 (a) f (n

(s 1)

(a)

a)

f 00 (a)

(s a)n 1 (n 1)!

(s

a)2 2!

Rn (a; s);

270

15. Approximation of Fuzzy Neural Network Operators, Univariate Case

where Rn (a; s) := (F R)

Z

Z

s

Z

s1

1

f (n) (sn )dsn dsn

1

: : : ds1 :

a

a

a

sn

Here Rn (a; s) 2 CF (T ) as a function of s. Note. (1) This formula is invalid when s < a, as it is based on Theorem 3.6 of [52]. (2) This Fuzzy Taylor formula is also valid with Fuzzy–Henstock integral remainder where now we can drop the assumptions that f (i) , i = 0; 1; : : : ; n are fuzzy continuous on T . The proof is totally the same, and it is now based on Theorem 3.6 of [52]. Clearly again the remainder Rn (a; s) exists in RF . Proof (of Theorem 15.14). By Corollary 15.9 we have Z s f (s) = f (a) (F R) f 0 (t)dt; a

0

with f being (FR)-integrable on T , f 0 (t) = f 0 (a)

(F R)

Z

t

f 00 ( )d ;

a

with f 00 being (FR)-integrable on T . Therefore Z s Z s Z s Z t (F R) f 0 (t)dt = (F R) f 0 (a)dt (F R) (F R) f 00 ( )d a

= f 0 (a)

(s

a)

(F R)

Z

a s

a

Z

a

a

t

f 00 ( )d

dt:

a

Clearly the last double integral belongs to RF by Lemma 15.12. That is Z s Z t 0 f (s) = f (a) f (a) (s a) (F R) f 00 ( )d dt: a

a

But also next similarly we have 00

00

f ( ) = f (a)

(F R)

Z

f 000 (`)d`;

a

with f 000 being (FR)-integrable on T , and Z t Z f 00 ( )d = f 00 (a) (t a) (F R) a

a

Furthermore, Z s Z t f 00 ( )d a

t

dt

00

= f (a)

a

(F R)

Z

Z

f 000 (`)d` d :

a

s

a s

a

Z

(t

a)dt Z

a

t

Z

a

f 000 (`)d` d

dt:

dt

15.3 Main Results

271

Clearly the last triple integral belongs to RF by Lemma 15.12. Hence it holds f (s)

(s a)2 f 0 (a) (s a) f 00 (a) 2 Z s Z t Z f 000 (`)d` d dt: (F R)

= f (a)

a

a

a

The remainder of the last formula by Lemma 15.12 is a continuous function in s. Etc.

15.3 Main Results We need (see also [6], [47]) De…nition 15.15. A function b : R ! R is said to be bell-shaped if b belongs to L1 and its integral is nonzero, if it is nondecreasing on ( 1; a) and nonincreasing on [a; +1), where a belongs to R. In particular b(x) is a nonnegative number and at a b takes a global maximum; it is the center of the bell-shaped function. A bell-shaped function is said to be centered if its center is zero. The function b(x) may have jump discontinuities. In this article we consider only centered bell-shaped functions of compact support RT [ T; T ], T > 0. Denote I := T b(t)dt. Notice that I > 0. Examples 15.16.

(1) b(x) can be the characteristic function on [ 1; 1]. (2) b(x) can be the hat function on [ 1; 1], i.e., 8 1 x 0; < 1 + x; 1 x; 0 < x 1; b(x) = : 0; elsewhere.

Here we consider functions f : R ! RF that are either continuous and B U bounded, i.e., f 2 CF (R), or uniformly continuous, i.e., f 2 CF (R), both in the fuzzy sense.

In this chapter we study …rst the pointwise convergence with rates over the real line, to the fuzzy unit operator of the fuzzy univariate Cardaliaguet– Euvrard neural network operators, 2

(Fn (f ))(x) := fn (x) :=

n X

k= n2

f

k n

b n1

x In

k n

;

(15.2)

272

15. Approximation of Fuzzy Neural Network Operators, Univariate Case

where 0 < < 1 and x 2 R, n 2 N. The above are linear operators over R. These operators in the ordinary real case were thoroughly studied in [6], [47]. So as in [6], we consider without loss of generality and for simplicity that n max(T + jxj, T 1= ). In this case we have that n2 nx T n nx+T n n2 , and card(k) 1. Furthermore, it holds card(k) ! +1, as n ! +1. Set b := b(0), the maximum of b(x). Denote by [ ] the integral part of a number. Next we present the …rst main result. Theorem 15.17. Let x 2 R, T > 0, and n 2 N such that n T 1= ). Then [nx+T n ]

X

D(fn (x); f (x))

b(n1

k=dnx T n e

b I

D(f (x); o~) +

k n

x In 2T +

max(T + jxj,

1

1 n

(F )

!1

(15.3) f;

T n1

:

Proof. As in [6] we have that [nx+T n ]

X

b n1

x In

k=dnx T n e

k n

b I

2T +

1 n

:

(15.4)

Next we estimate

D(fn (x); f (x))

0

= D@

2

n X

k=

0

f

n2

b n1

k n

dnx T n e 1

= D@

X

k n

b n1

x In

f

k n

b n1

x In

2

k=[nx+T n ]+1 [nx+T n ]

X

= D@

[nx+T n ]

X

k=dnx T n e

b n

k n

f

k=dnx T n e

0

k n

f

k= n2 n X

x In

f

k n

1

b n

x In 1

x In

1

; f (x)A k n

k n

1

k n

k n

; f (x)A

1

; f (x)A

15.3 Main Results

273

(by b having compact support [ T; T ]) 0

[nx+T n ]

X

= D@

k=dnx T n e [nx+T n ]

X

f (x)

b n1

k n

f

b n1

k=dnx T n e [nx+T n ]

X

f (x)

k=dnx T n e

0

[nx+T n ]

X

+ D @f (x)

=:

b n

1

x In

k n

x In

k n

b n

1

k=dnx T n e

;

k n

x In

; f (x) 1

x In

A

1

k n

; f (x)A

where

:=

0

D@

[nx+T n ]

X

f

k=dnx T n e [nx+T n ]

X

f (x)

k n

b n1

b n1

x In

k=dnx T n e

Here we observe that 0

D @f (x)

[nx+T n ]

X

b n

1

x In

k=dnx T n e [nx+T n ]

X

b n1

k=dnx T n e

x In

The last is true by Lemma 2.2 of [31]. Next we see that 0 [nx+T n ] X k = D@ f n k=dnx T n e

[nx+T n ]

X

k=dnx T n e

f (x)

b n

1

k n

x In

;

1

k n

A:

; f (x)

k n

b n1

k n

x In

(15.5)

1

1A

1 D(f (x); o~):

k n

x In k n

1 A

;

(15.6)

274

15. Approximation of Fuzzy Neural Network Operators, Univariate Case [nx+T n ]

X

k=dnx T n e

b n1

f (x)

[nx+T n ]

b n1

k n

D f

X

b n1

x In

k n

[nx+T n ]

b n1

x In

k n

=

k=dnx T n e

X

k=dnx T n e

0 @

[nx+T n ]

X

b n1

That is, we have that 0 [nx+T n X @

]

b n1

k=dnx T n e

x In

k n

D f

(F )

!1

f;

) A ! (F f; 1

Finally, applying (15.5), (15.6) and (15.7) into [nx+T n ]

D(fn (x); f (x))

X

b n1

k=dnx T n e

0

[nx+T n ]

X

+@

b n

k=dnx T n e

x

x In

T n1

T

:

:

n1

(15.7)

, we obtain

k n

x In 1

k n

) A ! (F f; 1

1

k n

; f (x)

1

k n

x In

k=dnx T n e

;

!

k n

x In

k n

x In

1 k n

Using now (15.4) into (15.8) we get (15.3).

D(f (x); o~) (15.8)

1

) A ! (F f; 1

T n1

:

Remark 15.18. By Lemma 2.1 of [9], p. 64 we obtain that [nx+T n ]

lim

n!+1

X

b n1

k=dnx T n e

x In (F )

U any x 2 R. Let f 2 CF (R), then lim ! 1 n!+1

k n

f; n1T

1 = 0;

= 0. Therefore from

(15.3) we get

lim D(fn (x); f (x)) = 0:

n!+1

That is, fn (x) ! f (x), pointwise with rates, as n ! +1, where x 2 R. NU We denote by CF (R) := ff : R ! RF , such that all the derivatives (i) f : R ! RF , i = 0; 1; : : : ; N exist and all are uniformly continuous in the

15.3 Main Results

275

NB fuzzy senseg, N 2 N. Also denote by CF (R) := ff : R ! RF , such that all f (i) : R ! RF , i = 0; 1; : : : ; N exist and all are bounded and continuous in the fuzzy senseg, N 2 N. Now we present the second main result.

Theorem 15.19. Let x 2 R, T > 0, n 2 N such that n NU NB T 1= ). Let f 2 CF (R), or f 2 CF (R), N 2 N. Then [nx+T n ]

X

D(fn (x); f (x))

b n1

k=dnx T n e (F )

T n1

+ !1

f;

b + I

1 2T + n

k n

x In

(F )

+ !1 8
j=1

2N T N b N !In(1 )N

(F )

(F )

2T +

f (N ) ;

+3! 1

f;

f (j) ; 1 n

T n1

D(f (x); o~)

T n1

2j T j j!n(1 )j

D(f (j) (x); o~) + ! 1 +

1

max(T + jxj,

T n1

(15.9)

D(f (N ) (x); o~)

:

Remark 15.20. By Lemma 2.1 of [9], p. 64 we obtain that [nx+T n ]

X

lim

n!+1

b n1

k=dnx T n e

x In

k n

1 = 0;

NU any x 2 R. Let f 2 CF (R), then (F )

lim ! 1

n!+1

f (j) ;

T

= 0;

n1

j = 0; 1; : : : ; N:

Therefore from (15.9) we get lim D(fn (x); f (x)) = 0:

n!+1

That is, fn (x) ! f (x), pointwise with rates, as n ! +1, x 2 R. Proof (of Theorem 15.19). Because b is of compact support [ T; T ] we have that [nx+T n ]

fn (x) =

X

k=dnx T n e

f

k n

b n1

x In

k n

:

276

15. Approximation of Fuzzy Neural Network Operators, Univariate Case

Clearly the terms in the sum (15.2) are nonzero i¤ n1 T n1

k n

x

k n

x

T , i¤

T : n1

k Hence in here x n1T n , x 2 R. Using the fuzzy Taylor formula (Theorem 15.14) we have

f

k n

=

N 1 X

k n

T

f (j) x

n1

j=0

j

T n1

x+ j!

RN

k T ; n1 n

x

;

where RN

x

T

;

n1

k n

:

Z

= (F R)

x

Z

sN

T n1

1

f

x

Z

k n

T n1

x

(N )

T n1

s1

(sN )dsN

!

dsN

!

1

ds1 :

Then f

k n

b n1

k n

x In

N 1 X

=

T

f (j) x

RN

b n1

x In

T n1

T n1

;

k n

k n

Therefore we have [nx+T n ]

X

k=dnx T n e [nx+T n ]

=

b n1

k n

f X

N 1 X

f

(j)

k=dnx T n e j=0

b n1

b n

x In 1

x In

k n

x In x

k n

T n1

[nx+T n ]

X

k n

k n

k=dnx T n e

:

j

k n

x In x

x+ j!

n1

j=0

b n1

k n

RN

x

x+ j!

T n1

T k ; n1 n

j

:

15.3 Main Results

277

That is,

fn (x)

N 1 X

=

[nx+T n ]

X

x In [nx+T n ] X

j

T n1

x+ j!

n1

j=0 k=dnx T n e

b n1

k n

T

f (j) x k n

RN

T

x

;

n1

k=dnx T n e

b n1

k n

x In

k n

:

Next we estimate

D(fn (x); f (x)) b n

1

x In

0

+ D @f

0

[nx+T n ]

D @fn (x); !

k n

X

[nx+T n ]

X

n1

T n1

x

k=dnx T n e

T

x

f

b n1

k=dnx T n e

T n1

k n

x In

T 1 +D f x ; f (x) n1 0 N 1 [nx+T n ] (by Lemma 1:2) X X @ D f (j) x j=1 k=dnx T n e & (properties of D) f

x

k n

x+ j!

b n1

D f N X1

X

j=1 k=dnx T n e

X

b n1

k=dnx T n e

T n1

RN

x

k=dnx T n e

(F )

x+ j!

X

[nx+T n ]

f;

; o~ + ! 1 k n

T n1

[nx+T n ]

k n

x In

; o~ +

T n1

[nx+T n ]

b n1

!

k n

x In x

j

T n1

;

j

x In

T n1

b n1

x In

k n

k n

1

T n1

;

k n

278

15. Approximation of Fuzzy Neural Network Operators, Univariate Case

T n1

D f (j) x

; o~

[nx+T n ]

X

+

b n1

k=dnx T n e [nx+T n ]

X

+

b n1

(F )

N X1

X

2T +

b n1

[nx+T n ]

X

b n1

(F )

f;

; o~

f;

T n1

f (j) ;

T n1

(F )

T n1

(F )

D(f (j) (x); o~) + ! 1 k n

D RN

k n

x In

k=dnx T n e

+ !1

k n

D(f (x); o~) + ! 1

1

1 n

x In

k=dnx T n e

+

;

n1

n1

[nx+T n ]

+

T

x

T

f;

2j T j b j!n(1 )j I

j=1

D RN

k n

x In

k=dnx T n e

+ !1

k n

x In

T

x

;

n1

k n

; o~

(F )

1

D(f (x); o~) + ! 1

f;

T n1

=: ( ):

In the following we work on

D RN

T

x

;

n1

= D RN

+D f

;

n1

k n

f (N ) x k n

T

T

k ; n

n1 x

T n1

;f

(N )

k n

x

k n

T n1 N

x + n1T N!

n1

x

(N )

; o~

T

x

f (N ) x

D RN

k n

N

;

! k n

T n1

x + n1T N!

x + n1T N!

N

!

; o~ :

x + n1T N!

N

!

15.3 Main Results

279

k n

N

Hence

D RN

T

x

D RN +

T

x

2N T N N !nN (1

k n

;

n1

n1

; o~ k ; n

;f

(N )

T

x

n1

(F )

D(f (N ) (x); o~) + ! 1

)

x + n1T N!

f (N ) ;

T n1

:

(15.10)

Finally we estimate

D RN

T

x

n1 Z = D (F R)

k ; n

dsN

=

Z

T n1

Z

T n1

x

n1 Z

T n1

k n T n1

x

sN

T n1

x

Z

1

f (N ) (sN )dsN Z

s1 T n1

x

N

x + n1T N!

k n

T

s1

x

) ds1 ) ;

1

f (N ) x dsN

;f

k n

x

(N )

sN

x

!

1

T n1

1dsN

) ds1

1

D (F R)

Z

x

dsN

Z

k n T n1

T n1

x

) ds1 ; (F R)

1

Z

k n T n1

x

Z

sN

x

1

f T n1

(N )

x

Z

s1

T n1

sN

f

x

Z

(N )

(sN )dsN

!

!

T n1

s1

x

dsN

1

T 1

!n

dsN

1

ds1

!

!

!

!

280

15. Approximation of Fuzzy Neural Network Operators, Univariate Case

Z

(by Lemmas 15:10;15:13)

Z

k n T n1

x

Z

Z

k n T n1

x

T

dsN 1 Z nk x

) ds1 Z s1

T n1

x

(F ) !1

f

2N T N N !nN (1

(N )

Z

T n1

x

Z

T n1

x+

(F )

)

sN

x

k ; n

sN

(F ) !1

T n1

1

1dsN

T n1

k n

T 2T n1

!

dsN f

ds1

1

(N )

dsN

1

T n1

; sN

x+

T

dsN

n1

ds1

1

x + n1T N!

n1

f (N ) ;

!1

1

sN

x

dsN

n1

s1

x

T n1

x

D f (N ) (sN ); f (N ) x

Z

s1

N (F )

f (N ) ;

!1

2T n1

:

That is, we get that D RN

T

x

k ; n

n1

2N T N N !nN (1

;f

(F )

)

(N )

f (N ) ;

!1

x

n1

2T n1

N

x + n1T N!

k n

T

:

!

(15.11)

Consequently we obtain D RN

k T ; n1 n

x

2N T N N !nN (1

; o~

(15.12) (F )

)

D(f (N ) (x); o~) + 3! 1

f (N ) ;

T n1

:

Therefore, by (15.12) and (15.4), we have [nx+T n ]

X

k=dnx T n e

b n1

2N T N b N !InN (1

x In )

k n

2T +

D RN 1 n

x

T k ; n1 n

; o~ (F )

D(f (N ) (x); o~) + 3! 1

(15.13) f (N ) ;

T n1

At last using (15.13) into ( ) we have completed the proof of the theorem. Corollary 15.21 (to Theorem 15.17). Let b(x) be a centered bell-shaped continuous function on R of compact support [ T; T ], T > 0. Let x 2

:

!

15.3 Main Results

[ T ; T ], T > 0, and n 2 N be such that n 0 < < 1. Consider p 1. Then kD(fn (x); f (x))kp;[ [nx+T n ]

X

b n1

+

b I

2T +

k n

x In

k=dnx T n e

max(T + T ; T

(kD(f (x); o~)k1;[

T ;T ]

281 1=

),

T ;T ] )

1 p;[ T ;T ]

1 n

21=p T

1=p

(F )

!1

f;

T n1

;

(15.14)

RT U where I := b(t)dt. From (15.14), when f 2 CF (R), we get the Lp T convergence of fn to f with rates. Proof. Since f is fuzzy continuous on [ T ; T ] it is fuzzy bounded there (Lemma 15.11), therefore kD(f (x); o~)k1;[

T ;T ]

< +1:

So from Theorem 15.17, we have that D(fn (x); f (x))

kD(f (x); o~)k1;[ [nx+T n ]

X

T ;T ]

b n1

k=dnx T n e

+

b I

2T +

1 n

k n

x In (F )

!1

f;

1

T n1

:

(15.15)

Inequality (15.14) now comes by integration of (15.15) and the properties of Lp norm. As in [9], p. 75, using the bounded convergence theorem we get that [nx+T n ]

X

lim

n!1

k=dnx T n e

b n1

k n

x In

1

= 0: p;[ T ;T ]

Corollary 15.22 (to Theorem 15.19). Let b(x) be a centered bell-shaped continuous function on R of compact support [ T; T ], T > 0. Let x 2 [ T ; T ], T > 0, and n 2 N be such that n

max(T + T ; T

1=

);

0<

< 1:

282

15. Approximation of Fuzzy Neural Network Operators, Univariate Case

Consider p

1. Then

kD(fn (x); f (x))kp;[

T ;T ] (F )

kD(f (x); o~)k1;[ [nx+T n ]

X

T ;T ]

b n1

f;

T

(F )

T n1

1

b 21=p T 1=p + I

n1

2N T N b N !In(1 )N

+ 3! 1

f;

p;[ T ;T ]

T ;T ]

2T +

f (N ) ;

1 2T + n

(F )

kD(f (j) (x); o~)kp;[ +

k n

x In

k=dnx T n e (F ) + !1

+ !1

T n1

f (j) ;

+ !1

T n1

8
n kD(f (N ) (x); o~)kp;[

1 n

21=p T

1=p

j=1

2j T j j!n(1 )j

21=p T

1=p

T ;T ]

:

(15.16)

NU When f 2 CF (R), from inequality (15.16) we obtain the Lp convergence of fn to f with rates.

Proof. Similar to Corollary 15.21. We need (see also [6], [47]) De…nition 15.23. Let the nonnegative function S : R ! R, S has compact support [ T; T ], T > 0, and is nondecreasing there and it can be continuous only on either ( 1; T ] or [ T; T ]. S can have jump discontinuities. We call S the “squashing function”. U (R)), or Let f : R ! RF be either fuzzy uniformly continuous (f 2 CF B fuzzy continuous and bounded (f 2 CF (R)). Assume that Z T I := S(t)dt > 0: T

Clearly max S(x) = S(T ):

x2[ T;T ]

For x 2 R we de…ne the “univariate fuzzy squashing operator” 2

(Gn (f ))(x) :=

n X

k= n2

f

k n

S n1

x I n

k n

;

(15.17)

0 < < 1 and n 2 N: n max(T + jxj; T 1= ). The above neural network operators are also linear operators over R. These operators in the ordinary real case were thoroughly studied in [6], [47].

15.3 Main Results

283

It is clear to see again that [nx+T n ]

X

(Gn (f ))(x) =

S n1

k n

f

k=dnx T n e

k n

x I n

:

(15.18)

Here we study the pointwise convergence with rates of (Gn (f ))(x) ! f (x), as n ! +1, x 2 R. Theorem 15.24. Under the above terms and assumptions we obtain [nx+T n ]

X

D(Gn (f )(x); f (x))

S n1

k=dnx T n e

+

S(T ) I

2T +

k n

x I n

1 n

(F )

!1

1 D(f (x); o~) f;

T n1

:

(15.19)

Proof. Notice that [nx+T n ]

X

1

(2T n + 1):

(15.20)

k=dnx T n e

We have that D((Gn (f ))(x); f (x)) = 0 [nx+T n ] X k @ D f n k=dnx T n e 0 [nx+T n ] X k D@ f n

S n1

x I n

S n1

x I n

x I n

k n

k=dnx T n e

[nx+T n ]

f (x) 0

S n

1

k=dnx T n e

+ D @f (x) [nx+T n ]

X

X

k=dnx T n e

[nx+T n ]

X

S n

k=dnx T n e

S n1

x I n

k n

1

x I n

1

k n

k n

; f (x)A ;

1 A k n

1

; f (x)A

1 D(f (x); o~) +

=: ( );

284

15. Approximation of Fuzzy Neural Network Operators, Univariate Case

where

0

[nx+T n ]

X

D@

:=

S n1

k n

f

k=dnx T n e

[nx+T n ]

X

f (x)

S n1

k=dnx T n e

k n

x I n

; 1

k n

x I n

A:

Thus [nx+T n ]

X

k=dnx T n e

S n1

f (x)

[nx+T n ]

S n1

x I n

k n

[nx+T n ]

S n1

x I n

k n

k=dnx T n e

X

k=dnx T n e

0 @

[nx+T n ]

X

S n1

k=dnx T n e

S(T ) I n

(F ) !1

f;

(F )

!1

0

T

;

f;

; f (x) k n

1

x

T n1 1

) A ! (F f; 1

[nx+T n ]

X

@

n1

k n

D f

k n

x I n

S(T ) (F ) T ! f; 1 I n 1 n

k n

x I n

!

k n

x I n

X

=

(15:20)

S n1

k n

D f

k=dnx T n e

1A

(2T n + 1):

I.e., S(T ) I

2T +

1 n

(F )

!1

f;

T n1

:

(15.21)

Using (15.21) into ( ) we have established (15.19). Remark 15.25. From Lemma 2.2, p. 79, [9], we have that [nx+T n ]

lim

n!+1

X

k=dnx T n e

S n1

x I n

k n

1 = 0:

(15.22)

U Let f 2 CF (R), then from (15.19) as n ! +1, we obtain the pointwise convergence with rates of (Gn (f )(x)) to f (x), where x 2 R.

15.3 Main Results

285

As a related …nal main result we give Theorem 15.26. Let x 2 R, T > 0, n 2 N such that n NU NB T 1= ). Let f 2 CF (R) or f 2 CF (R), N 2 N. Then [nx+T n ]

X

D(Gn (f )(x); f (x))

S n1

k=dnx T n e

S(T ) I (F )

+ !1

2T + f (j) ;

k n

x I n

1

T T (F ) f; 1 + !1 f; 1 n n 8
(F )

D(f (x); o~) + ! 1 +

max(T + jxj,

1 n T

+

n1

(F )

D(f (N ) (x); o~) + 3! 1

1 2N T N S(T ) 2T + (1 )N n N !I n T f (N ) ; 1 : (15.23) n

NU Note. When f 2 CF (R) and n ! +1 from (15.23) we get the pointwise convergence with rates of (Gn f )(x) ! f (x), x 2 R.

Proof (of Theorem 15.26). Again we see, that the terms in the sum (15.18) are nonzero i¤ n1 x nk T , i¤ T n1 Therefore here x n1T (Theorem 15.14) we have

f

k n

=

N 1 X

k n,

k n

x+ j!

n1

j=0

T : n1

x 2 R. Using again the fuzzy Taylor formula

T

f (j) x

k n

x

j

T n1

RN

T

x

n1

where RN (F R)

Z

x

k n T n1

Z

s1

x

T n1

T

x

;

n1 Z

sN

x

k n

1

f T n1

:= (N )

(sN )dsN

!

!

ds1 :

;

k n

;

286

15. Approximation of Fuzzy Neural Network Operators, Univariate Case

Thus [nx+T n ]

X

k=dnx T n e [nx+T n ]

N 1 X

X

=

S n1

k n

f

X

RN

T

x

n1

k=dnx T n e

;

j

T n1

x+ j!

n1

[nx+T n ]

k n

x I n

k n

T

f (j) x

k=dnx T n e j=0

S n1

k n

x I n

S n1

k n

k n

x I n

Hence

(Gn f )(x)

=

[nx+T n ]

N 1 X

X

f

(j)

x

x+ j!

n1

j=0 k=dnx T n e

S n1

k n

T

j

T n1

k n

x I n [nx+T n ] X

RN

T

x

;

n1

k=dnx T n e

S n1

k n

k n

x I n

:

Next we estimate D((Gn (f ))(x); f (x)) 0 [nx+T n X @ D (Gn (f ))(x); 0

+ D @f

]

f

x

k=dnx T n e

x

T

n1

[nx+T n ]

X

T x 1 +D f n1 0 N 1 [nx+T n ] X X @ D f (j) x j=1 k=dnx T n e

x I n [nx+T n ] X

k=dnx T n e

S n1

x I n

x I n

k n

n1 S n1

k=dnx T n e

f

S n1

T

T n1

x

1

k n

A

;

; f (x) k n

T n1

x+ j!

T n1

j

k n

RN

x

T k ; n1 n

S n

1

x I n

k n

1

; o~A

:

15.3 Main Results [nx+T n ]

X

+

S n1

k=dnx T n e (F )

+! 1

N X1

f;

[nx+T n ]

X

k n

x+ j!

T

D f (j) x X

S n1

k=dnx T n e [nx+T n ]

X

S n1

k=dnx T n e (F )

+ !1 8
f;

2 T j!n(1

S(T ) I

j

2T +

S n1

k n

x I n

k n

x I n

x I n

D RN

k n

T k ; n1 n

x

D(f (x); o~) + ! 1

f (j) ;

[nx+T n ]

S n1

k=dnx T n e

X

k=dnx T n e

x I n

k n

x I n

k n

(F )

D(f (x); o~) + ! 1

f;

D RN

x

f;

T n1

9 =

T n1

;

1 n S n1

; o~

(F )

1

(F )

X

+

j

T n1

D(f (j) (x); o~) + ! 1

)j

[nx+T n ]

+

; o~

n1

T n1

j

j=1

T

x

; o~

n1

[nx+T n ]

+

D f

1

T n1

j=1 k=dnx T n e

+

k n

x I n

287

T

;

n1

k n

; o~

1

T n1

+ !1

D RN

x

(F )

f;

T n1

=: ( ):

Using (15.12) and (15.20) we get [nx+T n ]

X

k=dnx T n e

S n1

x I n

2N T N S(T ) N !I nN (1 )

k n

2T +

1 n

T k ; n1 n

; o~ (F )

D(f (N ) (x); o~) + 3! 1

(15.24) f (N ) ;

T n1

Finally using (15.24) into ( ) we obtain (15.23). Note. In Chapter 16, see Theorems 16.13, 16.14 and Corollary 16.15, we present an improvement over Theorems 15.19, 15.26 and Corollary 15.22

:

288

15. Approximation of Fuzzy Neural Network Operators, Univariate Case

here, respectively. The reason for this improvement and simpli…cation is that there, in Chapter 16, we use real analysis results, however here we use only the fuzzy method and setting. We feel both have their own merits and so we present them.

16 HIGHER DEGREE OF FUZZY APPROXIMATION BY FUZZY WAVELET TYPE AND NEURAL NETWORK OPERATORS

In this chapter are studied in terms of fuzzy high approximation to the unit several basic sequences of fuzzy wavelet type operators and fuzzy neural network operators. These operators are fuzzy analogs of earlier studied real ones. The produced results generalize earlier real ones into the fuzzy setting. Here the high order fuzzy pointwise convergence with rates to the fuzzy unit operator is established through fuzzy inequalities involving the fuzzy modulus of continuity of the N th order (N 1) H-fuzzy derivative of the engaged fuzzy number valued function. At the end we present a related Lp result for fuzzy neural network operators. This chapter is based on [16].

16.1 Introduction We need the following results. They motivate this chapter and we generalize them into the Fuzzy setting. Theorems 16.1–16.4 deal with wavelet type operators. Theorem 16.1 ([9], [8]). Let f 2 C N (R), N ' be a bounded function of compact support

1, x 2 R and k 2 Z. Let [ a; a], a > 0 such that

G.A. Anastassiou: Fuzzy Mathematics: Approximation The., STUDFUZZ 251, pp. 289–308. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

290

16. Approximation by Fuzzy Wavelet Type and Neural Network Operators

P1

j= 1

'(x

j) = 1 all x 2 R. Suppose ' (Bk (f ))(x) :=

1 X

0. Call

j 2k

f

j= 1

'(2k x

j):

(16.1)

Then j(Bk (f ))(x)

N X aN jf (i) (x)j ai a + ! f (N ) ; k ; ki kN N ! 1 i! 2 2 2 i=1

f (x)j

(16.2)

which is attained by constant function. Theorem 16.2 ([9], [8]). Same assumptions as in Theorem 16.1. AddiR +1 tionally, suppose that ' is Lebesgue measurable (then '(x)dx = 1). 1 De…ne 2k=2 '(2k x j) all k; j 2 Z; Z 1 f (t)'kj (t)dt;

'kj (x) := hf; 'kj i :=

(16.3)

1

and (Ak (f ))(x) := =

1 X

j= 1

1 X

hf; 'kj i'kj (x)

j= 1

Z

1

u '(u 2k

f

1

j)du '(2k x

j): (16.4)

Then j(Ak (f ))(x)

N X jf (i) (x)j

f (x)j

i!

i=1

ai

+

2i(k 1)

2N N !2N (k

1)

! 1 f (N ) ;

a 2k 1

;

(16.5)

which is attained by constants. Theorem 16.3 ([9], [8]). Same assumptions as in Theorem 16.1. De…ne 1 X

Ck (f )(x) :=

k kj (f )'(2 x

j= 1 1 X

:=

2

j= 1

k

Z

2

k

2

j) !

(j+1)

f (t)dt '(2k x kj

j):

(16.6)

That is, kj (f )

k

:= 2

Z

2

2

k

(j+1) k

f (t)dt = 2 kj

Z

0

2

k

f

t+

j 2k

dt:

(16.7)

16.1 Introduction

291

Then (x 2 R, k 2 Z) j(Ck (f ))(x)

f (x)j

N X jf (i) (x)j(a + 1)i

i!2ki

i=1

+

(a + 1)N a+1 ; ! 1 f (N ) ; k kN 2 N! 2 (16.8)

which is attained by constants. Theorem 16.4 ([9], [8]). Same assumptions as in Theorem 16.1. De…ne (k; j 2 Z, x 2 R) (Dk f )(x) :=

1 X

k kj (f )'(2 x

j);

(16.9)

j= 1

where kj (f ) :=

n 2 N, wr~

0,

Pn

r=0

n X

wr~f

r~=0

j r~ + k 2k 2 n

;

(16.10)

wr~ = 1. Then

N X jf (i) (x)j (a + 1)i (a + 1)N (a + 1) ! 1 f (N ) ; + ; ki kN i! 2 2 N! 2k i=1 (16.11) which is attained by constants.

j(Dk f )(x)

f (x)j

Example of '’s: (i) '(x) :=

1 1 2;2

(x) =

8 > > < 1; > > :

0;

the characteristic function; (ii) 8 < 1 x; 1 + x; '(x) := : 0;

x2

1 1 ; 2 2

(16.12)

elsewhere;

0

x 1 1 x 0 elsewhere;

(16.13)

the hat function. The next in this section come from [9], [6], [47]. Here we consider functions f : R ! R that are continuous. We mention the pointwise convergence with rates over the real line, to the unit operator, of the univariate Cardaliaguet–Euvrard neural network operators (see [9], [6], [47]) 2

n X f (k=n) (Fn (f ))(x) := fn (x) := b n1 In 2 k= n

x

k n

;

(16.14)

292

16. Approximation by Fuzzy Wavelet Type and Neural Network Operators

RT where 0 < < 1 and x 2 R, n 2 N, I := T b(t)dt. Denote by [ ] the integral part of a number and by d e its ceiling. Set b := b(0) the maximum of b(x) which is a centered bell-shaped function, of compact support [ T; T ], T > 0, see De…nition 4.1 of [17]. We have Theorem 16.5. Let x 2 R, T > 0, and n 2 N such that n max(T + jxj; T 1= ). Let f 2 C N (R), N 2 N, such that f (N ) is a uniformly continuous function. Then [nx+T n ]

jfn (x)

X

f (x)j =

f (k=n) b n1 I n

k=dnx T n e

[nx+T n ]

X

jf (x)j +

k=dnx T n

b I

2T +

+ ! 1 f (N ) ;

1 n T

k n

x

f (x)

1 k b n1 x 1 I n n e 1 0 N (j) j X jf (x)j T A @ (16.15) nj(1 ) j! j=1 TN N ! nN (1

n1

)

b I

2T +

1 n

:

Let S be a squashing function of compact support [ T; T ], T > 0, see De…nition 4.2 of [17]. Let f : R ! R be continuous. Suppose that I :=

Z

T

S(t)dt > 0: T

For x 2 R we de…ne the “univariate squashing neural network operator" (see [9], [6], [47]) 2

n X f (k=n) S n1 (Gn (f ))(x) := I n 2

k n

x

k= n

0<

< 1 and n 2 N: n

max(T + jxj; T

[nx+T n ]

(Gn (f ))(x) =

X

k=dnx T n e

1=

f (k=n) S n1 I n

;

(16.16)

). It is easy to see that x

k n

:

(16.17)

Here we mention the pointwise convergence with rates of (Gn (f ))(x) ! f (x), as n ! +1, x 2 R. Theorem 16.6. Let x 2 R, T > 0, and n 2 N, such that n max(T + jxj; T 1= ). Let f 2 C N (R), N 2 N, such that f (N ) is a uniformly contin-

16.2 Main results

293

uous function. Then j(Gn (f ))(x)

f (x)j

jf (x)j

[nx+T n ]

X

1

k=dnx T n e

+

S(T ) I

S n1

n

2T +

+ ! 1 f (N ) ; S(T ) I

I

1 n

TN N ! nN (1

T 1 n

k n

1

0 1 N (j) j X jf (x)j T A @ j! nj (1 ) j=1

n1

2T +

x

)

(16.18)

:

In this chapter we study the same operators Bk , Ak , Ck , Dk Fn , Gn , k 2 Z, n 2 N, respectively, when they act on fuzzy real number valued functions, proving their high order approximation to the unit operator. So according to the context one can understand clearly, when these operators are applied to real valued functions, and when they are applied to fuzzy real valued functions. Here we apply real analysis results into the fuzzy setting. Remark 16.7. (i) Here CU (R) denotes the uniformly continuous functions (r) U (R), then f 2 CU (R), 8r 2 [0; 1]. Also one has from R into R. If f 2 CF (r) ! 1 (f ; ) < +1 for any > 0. (r) (ii) If f : R ! RF is fuzzy continuous then f : R ! R are continuous, 8r 2 [0; 1]. Note. Let f 2 C N (R; RF ), N 1. Then by Theorem 8 of [15], [71], we (r) N have f 2 C (R), for any r 2 [0; 1].

NU De…nition 16.8. Denote by CF (R) := ff : R ! RF j such that all fuzzy (i) derivatives f : R ! RF , i = 0; 1; : : : ; N exist and are fuzzy continuous, furthermore f (N ) is fuzzy uniformly continuous from R into RF g, N 1.

16.2 Main results We present the …rst main result. NU Theorem 16.9. Let f 2 CF (R), N 1, x 2 R and k 2 Z. Let the scaling function '(x) a real valued bounded function with supp '(x) [ a; a], 1 P 0 < a < +1, '(x) 0, such that '(x j) 1 on R. Consider the j= 1

294

16. Approximation by Fuzzy Wavelet Type and Neural Network Operators

fuzzy wavelet type operator 1 X

(Bk f )(x) :=

j 2k

f

j= 1

'(2k x

j):

(16.19)

Then D((Bk f )(x); f (x))

N X aN a D(f (i) (x); o~) ai (F ) + !1 f (N ) ; k ki kN i! 2 2 N! 2 i=1

=:

1 (x):

(16.20) As k ! +1 we get f ; ! 0 and D((Bk f )(x); f (x)) ! 0, i.e. lim Bk f = f , pointwise with rates. (F ) !1

(N )

a 2k

k!+1

Proof. Since ' is of compact support (16.19) is a …nite sum. Thus for r 2 [0; 1] we have [(Bk f )(x)]r

1 X

=

j= 1 1 X

=

j= 1

2

= 4

j 2k

f "

f

1 X

f

j= 1

1 X

f

j= 1

=

(r)

(Bk (f )

I.e.

(r)

(Bk f )

r

'(2k x

j 2k

(r)

j 2k

(r)

j)

; f

'(2k x

'(2k x + (r)

+

#

'(2k x

j)

j); 3

(r)

j 2k

(r)

j 2k

j)5

)(x); (Bk (f )+ )(x) :

= Bk (f

(r)

);

8r 2 [0; 1]:

(16.21)

We observe by Section 2 of [17] that D((Bk f )(x); f (x))

=

(r)

sup max j(Bk f )

(x)

f

(r)

(x)j;

r2[0;1]

(r)

(16:21)

=

j(Bk f )+ (x)

(r)

f+ (x)j

sup max j(Bk (f

(r)

))(x)

f

(r)

(x)j;

r2[0;1]

(r)

j(Bk (f+ ))(x)

(r)

f+ (x)j :

At this point we notice by Theorem 8 of [15] that for any r 2 [0; 1] the real (r) (r) function f 2 C N (R) and (f )(N ) 2 CU (R). Therefore we can apply

16.2 Main results

295

Theorem 16.1 (16.2) to obtain D((Bk f )(x); f (x)) N (r) X j(f (x))(i) j ai aN a (r) + kN ! (f )(N ) ; k ki i! 2 2 N ! 2 i=1

sup max r2[0;1]

N (r) X j(f+ (x))(i) j ai aN a (r) + kN ! 1 (f+ )(N ) ; k ki i! 2 2 N ! 2 i=1 N (i) X j(f (x))(r) j ai i! 2ki i=1

(by (33) of [15]) = sup max r2[0;1] N

+

a

2kN N !

! 1 (f

(N ) (r)

)

a 2k

;

N (i) X j(f+ (x))(r) j ai aN a (N ) + ! 1 (f+ )(r) ; k ki kN i! 2 2 N! 2 i=1

(by (29) of [15]) +

N X ai (i) (i) sup max j(f (x))(r) j; j(f+ (x))(r) j ki i!2 r2[0;1] i=1

a a aN (N ) (N ) sup max ! 1 (f )(r) ; k ; ! 1 f+ )(r) ; k 2kN N ! r2[0;1] 2 2 N X ai D(f (i) (x); o~) ki i!2 i=1

(by (25) & (26) of [15]) = +

aN a (F ) !1 f (N ) ; k : kN 2 N! 2

We have established (16.20). We continue with NU (R) \ Cb (R; RF ), N 1, x 2 R and k 2 Theorem 16.10. Let f 2 CF Z. Let the scaling function '(x) a real valued function with supp '(x) [ a; a], 0 < a < +1, ' is continuous on [ a; a], '(x) 0, such that 1 R1 P '(x j) = 1 on R (then 1 '(x)dx = 1). De…ne j= 1

'kj (t) :=

hf; 'kj i :=

2k=2 '(2k t j); for k; j 2 Z; t 2 R; Z j+a 2k f (t) 'kj (t)dt; (F R)

(16.22) (16.23)

j a 2k

and the fuzzy wavelet type operator (Ak f )(x) :=

1 X

hf; 'kj i

j= 1

'kj (x);

x 2 R:

(16.24)

;

296

16. Approximation by Fuzzy Wavelet Type and Neural Network Operators

Then N X D(f (i) (x); o~)

D((Ak f )(x); f (x))

i!

i=1

aN N !2N (k : 2 (x):

2i(k

(F )

+ =

ai

1)

f (N ) ;

!1

(16.25)

1)

a 2k 1

As k ! +1 we get lim Ak f = f , pointwise with rates in the D-metric. k!+1

Proof. Since ' is of compact support (16.24) is a …nite sum. For any r 2 [0; 1] by Theorem 16.9 and Lemma 2 of [15] we notice that "Z j+a # Z j+a 2k 2k (r) (r) (f (t) 'kj (t)) dt; (f (t) 'kj (t))+ dt [hf; 'kj i]r = =

"Z

j a 2k j+a 2k

(r)

(f (t))

j a 2k

'kj (t)dt;

Z

j a 2k j+a 2k

j a 2k

(r) (f (t))+ 'kj (t)dt

#

:

We see that 1 X

r

[(Ak f )(x)] =

=

1 X

j= 1

"Z

j+a 2k

(r)

(f (t))

j a 2k

2

=4

[hf; 'kj i]r 'kj (x)

j= 1

Z

1 X

j= 1

j+a 2k

j= 1

(f (t))

j+a 2k j a 2k

= (Ak (f

(r)

j+a 2k j a 2k

(r)

j a 2k

Z

1 X

'kj (t)dt;

Z

#

(r) (f (t))+ 'kj (t)dt

!

'kj (x)

'kj (t)dt 'kj (x); 3

!

(r)

(f (t))+ 'kj (t)dt 'kj (x)5 (r)

))(x); (Ak (f+ ))(x) :

That is we prove that (r)

(Ak f )

= Ak (f

(r)

);

8r 2 [0; 1]:

(16.26)

We observe from Section 2 of [17] that D((Ak f )(x); f (x))

=

(r)

sup max j(Ak f )

(x)

f

(r)

(x)j;

r2[0;1]

(r)

(16:26)

=

j(Ak f )+ (x)

(r)

f+ (x)j

sup max j(Ak (f

(r)

))(x)

r2[0;1]

(r)

j(Ak (f+ ))(x)

(r)

f+ (x)j :

f

(r)

(x)j;

16.2 Main results (r)

Here by Theorem 8 of [15] for any r 2 [0; 1] f 2 C N (R) and (f CU (R). Hence by applying Theorem 16.2 (16.5) we get

297

(r) (N )

)

2

D((Ak f )(x); f (x)) N (r) X j(f (x))(i) j

sup max r2[0;1]

i!

i=1 N

+

a N !2N (k

1)

N (r) X j(f+ (x))(i) j

i!

i=1

+

aN N !2N (k

! 1 (f

(r) (N )

)

1)

! 1 (f+ )(N ) ;

N (i) X j(f (x))(r) j

(by (29) of [15])

N X i=1

N

+

a N !2N (k

1)

(N ) (r)

ai 2i(k

i!

1)

)

a

;

ai i!2i(k 1)

(N ) (r)

;

(N ) (f+ )(r) ;

(i)

a 2k

1

!

(i)

(x))(r) j; j(f+ (x))(r) j

a ; 2k 1

a 2k 1

(by (24) & (25) of [15]) =

N X i=1

(F )

1)

!1

r2[0;1]

r2[0;1]

(N )

1)

sup max j(f )

ai 2i(k 1)

;

2k 1

aN + N !2N (k

sup max ! 1 (f

! 1 (f+ )(r) ;

aN + N !2N (k

i!

i=1

! 1 (f

N (i) X j(f+ (x))(r) j i=1

;

a 2k 1

(r)

1)

a 2k 1

ai

r2[0;1]

aN N !2N (k

;

2i(k 1)

(by (33) of [15]) = sup max +

ai 2i(k 1)

!1

ai i!2i(k 1)

f (N ) ;

a 2k 1

D(f (i) (x); o~) :

We have proven (16.25). Next we give Theorem 16.11. All assumptions here are as in Theorem 16.9. For k 2 Z, x 2 R we de…ne the fuzzy wavelet type operator ! Z 2 k 1 X j k (Ck f )(x) := 2 (F R) f t + k dt '(2k x j): 2 0 j= 1 (16.27)

298

16. Approximation by Fuzzy Wavelet Type and Neural Network Operators

Then N X D(f (i) (x); o~) (a + 1)i

D((Ck f )(x); f (x))

2ki

i!

i=1

(a + 1)N (F ) a+1 !1 f (N ) ; k kN 2 N! 2

=:

+

3 (x):

(16.28)

As k ! +1 we get lim Ck f = f , pointwise with rates in the D-metric. k!+1

Proof. Since ' is of compact support (16.27) is a …nite sum. So for r 2 [0; 1] we observe that #r " Z 2 k 1 X j [(Ck f )(x)]r = 2k (F R) f t + k dt '(2k x j) 2 0 j= 1 " #r Z 2 k (j+1) 1 X = 2k (F R) f (t)dt '(2k x j) j= 1 1 X

(by (23) of [15]) =

j= 1

2

"

2k

Z

k

2

(f )

Z

Z

2k

2

j= 1

(Ck (f

(r)

Z

(t)dt; 2k

2

2

k

(j+1)

kj

(r) (f )+ (t)dt

j)

j= 1

=

(j+1)

kj

2

'(2k x 2 1 X = 4 1 X

k

2

kj

(r)

k

2

(j+1)

(r)

(f )

2

kj

2

k

(j+1)

!

(t)dt '(2k x !

(r)

(f )+ (t)dt '(2k x

kj

(r)

))(x); (Ck (f+ ))(x) :

j); 3

j)5

That is, for any r 2 [0; 1] we proved that (r)

(Ck f )

= Ck (f

(r)

):

(16.29)

Then by Section 2 of [17] D((Ck f )(x); f (x))

=

(r)

sup max j((Ck f )(x))

f

(r)

(x)j;

f

(r)

(x)j;

r2[0;1]

(r)

(16:29)

=

j((Ck f )(x))+

(r)

f+ (x)j

sup max j(Ck (f

(r)

))(x)

r2[0;1]

(r)

j(Ck (f+ ))(x)

(r)

f+ (x)j :

Again by Theorem 8 of [15] for any r 2 [0; 1] the real functions f (r) N C (R) and (f )(N ) 2 CU (R). Hence by Theorem 16.3 (16.8) we get

(r)

2

#

16.2 Main results

D((Ck f )(x); f (x)) ( sup max

r2[0;1]

+

N (r) X j(f (x))(i) j (a + 1)i

2ki

i!

i=1

(a + 1)N a+1 (r) ! 1 (f )(N ) ; k 2kN N ! 2

N (r) X j(f+ (x))(i) j (a + 1)i

2ki

i!

i=1

(by (33) of [15]) = sup max r2[0;1]

+

(

N (i) X j(f+ (x))(r) j (a + 1)i

2ki

i!

N X (a + 1)i

(by (29) of [15])

i=1

i!2ki

;

(a + 1)N a+1 (r) + kN ! 1 (f+ )(N ) ; k 2 N! 2

2ki

i!

i=1

;

(a + 1)N a+1 (N ) + kN ! 1 (f+ )(r) ; k 2 N! 2 sup max j(f

(i)

r2[0;1]

(N )

(i)

;

a+1 2k

(by (25) & (26) of [15]) =

N X (a + 1)i i=1

N

+

!)

(x))(r) j; j(f+ (x))(r) j

(a + 1)N a+1 (N ) + kN sup max ! 1 (f+ )(r) ; k 2 N ! r2[0;1] 2 ! 1 (f+ )(r) ;

!)

N (i) X j(f (x))(r) j (a + 1)i

(a + 1)N a+1 (N ) ! 1 (f )(r) ; k 2kN N ! 2

i=1

299

i!2ki

a+1 (a + 1) (F ) ! f (N ) ; k 2kN N ! 1 2

D(f (i) (x); o~)

:

That is proving (16.28). The last fuzzy wavelet type result follows. Theorem 16.12. All assumptions here are as in Theorem 16.9. For k 2 Z, x 2 R we de…ne the fuzzy wavelet type operator (Dk f )(x) :=

1 X

kj (f )

'(2k x

j);

(16.30)

j= 1

where kj (f )

:=

n X

r~=0

wr~ f

r~ j + k k 2 2 n

;

n 2 N; wr~

0;

n X r~=0

wr~ = 1: (16.31)

300

16. Approximation by Fuzzy Wavelet Type and Neural Network Operators

Then N X D(f (i) (x); o~) (a + 1)i

D((Dk f )(x); f (x))

(16.32)

2ki

i!

i=1

(a + 1)N (F ) a+1 !1 f (N ) ; k kN 2 N! 2 : 4 (x):

+ =

As k ! +1 we get lim Dk f = f , pointwise with rates in the D-metric. k!+1

Proof. Since ' is of compact support (16.30) is a …nite sum. So for r 2 [0; 1] we observe that [(Dk f )(x)]r

=

1 X

[

j= 1

=

1 X

j= 1 k

=

'(2 x " 1 X

r k kj (f )] '(2 x n X

j) n X

j= 1 1 X

(Dk (f

f

wr~ f

r~=0

j= 1

=

wr~

r~=0

"

n X

j) r~ j + k 2k 2 n

(r)

j r~ + k k 2 2 n

(r)

wr~ f

r~=0

(r)

; f !

(r)

r~ j + k k 2 2 n

+

j r~ + k 2k 2 n

'(2k x

!

(r) +

j);

k

'(2 x

#

j)

(r)

))(x); (Dk (f+ ))(x) :

That is, we prove that (r)

(Dk f )

= Dk (f

(r)

);

8r 2 [0; 1]:

(16.33)

Next we see by Section 2 of [17] that D((Dk f )(x); f (x))

=

(r)

sup max j((Dk f )(x))

f

(r)

(x)j;

f

(r)

(x)j;

r2[0;1]

(r)

(16:33)

=

j((Dk f )(x))+

(r)

f+ (x)j

sup max j(Dk (f

(r)

))(x)

r2[0;1]

(r)

j(Dk (f+ ))(x)

(r)

f+ (x)j :

Using Theorem 16.4 (16.11) and acting as in the proof of Theorem 16.11 we obtain (16.32).

#!

16.2 Main results

301

In the next let b be a bell-shaped function as in De…nition 4.1 of [17]. Let f 2 C(R; RF ) we de…ne the fuzzy univariate Cardaliaguet–Euvrard neural network operators by 2

(Fn (f ))(x) :=

n X

b n1

k n

f

k= n2

k n

x In

;

(16.34)

where 0 < < 1 and x 2 R, n 2 N. Since b is of compact support [ T; T ], T > 0 we obtain [nx+T n ]

X

(Fn (f ))(x) =

b n1

k n

f

k=dnx T n e

k n

x In

;

(16.35)

for n T + jxj. We present the fuzzy pointwise convergence with rates to the unit of operators Fn . Theorem 16.13. Let x 2 R, T > 0, n 2 N such that n NU (R), N 1. Then jxj; T 1= ). Let f 2 CF [nx+T n ]

D((Fn (f ))(x); f (x))

X

D(f (x); o~)

b n1

0

1 2T + n

(F )

+ !1

f (N ) ;

T

@

N X

D(f (j) (x); o~)

j=1

n1

TN N !nN (1

)

Tj nj(1

b I

k n

x In

k=dnx T n e

b + I

max(T +

) j!

2T +

1 A

1 n

where b := b(0). By Lemma 2.1 ([9], p. 64) we have [nx+T n ]

lim

n!+1

X

b n1

k=dnx T n e

k n

x In

= 1;

U Since f (N ) 2 CF (R) we get that (F )

lim ! 1

n!+1

f (N ) ;

T n1

= 0:

Consequently from (16.36) as n ! +1 we derive D((Fn (f ))(x); f (x)) ! 0;

x 2 R:

1

x 2 R:

;

(16.36)

302

16. Approximation by Fuzzy Wavelet Type and Neural Network Operators

Proof of theorem 16.13. Notice that (16.35) is a …nite sum. Thus [nx+T n ]

[(Fn (f ))(x)]r

X

=

k=dnx T n e

2

= 4

[nx+T n ]

X

f

(r)

k=dnx T n e [nx+T n ]

X

(Fn (f

(r)

f+

(r)

x In

k n

x k b n1 n In

k n

b n1

; 3

k n

x k b n1 n In

k=dnx T n e

=

r

k n

f

5

(r)

))(x); (Fn (f+ ))(x) :

I.e. for any r 2 [0; 1] we have (r)

(Fn (f ))

= Fn (f

(r)

):

(16.37)

Next we observe from Section 2 of [17] that D((Fn f )(x); f (x))

(r)

=

sup max j((Fn f )(x)

f

(r)

(x)j;

r2[0;1]

(r)

j((Fn f )(x))+

(16:37)

=

(r)

f+ (x)j

sup max jFn (f

(r)

)(x)

f

(r)

(x)j;

r2[0;1]

(r)

(r)

f+ (x)j :

jFn (f+ )

(r)

NU 2 C N (R) and (R) we have for any r 2 [0; 1] that f Since f 2 CF (r) (N ) (f ) 2 CU (R). Therefore we can apply Theorem 16.5 (16.15) to obtain

D((Fn f )(x); f (x)) ( sup max

r2[0;1]

+

b I

2T +

+ ! 1 (f

[nx+T n ]

jf

(x)j

0

1 n

(r) (N )

)

(r)

@ ;

b n1

k=dnx T n e N (r) X j(f )(j) (x)jT j j=1

T

n1

[nx+T n ] (r)

jf+ (x)j

X

X

k=dnx T n e

nj(1

) j!

TN N !nN (1 b n1

x In

)

b I

k n

x In

1

1 A 2T + k n

1

1 n

;

16.2 Main results N (r) X j(f+ )(j) (x)jT j

b + I

1 2T + n

+ !1

(r) (f+ )(N ) ;

nj(1

j=1

n1

(by (33) of [15]) = sup max r2[0;1] [nx+T n ]

X

b n1

[nx+T n ]

b n1

) j!

TN N !nN (1 (

T

jf

303

b ) I

(r)

1 2T + n

)

(x)j

x nk 1 In k=dnx T n e 1 0 N (j) b T 1 @X j(f )(r) (x)jT j A (N ) + + ! 1 (f )(r) ; 1 2T + j(1 ) I n n n j! j=1 ! b TN 1 (r) 2T + ; jf+ (x)j n N !nN (1 ) I X

k=dnx T n e

+

b I

2T +

1

N (j) X j(f+ )(r) (x)jT j

1 n

nj(1

j=1

(N ) (f+ )(r) ;

+ !1

k n

x In

(by (29) of [15])

TN N !nN (1

T n1

sup maxfjf

(r)

r2[0;1] [nx+T n ]

X

b n1

k=dnx T n e

1 2T + n

b + I Tj nj(1 +

) j!

(

x In 8 N <X :

j=1

sup max ! 1 (f r2[0;1]

TN N !nN (1

)

b I

) j!

k n

1 2T + n

b ) I

!

(r) (x)j; jf+ (x)jg

1

sup maxfj(f

(j) (r)

r2[0;1]

(N ) (r)

2T +

)

1 n

!)

;

T n1

; !1

)

!

(j) (x)j; j(f+ )(r) (x)jg

(N ) (f+ )(r) ;

T n1

)!

304

16. Approximation by Fuzzy Wavelet Type and Neural Network Operators [nx+T n ]

(by (25) & (26) of [15]) = D(f (x); o~)

X

b n1

k=dnx T n e

b + I

1 2T + n

(F )

f (N ) ;

+ !1

0 N X @ D(f (j) (x); o~)

Tj nj(1

j=1

T

TN N !nN (1

n1

)

b I

) j!

2T +

1 n

x In

k n

1 A :

That is proving (16.36). In the following let S be a squashing function as in De…nition 4.2 of [17]. RT Clearly max S(x) = S(T ), x 2 [ T; T ]. Assume that I := T S(t)dt > 0. Let f 2 C(R; RF ) we de…ne the fuzzy univariate squashing neural network operators by 2

(Gn (f ))(x) :=

n X

k= n2

x 2 R, 0 < that

S n1

k n

f

< 1 and n 2 N: n

max(T + jxj; T

[nx+T n ]

X

(Gn (f ))(x) =

S n1

k n

f

k=dnx T n e

k n

x I n

1=

;

(16.38)

). It is easy to see k n

x I n

:

(16.39)

We present the fuzzy pointwise convergence with rates to the unit of operators Gn . Theorem 16.14. Let x 2 R, T > 0, n 2 N such that n NU jxj; T 1= ). Let f 2 CF (R), N 1. Then [nx+T n ]

D((Gn (f ))(x); f (x))

X

D(f (x); o~)

S n1

k=dnx T n e

S(T ) + I (F )

+ !1

1 2T + n f (N ) ;

T

0 @

n1

N X

D(f (j) (x); o~)

j=1

TN N !nN (1

)

S(T ) I

k n

x I n

Tj nj(1

) j!

2T +

[nx+T n ] n!+1

X

k=dnx T n e

S n1

x I n

k n

= 1;

x 2 R:

1

1 A

1 n

By Lemma 2.2 ([9], p. 79) we have lim

max(T +

: (16.40)

1

16.2 Main results

305

U Since f (N ) 2 CF (R) we get that (F )

lim ! 1

n!+1

f (N ) ;

T n1

= 0:

Consequently from (16.40) as n ! +1 we obtain D((Gn (f ))(x); f (x)) ! 0, x 2 R. Proof of theorem 16.14. Notice that (16.39) is a …nite sum. Then one …nds easily that (r) (r) (Gn (f )) = Gn (f ); (16.41) for all r 2 [0; 1]. Next we see again by Section 2 of [17] that D((Gn f )(x); f (x))

(r)

=

sup max j((Gn f )(x))

f

(r)

(x)j;

f

(r)

(x)j;

r2[0;1]

(r)

(r)

j((Gn f )(x))+

(16:41)

=

f+ (x)j

sup max j(Gn (f

(r)

))(x)

r2[0;1]

(r)

(r)

f+ (x)j :

j((Gn (f+ ))(x)

Then the proof follows like in the proof of Theorem 16.13. We make use (r) of Theorem 16.6 (16.18) for f . We again use in order (16.33), (16.29), (16.26) and (16.25) of [15] and …nally produce (16.40). Also we give Corollary 16.15 (to Theorem 16.13). Let b(x) be a centered bell-shaped continuous function on R of compact support [ T; T ], T > 0. Let x 2 [ T ; T ], T > 0, and n 2 N such that n Consider p

max T + T ; T

1=

;

0<

< 1:

1. Then we obtain

kD((Fn (f ))(x); f (x))kp;[

T ;T ] [nx+T n ]

kD(f (x); o~)k1;[ b + I (F )

+ !1

T ;T ]

b n1

k=dnx T n e

1 2T + n f (N ) ;

X

0 N X @ kD(f (j) (x); o~)kp;[

k n

x In

p;[ T ;T ]

Tj T ;T ]

j=1

T n1

T N 21=p T N !nN (1

1=p )

b I

1

2T +

1 n

nj(1 :

) j!

1 A

(16.42)

306

16. Approximation by Fuzzy Wavelet Type and Neural Network Operators

From inequality (16.42) as n ! +1 we obtain the Lp -fuzzy convergence of Fn (f ) to f with rates. Proof. Since f if fuzzy continuous on [ T ; T ] it is fuzzy bounded there (see Lemma 2 of [11]), therefore kD(f (x); o~)k1;[

T ;T ]

< +1:

So from Theorem 16.13, we have that D((Fn (f ))(x); f (x))

kD(f (x); o~)k1;[ [nx+T n ]

X

b n1

k=dnx T n e

+

b I

2T +

(F )

+ !1

1 n

f (N ) ;

(16.43)

T ;T ]

x nk 1 In 0 N X @ D(f (j) (x); o~) j=1

T n1

TN N !nN (1

)

T nj(1

b I

j ) j!

2T +

1 A

1 n

Inequality (16.42) now comes by integration of (16.43) and the properties of Lp norm. As in [9], p. 75, using the bounded convergence theorem we get that [nx+T n ]

lim

n!+1

X

k=dnx T n e

b n1

x In

k n

1

= 0: p;[ T ;T ]

Note. Theorems 16.13, 16.14 and Corollary 16.15 simplify and improve the author’s earlier corresponding results in [11], namely from there see Theorems 3, 5, and Corollary 2, respectively. A di¤erent direct fuzzy method, not using real analysis results, is applied there [11]. See also here Chapter 15. We also give the following related results Theorem 16.16. All assumptions as in Theorem 16.9. Let (Bk f )(x) as in (16.19), (Ck f )(x) as in (16.27), (Dk f )(x) as in (16.30). Then (i) E1k (x) = D ((Bk f ) (x) ; (Dk f ) (x))

1

(x) +

4

(x) ;

(16.44)

E2k (x) = D ((Bk f ) (x) ; (Ck f ) (x))

1

(x) +

3

(x) ;

(16.45)

(ii)

:

16.2 Main results

307

(iii) E3k (x) = D ((Ck f ) (x) ; (Dk f ) (x))

3

(x) +

4

(x) :

(16.46)

Notice E1k (x); E2k (x); E3k (x) ! 0; pointwise with rates, as k ! 1: Proof. We use the triangle inequality property of metric distance D and (16.20), (16.28) and (16.32) of Theorems 16.9, 16.11 and 16.12, respectively. Indeed we have (i) D ((Bk f ) (x) ; (Dk f ) (x))

D ((Bk f ) (x) ; f (x)) +D ((Dk f ) (x) ; f (x)) 1 (x) + 4 (x) ;

D ((Bk f ) (x) ; (Ck f ) (x))

D ((Bk f ) (x) ; f (x)) +D ((Ck f ) (x) ; f (x)) 1 (x) + 3 (x) ;

D ((Ck f ) (x) ; (Dk f ) (x))

D ((Ck f ) (x) ; f (x)) +D ((Dk f ) (x) ; f (x)) 3 (x) + 4 (x) :

(ii)

(iii)

Finally we give Theorem 16.17. All assumptions as in Theorem 16.10. Let (Ak f )(x) as in (16.24), (Bk f )(x) as in (16.19), (Ck f )(x) as in (16.27) and (Dk f )(x) as in (16.30). Then (i) E4k (x) = D ((Ak f ) (x) ; (Bk f ) (x))

1

(x) +

2

(x) ;

(16.47)

E5k (x) = D ((Ak f ) (x) ; (Ck f ) (x))

2

(x) +

3

(x) ;

(16.48)

E6k (x) = D ((Ak f ) (x) ; (Dk f ) (x))

2

(x) +

4

(x) :

(16.49)

(ii)

(iii)

308

16. Approximation by Fuzzy Wavelet Type and Neural Network Operators

Notice E4k (x); E5k (x); E6k (x) ! 0; pointwise with rates, as k ! 1: Proof. Similar to Theorem 16.16. It is based on (16.20) of Theorem 16.9, (16.25) of Theorem 16.10, (16.28) of Theorems 16.11, and (16.32) of Theorem 16.12. Of course we apply again the triangle inequality property of D: Note. We notice that (16.44) of Theorem 16.16 improves (13.33) of Theorem 13.21. Also (16.46) of Theorem 16.16 improves (13.34) of Theorem 13.22. Furthermore (16.45) of Theorem 16.16 improves (13.35) of Theorem 13.23. Similarly can be improved Corollaries 13.27, 13.28, 13.29, accordingly. The reason for this improvement is that in the proofs here we use real and fuzzy methods, while in Chapter 13 we use only fuzzy methods.

17 FUZZY RANDOM KOROVKIN THEOREMS AND INEQUALITIES

Here we study the fuzzy random positive linear operators acting on fuzzy random continuous functions. We establish a series of fuzzy random Shisha–Mond type inequalities of Lq -type 1 q < 1 and related fuzzy random Korovkin type theorems, regarding the fuzzy random q-mean convergence of fuzzy random positive linear operators to the fuzzy random unit operator for various cases. All convergences are with rates and are given using the above fuzzy random inequalities involving the fuzzy random modulus of continuity of the engaged fuzzy random function. The assumptions for the Korovkin theorems are minimal and of natural realization, ful…lled by almost all example fuzzy random positive linear operators. The astonishing fact is that the real Korovkin test functions assumptions are enough for the conclusions of the fuzzy random Korovkin theory. We give at the end applications. This chapter follows [22].

17.1 Introduction Motivation for this chapter are [4], [18], [24], [32], [17], [66], [96]. We introduce the concept of fuzzy random positive linear operator and we prove the results for a very large general class of such operators. Most of the G.A. Anastassiou: Fuzzy Mathematics: Approximation The., STUDFUZZ 251, pp. 309–345. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

310

17. Fuzzy Random Korovkin Theorems and Inequalities

summation and integration operators fall into this class. To do that we are greatly helped by the fuzzy Riesz representation theorem developed in [24]. The surprising fact is that the basic assumptions of real Korovkin theory for the test functions 1, id, id2 carry over here and they are the only ones needed. Of course a natural realization condition is required in the fuzzy random setting to prove the fuzzy random q-mean convergence. But …rst we establish a series of fuzzy random Shisha–Mond type inequalities for various important cases. These contain the fuzzy random modulus of continuity of the involved function. So this chapter is basically the study with rates and quantitavely for the fuzzy random q-mean convergence of a sequence of very general and abstract fuzzy random positive linear operators to the fuzzy random unit operator. Linearity and positivity here are the analogs of the real case. Finally we give applications to fuzzy random Bernstein operators. We need the following de…nitions. De…nition 17.1. (See also [66, De…nition 13.16, p. 654]). Let (X; B; P ) be a probability space. A fuzzy-random variable is a B-measurable mapping g : X ! RF , i.e., for any open set U RF , in the topology of RF generated by the metric D, we have g

1

(U ) = fs 2 X; g(s) 2 U g 2 B:

(17.1)

The set of all fuzzy-random variables is denoted by LF (X; B; P ). Let gn ; g 2 LF (X; B; P ), n 2 N, and 0 < q < +1. We say, gn (s) if lim

n!+1

Z

“q-mean” ! g(s); n ! +1

(17.2)

q

D(gn (s); g(s)) P (ds) = 0:

(17.3)

X

De…nition 17.2. (See [66, p. 654, De…nition 13.17].) Let (T; T ) be a topological space. A mapping f : T ! LF (X; B; P ) will be called fuzzy-random function (or fuzzy-stochastic process) on T . We denote f (t)(s) = f (t; s), t 2 T , s 2 X.

Remark 17.3. (See [66, p. 655].) Any usual fuzzy real function f : T ! RF can be identi…ed with the degenerate fuzzy-random function f (t; s) = f (t), 8t 2 T , s 2 X. Remark 17.4. (See [66, p. 655].) Fuzzy-random functions that coincide with probability one, for each t 2 T , will be considered equivalent. Remark 17.5. (see [66, p. 655].) Let f; g : T ! LF (X; B; P ). Then, f and k f are de…ned pointwise, i.e., (f (k

g)(t; s) = f (t; s) g(t; s); f )(t; s) = k f (t; s); t 2 T;

s 2 X;

k 2 R:

g

17.1 Introduction

311

De…nition 17.6. (See also [66, De…nition 13.18, pp. 655–656].) For a fuzzy-random function f : [a; b] ! LF (X; B; P ), we de…ne the (…rst) fuzzyrandom modulus of continuity ( Z 1=q

(F ) 1 (f;

)Lq

=

Dq f (x; s); f (y; s) P (ds)

sup

X

jx

yj

)

;

0< ;

1

; x; y 2 [a; b];

q < 1:

(17.4)

De…nition 17.7. Here, 1 q < 1. Let f : [a; b] ! LF (X; B; P ) be a fuzzy random function. We call f a (q-mean) uniformly continuous fuzzy random function over [a; b] i¤ 8" > 0 9 > 0: whenever jx yj , x; y 2 [a; b], implies that Z q D(f (x; s); f (y; s)) P (ds) ": (17.5) X

Uq We denote it as f 2 CF R ([a; b]). We need

(F ) 1 (f;

Uq Proposition 17.8. Let f 2 CF R ([a; b]). Then, > 0.

)Lq < 1, any

Proof. Let "0 > 0 be arbitrary, but …xed. Then, there exists yj 0 , x; y 2 [a; b] which implies Z q D(f (x; s); f (y; s)) P (ds) "0 < 1:

0

> 0 : jx

X

(F )

1=q

That is, 1 (f; 0 )Lq "0 < 1. Let now > 0 arbitrary, x; y 2 [a; b], such that jx yj . Choose n 2 N: n 0 and set xi := x + (i=n)(y x), 0 i n. Then, D f (x; s); f (y; s)

D f (x; s); f (x1 ; s)

+ D f (x1 ; s); f (x2 ; s) +

+ D f (xn

1 ; s); f (y; s)

Consequently, Z

1=q

q

D(f (x; s); f (y; s)) P (ds)

X

Z

q

X

+ n

1=q

D(f (x; s); f (x1 ; s)) P (ds) +

Z

X (F ) 1 (f; 0 )Lq

D(f (xn 1=q

n"0

1 ; s); f (y; s))

< 1;

q

1=q

P (ds)

:

312

17. Fuzzy Random Korovkin Theorems and Inequalities

since jxi xi+1 j (1=n)jx (F ) 1=q n"0 < 1. 1 (f; )Lq

yj

(1=n)

0,

0

i

n. Therefore,

Proposition 17.9. Let f; g : [a; b] ! LF (X; B; P ) be fuzzy random functions, [a; b] R. The following hold. (F ) 1 (f;

(i)

(ii) lim

)Lq be nonnegative and nondecreasing in (F )

#0

(iii)

(F ) 1 (f; 1

+

(iv)

(F ) 1 (f; n

)Lq

(v) (vi)

(F ) 1 (f; 0)Lq

! 1 (f; )Lq =

(F ) 1 (f; 1 )Lq

2 )Lq

n

(F ) 1 (f;

(F ) 1 (f;

)Lq ,

Uq = 0, i¤ f 2 CF R ([a; b]).

+

(F ) 1 (f; 2 )Lq ,

(F )

(F )

g; )Lq 1 (f; )Lq + a fuzzy random function.

(vii)

(F ) 1 (f;

1; 2

> 0.

> 0, n 2 N.

)Lq d e 1 (f; )Lq ( + 1) where d e is the ceiling of the number. (F ) 1 (f

> 0.

(F ) 1 (g;

(F ) 1 (f;

)Lq ,

)Lq ,

> 0,

> 0,

> 0. Here, f

g is

Uq )Lq is continuous on R+ , for f 2 CF R ([a; b]).

Proof. The proof is obvious. Proposition 17.10 (see [17]). Let f; g be fuzzy random variables from X ! RF . Then, we have the following. (i) Let c 2 R, then c (ii) f

f is a fuzzy random variable.

g is a fuzzy random variable.

For the de…nition of general fuzzy integral we follow [75] next. De…nition 17.11. Let ( ; ; ) be a complete -…nite measure space. We call F : ! RF measurable i¤ 8 closed B R the function F 1 (B) : ! [0; 1] de…ned by F

1

(B)(!) := sup F (!)(x); all ! 2 x2B

is measurable, see [46], [75]. Notice here that the concept of measurability is di¤erent than the Bmeasurability of De…nition 17.1. Theorem 17.12 ([75]). For F : ! RF , F (!) = 0 r 1 , the following are equivalent. (1) F is measurable, (2) 8r 2 [0; 1], F

(r)

(r)

, F+ are measurable.

(F

(r)

(r)

(!); F+ (!)) j

17.1 Introduction (r)

313

(r)

Following [75], given that for each r 2 [0; 1], F , F+ are integrable we have that the parametrized representation Z Z (r) (r) F d ; F+ d j0 r 1 A

A

is a fuzzy real number for each A 2 . The last fact leads to

De…nition 17.13 ([75]). A measurable function F : F (!) = (F

(r)

(r)

(!); F+ (!)) j 0

r

! RF , 1

(r)

is called integrable if for each r 2 [0; 1], F are integrable, or equivalently, (0) if F are integrable. In this case, the fuzzy integral of F over A 2 is de…ned by Z Z Z (r) (r) j0 r 1 : F+ d F d := F d ; A

A

A

By [14], F is integrable i¤ ! ! kF (!)kF is real-valued integrable. We need also

Theorem 17.14 ([75]). Let F; G :

! RF be integrable. Then

(1) Let a; b 2 R, then a F + b G is integrable and for each A 2 , Z Z Z (a F b G) d = a Fd b Gd ; A

A

A

(2) D(F; G) is a real-valued integrable function and for each A 2 , D

Z

Fd ;

A

In particular,

Z

Z

Gd

A

Z

A

D(F; G) d :

A

Fd F

Z

A

kF kF d :

We need De…nition 17.15. Let U open or compact (M; d) metric space and f : U ! RF . We say that f is fuzzy continuous at x0 2 U i¤ whenever xn ! x0 , then D(f (xn ); f (x0 )) ! 0. If f is continuous for every x0 2 U , we then call f a fuzzy continuous real number valued function. We denote the related space by CF (U ). Similarly one de…nes CF ([a; b]), [a; b] R.

314

17. Fuzzy Random Korovkin Theorems and Inequalities

De…nition 17.16. Let L : CF (U ) ,! CF (U ), where U is open or compact (M; d) metric space, such that L(c1

f

c2

g) = c1

L(f )

c2

L(g);

8c1 ; c2 2 R:

We call L a fuzzy linear operator. We give the following example of a fuzzy linear operator, etc. De…nition 17.17. Let f : [0; 1] ! RF be a fuzzy real function. The fuzzy algebraic polynomial de…ned by Bn(F ) (f )(x) =

n X

n k x (1 k

k=0

x)n

k

f

will be called the fuzzy Bernstein operator. Here summation. We also need

k n P

;

8x 2 [0; 1];

stands for the fuzzy

De…nition 17.18. Let f; g : U ! RF , U (M; d) metric space. We denote (r) (r) (r) (r) f % g, i¤ f (x) % g(x), 8x 2 U , i¤ f+ (x) g+ (x) and f (x) g (x), (r) (r) (r) (r) g , 8r 2 [0; 1], where g+ and f 8x 2 U , 8r 2 [0; 1], i¤ f+ (r) (r) [f (x)]r = [f (x); f+ (x)]. We give De…nition 17.19. Let L : CF (U ) ,! CF (U ) be a fuzzy linear operator, U open or compact (M; d) metric space. We say that L is positive, i¤ whenever f; g 2 CF (U ) are such that f % g then L(f ) % L(g), i¤ (r)

(r)

(L(g))+

(L(f ))+ and (r)

(L(f ))

(r)

(L(g))

;

8r 2 [0; 1]:

Here we denote (r)

[L(f )]r = (L(f ))

(r)

; (L(f ))+ ;

8r 2 [0; 1]:

An example of a fuzzy positive linear operator is the fuzzy Bernstein operator on the domain [0; 1], etc. For more see [18], [24], [32]. We mention Assumption 17.20 (see [24]). Let L be a fuzzy positive linear operator from CF (K), K compact (M; d) metric space, into itself. Here we assume ~ from C(K) into itself with that there exists a positive linear operator L the property (r) ~ (r) ); (17.6) (Lf ) = L(f

17.2 Auxilliary Material

315

respectively, for all r 2 [0; 1], 8f 2 CF (K). As an example again we mention the fuzzy Bernstein operator and the real Bernstein operator ful…lling the above assumption on [0; 1], etc. We apply the following Fuzzy Riesz Representation Theorem. Theorem 17.21 (see [24]). Let L be a fuzzy positive linear operator from CF (K) into itself as in Assumption 17:20, K compact (M; d) metric space. Then for each x 2 K there exists a unique positive …nite completed Borel measure x on K such that Z (Lf )(x) = f (t) x (dt); 8f 2 CF (K): K

17.2 Auxilliary Material We apply in proofs Remark 17.22. Let f : [a; b] ! LF (X; B; P ), [a; b] function. Then by Proposition 17.9(v) we get (F ) 1 (f; jx

jx

yj)Lq

yj

(F ) 1 (f;

)Lq ;

R be a fuzzy random

8x; y 2 [a; b] any

> 0:

(17.7) The main function space we are going to work on in the chapter is de…ned as follows. De…nition 17.23. Let (X; B; P ) be a probability space, [a; b] R, and the fuzzy random function f : [a; b] X ! RF such that f (t; !) is fuzzy continuous in t 2 [a; b] uniformly with respect to ! in X. I.e. 8" > 0 9 > 0 such that whenever jx yj ; x; y 2 [a; b], then D f (x; !); f (y; !)

";

8! 2 X:

U We denote the space of all these functions by CF R ([a; b]). U One can easily see that if f 2 CF R ([a; b]) then for each ! 2 X we have that f ( ; !) 2 CF ([a; b]) and f is q-mean uniformly continuous in t 2 [a; b], Uq i.e. f 2 CF q < +1, see De…nition 17.7. R ([a; b]), any 1

We mention

U U De…nition 17.24. Let L : CF R ([a; b]) ,! CF R ([a; b]) such that

L (c1

f1

c2

f2 ) = c1

L (f1 )

c2

L (f2 );

U We call L a fuzzy random linear operator on CF R ([a; b]).

The following motivate this chapter.

8c1 ; c2 2 R:

316

17. Fuzzy Random Korovkin Theorems and Inequalities

Example 17.25 (see [66], p. 656). For f : [0; 1] ! LF (X; B; P ), the fuzzy random polynomials de…ned by Bn(F ) (f )(x; !) :=

n X

k=0

n k x (1 k

x)n

k

k ; ! ; x 2 [0; 1]; ! 2 X n

f (F )

will be called a Bernstein-type. Clearly Bn ( )(x; !) is a fuzzy random linear operator, n 2 N. We have Theorem 17.26 (see [66], p. 656). For f : [0; 1] ! LF (X; B; P ) we have the estimate Z 1 3 (F ) f; p ; (17.8) D Bn(F ) (f )(x; !); f (x; !) P (d!) 1 2 n L1 X 8x 2 [0; 1], n 2 N. If moreover f satis…es the condition lim #0

(F ) 1 (f;

)L1 = 0;

then Bn(F ) (f )(x; !) uniformly with respect to x 2 [0; 1]. We mention

“1-mean” ! f (x; !); n ! +1

U U De…nition 17.27. Let L : CF R ([a; b]) ,! CF R ([a; b]) be a fuzzy random linear operator. We call L a positive fuzzy random linear operator i¤ whenU ever we have f; g 2 CF R ([a; b]) such that f % g, i.e. f (x; !) % g(x; !) for all (x; !) 2 [a; b] X then L f % L g, i.e. (L f )(x; !) % (L g)(x; !) for all (r) (r) (x; !) 2 [a; b] X, i¤ (L f )+ (x; !) (L g)+ (x; !) and (r)

(L f )

(x; !)

(r)

(L g)

(x; !);

8r 2 [0; 1]; 8(x; !) 2 [a; b]

X:

Here we denote (r)

(r)

[L (f )(x; !)]r = (L f ) (x; !); (L f )+ (x; !) ; 8(x; !) 2 [a; b] X:

8r 2 [0; 1]; (F )

An example of a positive fuzzy random linear operator is Bn ( )(x; !), etc. We give the useful Remark 17.28. Let L be a fuzzy positive linear operator from CF ([a; b]) ~ from into itself. We suppose that there exists a positive linear operator L C([a; b]) into itself with the property (r)

(Lf )

~ (r) ); = L(f

(17.9)

17.2 Auxilliary Material

317

respectively, 8r 2 [0; 1], 8f 2 CF ([a; b]). Then by Theorem 17.21, 8t 2 [a; b] there exists a unique positive …nite completed Borel measure t on [a; b] such that Z (Lf )(t) = f (s) t (ds); 8f 2 CF ([a; b]): (17.10) [a;b]

U Consequently for f 2 CF R ([a; b]) and since f ( ; !) 2 CF ([a; b]), 8! 2 X, we get that Z L f ( ; !) (t) = f (s; !) t (ds); 8t 2 [a; b]; 8! 2 X: (17.11) [a;b]

Of course here by (17.9) we have (r)

L(f ( ; !))

~ f (r) ( ; !) (t); (t) = L

8t 2 [a; b]; 8! 2 X; 8r 2 [0; 1]: (17.12)

We call mt :=

t ([a; b])

0:

(17.13)

By setting M (f )(t; !) := L f ( ; !) (t); that is M (f )(t; !) =

Z

(17.14)

f (s; !) t (ds);

(17.15)

[a;b]

from Theorem 17.14 (1) we have that M (c1

f

c2

g)(t; !) = c1

M (f )(t; !)

c2

M (g)(t; !);

(17.16)

U 8(t; !) 2 [a; b] X, 8f; g 2 CF R ([a; b]), 8c1 ; c2 2 R. Let CF R ([a; b]) := ff : [a; b] X ! RF : such that f (t; !) is fuzzy continuous in t 2 [a; b] and B-measurable in ! 2 Xg. Additionally we assume here that M (f )(t; !) is B-measurable in ! 2 X. Then U 8f 2 CF R ([a; b]):

M (f ) 2 CF R ([a; b]);

U That is M is a fuzzy random linear operator from CF R ([a; b]) into CF R ([a; b]). Thus by (17.12) we have (r)

(M (f ))

(t; !) = L(f ( ; !))

(r)

~ f (r) ( ; !) (t): (t) = L

U Let f; g 2 CF ([a; b]) such that f % g i¤ f 8r 2 [0; 1]. Then

and

(r)

g

(r)

~ f (r) ( ; !) (t) L

~ g (r) ( ; !) (t) L

~ f (r) ( ; !) (t) L +

~ g (r) ( ; !) (t): L +

(17.17) (r)

and f+

(r)

g+ ,

318

17. Fuzzy Random Korovkin Theorems and Inequalities (r)

(r)

(r)

(r)

That is (M (f )) (M (g)) and (M (f ))+ (M (g))+ , 8r 2 [0; 1]. Hence M is a positive fuzzy random linear operator. For example we notice that Bn(F ) f ( ; !) (x) =

n X

k=0

n k x (1 k

x)n

k

f

k ;! n

= Bn(F ) (f )(x; !);

(17.18) U 8x 2 [0; 1], 8! 2 X, 8f 2 CF R ([0; 1]), the last ful…lls all the above theory. So fuzzy operators like L, M are quite common, e.g. summation, integral operators in the fuzzy sense, therefore we study their approximation properties next. ~ from C([a; b]) Clearly, by Theorem 5 of [24], any positive linear operator L into itself induces a unique positive fuzzy linear operator L acting on CF ([a; b]), which in turn generates by (17.14) a unique positive fuzzy ranU dom linear operator M acting on CF R ([a; b]), so the class of M ’s is very rich.

17.3 Main Results We will use the following Proposition 17.29. Let (X; B; P ) be a probability space, [a; b] R, f 2 U CF R ([a; b]). Let L a fuzzy positive linear operator from CF ([a; b]) into itself ~ from C([a; b]) into itself for which there exists a positive linear operator L such that (r)

(Lg)

~ (r) ); = L(g

(17.19)

respectively, 8r 2 [0; 1], 8g 2 CF ([a; b]). We consider the positive fuzzy U random linear operator M acting on CF R ([a; b]) de…ned by U X; 8f 2 CF R ([a; b]): (17.20) We assume that M (f )(t; !) is B-measurable in ! 2 X. That is M (f ) 2 CF R ([a; b]). Then

M (f )(t; !) := L f ( ; !) (t);

D M (f )(t; !); f (t; !)

Z

8(t; !) 2 [a; b]

D f (s; !); f (t; !)

t (ds)

(17.21)

[a;b]

+ jmt where

t

1jD f (t; !); o~ ;

is as in (17.10) and mt as in (17.13).

8(t; !) 2 [a; b]

X;

17.3 Main Results

319

Proof. We observe that the B-measurable function (see Remark 13.39, p. 654, [66]) D M (f )(t; !); f (t; !)

(17:15)

=

Z

D

f (s; !) t (ds); f (t; !)

[a;b]

D

Z

f (s; !) t (ds); f (t; !)

mt

[a;b]

= D

Z

f (s; !) t (ds);

[a;b]

Z

!

!

+ D f (t; !) !

f (t; !) t (ds)

[a;b]

(by Theorem 17.14(2) and Lemma 1.2) Z D f (s; !); f (t; !) t (ds) + jmt

mt ; f (t; !)

+ D f (t; !)

mt ; f (t; !)

1jD f (t; !); o~ :

[a;b]

Here notice that f (t; !)

mt

(r)

(r)

mt (f (t; !)) ; mt (f (t; !))+ j 0 r 1 ( Z Z (r) (r) = f (t; !) d t; f (t; !) + d t =

=

[a;b]

Z

[a;b]

j0

r

1

)

f (t; !) t (ds):

[a;b]

By Remark 2 of [24] we trivially see that ~ mt = L(1)(t)

0;

8t 2 [a; b]:

(17.22)

We give the …rst main result. Theorem 17.30. Assume all terms and assumptions of Proposition 17.29 and Z D f (t; !); o~ dP (!) < 1; 8t 2 [a; b]: X

Then Z

X

D M (f )(t; !); f (t; !) dP (!) Z ~ jL(1)(t) 1j D f (t; !); o~ dP (!) ~ + L(1)(t) +

q

(17.23)

X

~ L(1)(t)

(F ) 1

~ f; (L((

t)2 )(t))1=2

L1

;

8t 2 [a; b];

320

17. Fuzzy Random Korovkin Theorems and Inequalities

and Z

sup t2[a;b]

D M (f )(t; !); f (t; !) dP (!)

X

~ kL(1)

1k1 sup

t2[a;b]

q ~ L(1)k 1

~ + kL(1) +

Z

(17.24)

D f (t; !); o~ dP (!)

X

(F ) 1

~ f; kL((

t)2 )(t)k1=2 1

L1

:

Proof. Integrating (17.21) we obtain Z

D M (f )(t; !); f (t; !) dP (!)

X

Z

X

Z

!

D f (s; !); f (t; !)

[a;b]

+ jmt

Z

1j

t (ds)

D f (t; !); o~ dP (!)

dP (!) =

X

(by D 0 and the facts that D f (s; !); f (t; !) is continuous in s 2 [a; b], by Lemma 1 of [11], also it is a real random variable in !, by Remark 13.39 of [66], p. 654 and thus by Proposition 3.3(i) of [17] it is jointly measurable in (s; !), and then being able to use Tonelli–Fubini’s theorem, p. 104 of [62] and thus see both double integrals make sense) Z

Z

[a;b]

D f (s; !); f (t; !) dP (!)

X

+ jmt

1j

Z

D f (t; !); o~ dP (!)

X

(h > 0) js (F ) f; 1

Z

[a;b]

+ jmt

1j

Z

tj h

h

t (ds) L1

D f (t; !); o~ dP (!)

X

(F ) 1 (f; h)L1

+ jmt

t (ds)

1j

Z

Z

js

[a;b]

X

tj h

t (ds)

D f (t; !); o~ dP (!)

(by (17.7))

17.3 Main Results

jmt

Z

D f (t; !); o~ dP (!) ! Z js tj (F ) + 1+ t (ds) 1 (f; h)L1 h [a;b] Z = jmt 1j D f (t; !); o~ dP (!) X ! Z 1 (F ) + mt + js tj t (ds) 1 (f; h)L1 h [a;b] jmt 0

(by choosing

1j

321

X

(by Cauchy–Schwarz inequality) Z 1j D f (t; !); o~ dP (!) X

+ @mt +

h :=

Z

(s

1p mt h

2

t)

Z

(s

[a;b]

t (ds)

[a;b]

t)2

!1=2

!1=2 1 A t (ds)

~ = L((

t)2 )(t)

for > 0 it is enough to assume t ([a; b] ftg) > 0) Z jmt 1j D f (t; !); o~ dP (!)

(F ) 1 (f; h)L1

1=2

> 0;

X

+ (mt +

p

mt )

(F ) 1

~ f; L((

t)2 )(t)

1=2 L1

;

by using (17.22) we have established (17.23). One can easily see that if ~ L((

t)2 )(t) = 0

then again (17.23) is valid. Clearly by Remark 13.39, p. 654, [66] D(f (t; !); o~) is a real random variable in ! 2 X, for each t 2 [a; b]. Next we notice that jD(f (x; !); o~) D(f (y; !); o~)j

D(f (x; !); f (y; !)) 8x; y 2 [a; b]; 8! 2 X:

Hence 8" > 0 9 > 0 such that whenever x; y 2 [a; b] with jx yj Z Z D f (x; !); o~ P (d!) D f (y; !); o~ P (d!) X X Z D f (x; !); f (y; !) P (d!) "; X

then

322

17. Fuzzy Random Korovkin Theorems and Inequalities

U1 U because f 2 CF R ([a; b]) by f 2 CF R ([a; b]). Therefore the function

Z

F (x) :=

D f (x; !); o~ P (d!);

X

is continuous in x 2 [a; b] and hence is bounded, i.e. kF (x)k1 < 1, making (17.24) valid. We need the following Proposition 17.31. All here as in Proposition 17.29 and Z

D(f (t; !); o~) P (d!) < 1;

Z

D(M (f )(t; !); f (t; !) dP (!)

q

X

q > 1; 8t 2 [a; b]:

Then 1=q

q

X

jmt +

1j

Z

1=q

q

D(f (t; !); o~) P (d!)

X

1 mt

1 q

Z

1+

[a;b]

js

tj h

(17.25) !1=q

q

(F ) 1 (f; h)Lq ;

d t (s)

h > 0; 8t 2 [a; b]: Proof. Let q > 1 then by (17.21) we have Z

q

1=q

D M (f )(t; !); f (t; !) dP (!)

X

Z

X

Z

q

D f (s; !); f (t; !)

t (ds)

!1=q

P (d!)

[a;b]

+ =: ( );

where := jmt

1j

Z

X

q

D(f (t; !); o~) P (d!)

1=q

:

(17.26)

17.3 Main Results 1 p

Let p > 1 such that Z

1=p mt

( )

+

= 1. Hence by Hölder’s inequality we have

Z

X

=

1 q

q

D f (s; !); f (t; !)

!1=q

t (ds)

P (d!)

[a;b]

+

(by Tonelli-Fubini’s theorem as in the proof of Theorem 17.30) !1=q Z Z 1=p mt + Dq f (s; !); f (t; !) P (d!) t (ds) [a;b]

X

(let h > 0) Z

1=p mt

(F ) 1

js

f;

[a;b]

(17:7)

323

Z

1=p mt

1+

js

[a;b]

tj h

t (ds)

h Lq

!1=q

q

tj

t (ds)

h

!1=q

q

+

(F ) 1 (f; h)Lq

+ :

We examine two cases and we give Theorem 17.32. Here we assume all as in Proposition 17.31. 1) Let q 2 N f1g. Then Z

1=q

Dq M (f )(t; !); f (t; !) P (d!)

X

~ jL(1)(t)

Z

1j

1=q

Dq f (t; !); o~ P (d!)

X

~ + (L(1)(t))

1

1 q

q X q ~ (L(1)(t))1 k

k q

k=0 (F ) 1

~ f; ((L(1

tjq ))(t))1=q

!1=q ;

(17.27)

Lq

8t 2 [a; b]: 2) Let q > 1 real. Then Z

1=q

Dq M (f )(t; !); f (t; !) P (d!)

X

~ jL(1)(t) 1

+2

Z

1j

X

1 q

1 q1 ~ ~ (L(1)(t)) (L(1)(t) + 1)1=q (F ) ~ f; ((L(j tjq ))(t))1=q q ; 1

8t 2 [a; b]:

1=q

Dq f (t; !); o~ P (d!)

L

(17.28)

324

17. Fuzzy Random Korovkin Theorems and Inequalities

When q 2 N have 3) Let q 2 N

f1g then (17.27) is sharper than (17.28). Furthermore we f1g. Then Z

sup t2[a;b]

1=q

Dq M (f )(t; !); f (t; !) P (d!)

X

~ kL(1) +

Z

1k1 sup

t2[a;b]

1 ~ kL(1)k 1 (F ) 1

1=q

Dq f (t; !); o~ P (d!)

X

q X

1 q

q ~ (L(1))1 k

k=0

~ f; k(L(j

k q

tjq ))(t)k1=q 1

1 Lq

!1=q

:

(17.29)

4) Let q > 1 real. Then Z

sup t2[a;b]

1=q

Dq M (f )(t; !); f (t; !) P (d!)

X

~ kL(1)

1k1 sup

t2[a;b]

1 q

+ 21

1=q

Dq f (t; !); o~ P (d!)

X

1 ~ kL(1)k 1 (F ) 1

When q 2 N

Z 1 q

~ kL(1) + 1k1=q 1 ~ f; k(L(j tjq ))(t)k1=q 1

Lq

:

(17.30)

f1g inequality (17.29) is sharper than (17.30).

Proof. 1) We notice that Z

1+

[a;b]

= mt +

js

tj h

q

d t (s) =

Z

q X q js tjk hk k

[a;b]

Z q 1 X q 1 js k hk [a;b]

k=0

tjk d t (s) +

k=1

Also we see for k = 1; : : : ; q Z

[a;b]

js

k

tj d t (s)

1 hq

Z

!

d t (s)

js

tjq d t (s):

!k=q

:

[a;b]

1 that

1 mt

k q

Z

[a;b]

js

q

tj d t (s)

17.3 Main Results

325

Hence Z

1+

js

[a;b]

q

tj

d t (s)

h

!k=q Z q 1 X q 1 1 kq mt + m js tjq d t (s) k hk t [a;b] k=1 Z 1 + q js tjq d t (s) h [a;b] !k=q Z q 1 (k=q) X q mt q = js tj d t (s) hk k [a;b] k=0

by choosing Z

h :=

[a;b]

!1=q

q

js

tj d t (s) 1=q

~ (L(j tjq ))(t) q X 1 k q mt q : k

= =

>0 (17.31)

k=0

That is, we got that Z

1+

js

[a;b]

tj h

q

d t (s)

q X 1 q mt k

k q

:

k=0

Hence proving (17.27) with the use of (17.25). The inequality (17.27) is true easily if our choice is easily hq :=

Z

[a;b]

tjq d t (s) = 0:

js

2) The function xq is convex for x js tj h

1+

2

!q

0, q > 1. Therefore

1+

js tjq hq

2

;

h > 0:

I.e. 1+

js

tj h

q

2q

1

1+

js

tjq hq

;

8s; t 2 [a; b]:

326

17. Fuzzy Random Korovkin Theorems and Inequalities

Hence Z

1+

js

tj

2

1

1 q

d t (s)

h

[a;b]

Z

!1=q

q

1+

js

[a;b]

= 21

1 q

d t (s)

hq

Z

1 mt + q h

!1=q

tjq

[a;b]

!!1=q

tjq d t (s)

js

by choosing again Z

h :=

[a;b]

=

21

js

tj

1 q

!1=q

q

js

tj d t (s)

>0

(mt + 1)1=q :

That is, we got that Z

1+

[a;b]

h

!1=q

q

21

d t (s)

1 q

(mt + 1)1=q :

Using (17.25) and the last estimate we obtain (17.28). Again if our above choice is h = 0 then (17.28) is still valid. When q 2 N f1g and mt > 0 we would like to prove that !1=q q X 1 1 k q q mt 21 q (mt + 1)1=q ; (17.32) k k=0

hence (17.27) is better than (17.28). Notice that (17.32) is trivially true when mt = 0. Equivalently we need valid q X 1 q mt k

k q

2q

1

(mt + 1)

2q

1

(1 + mt 1 ):

k=0

q X q mt k

, k=q

k=0

By calling z := mt

1

> 0, equivalently we need true q X q k=q z k

2q

1

(1 + z)

2q

1

(1 + z):

k=0

(1 + z

1=q q

)

,

17.3 Main Results

327

The last is true by the convexity of z q , z 0, q 2 N f1g. If mt = 0, then both (17.27) and (17.28) are trivially the same. Deriving (17.29), (17.30) from (17.27), (17.28), respectively, it is easy. Clearly Dq (f (t; !); o~) is a real random variable in ! 2 X, 8t 2 [a; b]. Additionally we notice that 8" > 0 9 > 0 such that whenever x; y 2 [a; b] with jx yj then Z

Z

1=q q

D (f (x; !); o~)dP (!)

X

1=q

Dq (f (y; !); o~)dP (!)

X

Z

1=q

Dq (f (x; !); f (y; !))dP (!)

U "; by f 2 CF R ([a; b]):

X

Hence proving that the function Z G(x) := Dq (f (x; !); o~)dP (!)

1=q

;

X

is continuous in x 2 [a; b]. Therefore kGk1 < 1, making the inequalities (17.29), (17.30) valid. Similar general results using a di¤erent initial estimate follow. Lemma 17.33. Let f : [a; b] ! LF (X; B; P ) be fuzzy random function, 1 q < 1, > 0. Then (F ) 1 (f; jx

yj)Lq

1+

y)2

(x

(F ) 1 (f;

2

)Lq ;

8x; y 2 [a; b]: (17.33)

Proof. We have by (17.7) that (F ) 1 (f; jx

Let jx

yj > , thus

yj)Lq jx yj

1+

yj (F ) 1 (f; jx

yj

(F ) 1 (f;

)Lq :

(17.34)

> 1. Then

R.H.S.(17:34) Let jx

jx

1+

(x

y)2

(F ) 1 (f;

2

)Lq :

then yj)

(F ) 1 (f;

)Lq

1+

(x

y)2 2

(F ) 1 (f;

)Lq :

We present Theorem 17.34. Assume all terms and assumptions of Proposition 17.29 and Z D f (t; !); o~ dP (!) < 1; 8t 2 [a; b]: X

328

17. Fuzzy Random Korovkin Theorems and Inequalities

Then 1) Z

D M (f ); (t; !); f (t; !) dP (!) Z ~ jL(1)(t) 1j D f (t; !); o~ dP (!)

X

X

~ + (L(1)(t) + 1)

(F ) 1

~ f; L((

t)2 )(t)

1=2 L1

; (17.35)

8t 2 [a; b]; and 2) sup t2[a;b]

Z

D M (f )(t; !); f (t; !) dP (!)

X

~ kL(1)

1k1 sup

t2[a;b]

~ + kL(1) + 1k1

(F ) 1

Z

D f (t; !); o~ dP (!)

X

~ f; kL((

t)2 )(t)k1=2 1

L1

: (17.36)

Proof. Initially from the proof of Theorem 17.30 we have Z Z D M (f )(t; !); f (t; !) dP (!) jmt 1j D f (t; !); o~)dP (!) X X Z (F ) + tj)L1 t (s) 1 (f; js [a;b]

(let h > 0) Z (17:33) jmt 1j D f (t; !); o~)dP (!) X ! Z (s t)2 (F ) + 1+ t (ds) 1 (f; h)L1 2 h [a;b] Z = jmt 1j D f (t; !); o~)dP (!) X ! Z 1 (F ) 2 (s t) t (ds) + mt + 2 1 (f; h)L1 h [a;b]

taking h :=

Z

(s

[a;b]

= jmt 1j

Z

X

2

t)

t (dt)

!1=2

~ = L((

t)2 )(t)

1=2

>0 (17:37))

D f (t; !); o~ dP (!) +(mt +1)

(F ) 1

~ f; L((

t)2 )(t)

1=2 L1

;

17.3 Main Results

329

8t 2 [a; b]. That is proving (17.35). The above choice (17.37) of h if h = 0 makes again (17.35) valid. Inequality (17.36) is now clear. Finally we get the very useful Theorem 17.35. Assume all terms and assumptions of Proposition 17.29 and Z D f (t; !); o~ dP (!) < 1; 8t 2 [a; b]: X

Then 1) Z

X

D M (f )(t; !); f (t; !) dP (!) Z ~ jL(1)(t) 1j D f (t; !); o~ dP (!) X

+ min (F ) 1

~ L(1)(t) + ~ f; (L((

q

~ ~ L(1)(t) ; (L(1)(t) + 1)

t)2 )(t))1=2

L1

; 8t 2 [a; b];

(17.38)

and 2)

sup t2[a;b]

Z

D M (f )(t; !); f (t; !) dP (!)

X

~ kL(1)

1k1 sup

t2[a;b]

~ + min kL(1) + (F ) 1

~ f; kL((

Z

D f (t; !); o~ dP (!)

X

q ~ ~ L(1)k 1 ; kL(1) + 1k1 t)2 )(t)k1=2 1

L1

:

(17.39)

Proof. Obvious from Theorems 17.30 and 17.34. The corresponding results for q > 1 follow. Theorem 17.36. Here we assume all as in Proposition 17.31. Let q 2 N f1g. Then

330

17. Fuzzy Random Korovkin Theorems and Inequalities

1) Z

1=q

Dq M (f )(t; !); f (t; !) dP (!)

X

~ jL(1)(t)

Z

1j

1=q

q

D(f (t; !); o~) dP (!)

X q X q k

1 q

1 ~ + (L(1)(t))

~ (L(1))(t)

1

k q

k=0 (F ) 1

~ f; ((L(

t)2q )(t))1=2q

Lq

!1=q

;

(17.40)

8t 2 [a; b]: And also holds 2) sup t2[a;b]

Z

1=q

q

D M (f )(t; !); f (t; !) dP (!)

X

~ kL(1)

Z

1k1 sup

t2[a;b]

(F ) 1

k q

k=0

~ f; k(L(

(17.41)

X

q X q ~ (L(1))1 k

1 q

1 ~ + kL(1)k 1

1=q

Dq f (t; !); o~ dP (!)

2q

t)

)(t)k1=2q : 1 Lq

1

!1=q

Let q > 1 real. Then 3) Z

1=q

q

D M (f )(t; !); f (t; !) dP (!)

X

~ jL(1)(t) 1 q

+ 21

1j

Z

1=q

q

D(f (t; !); o~) dP (!)

X 1

~ (L(1)(t))

1 q

~ (L(1)(t) + 1)1=q

(F ) 1

(17.42) ~ f; ((L(

t)2q )(t))1=2q

Lq

8t 2 [a; b]. And also holds 4) sup t2[a;b]

Z

1=q

q

D M (f )(t; !); f (t; !) dP (!)

X

~ kL(1) + 21

1 q

1k1 sup

t2[a;b] 1

1

Z

1=q

Dq (f (t; !); o~)dP (!)

(17.43)

X

q 1=q ~ ~ kL(1)k 1 kL(1) + 1k1

(F ) 1

~ f; k(L(

t)2q )(t)k1=2q 1

Lq

:

;

17.3 Main Results

331

When q 2 N f1g then inequality (17.40) is sharper than (17.42) and inequality (17.41) is sharper than (17.43). Note 17.37. We see later that inequalities (17.40)–(17.43) and/or inequalities (17.27)–(17.30) can be used to prove “q-mean”convergence with rates of a sequence of M ’s to unit operator I. Proof. Initially from the proof of Proposition 17.31 we obtain Z

1=q

q

D(M (f )(t; !); f (t; !) dP (!)

X

Z

+

Z

X

D f (s; !); f (t; !)

( as in (17.26)) Z 1

1 mt

+

t (dt)

[a;b]

q

[a;b]

(F ) 1 (f; js

tj)Lq

q

!q

!1=q

dP (!) !1=q

d t (s)

(let h > 0) (by (17.33)) +

1 mt

1 q

Z

1+

(s

[a;b]

=

t)2 h2

!1=q

q

d t (s)

(F ) 1 (f; h)Lq

: ( ):

1) Let q 2 N

f1g. We observe that

! q X q (s t)2k d t (s) 1+ d t (s) = h2 h2k [a;b] [a;b] k=0 k ! ! Z Z q 1 X 1 q 1 = mt + (s t)2k d t (s) + 2q (s t)2q d t (s) : h k h2k [a;b] [a;b]

Z

t)2

(s

Z

q

k=1

For k = 1; : : : ; q Z

q k

1,

(s

> 1 and by Hölder’s inequality we have

2k

t) d t (s)

1 mt

[a;b]

k q

Z

(s

!k=q

2q

t) d t (s)

[a;b]

:

Hence Z

[a;b]

1+

(s

t)2 h2

q

d t (s)

q 1 (k=q) X q mt k h2k

k=0

Z

[a;b]

(s

2q

!k=q

t) d t (s)

332

17. Fuzzy Random Korovkin Theorems and Inequalities

by choosing Z

h :=

!1=2q

2q

(s

t) d t (s)

[a;b]

~ (L(( t)2q ))(t) q X 1 k q mt q : k

=

1=2q

>0 (17.44)

k=0

That is Z

1+

q X 1 q mt k

d t (s)

h2

[a;b]

!1=q

q

t)2

(s

k q

k=0

!1=q

:

(17.45)

Thus by (17.44) and (17.45) we have ~ + L(1)(t)

( )

q X q ~ (L(1)(t))1 k

1 q

1

k q

k=0 (F ) 1

~ f; ((L((

t)2q ))(t))1=2q

Lq

!1=q

; 8t 2 [a; b]:

That is establishing (17.40). When the choice (17.44) of h = 0 then again (17.40) is trivially valid. 2) Let now q > 1 real, then again by convexity of xq , x 0 we have 1+

t)2

(s

q

2q

h2

1

1+

t)2q

(s

; h > 0; 8s; t 2 [a; b]:

h2q

Hence ( )

+

1 q

1 mt

21

1 q

Z

1+

h2q

[a;b]

=

1

+2

1 q

1 mt

1 q

"

1 mt + 2q h

t)2q

(s

Z

(s

!1=q

d t (s) 2q

#1=q

t) d t (s)

[a;b]

(F ) 1 (f; h)Lq

(F ) 1 (f; h)Lq

(let h > 0 as in (17.44)) =

+ 21

=

+ 21

1 q

1

1

(F )

mt q (mt + 1)1=q 1 (f; h)Lq 1 1 q1 ~ ~ q (L(1)(t)) (L(1)(t) + 1)1=q

(F ) 1

~ f; ((L((

t)2q ))(t))1=2q

8t 2 [a; b]. That is proving (17.42). When the choice (17.44) for h = 0 then inequality (17.42) is trivially valid. Inequalities (17.41) and (17.43) derive easily from (17.40) and (17.42), respectively, and they are valid, similarly, as inequalities (17.29) and (17.30). The comparison of inequalities is the same as in Theorem 17.32.

Lq

;

17.3 Main Results

333

Finally we derive Theorem 17.38. Here we suppose all as in Proposition 17.31. Let q 2 N f1g. Then 1) Z

1=q

Dq M (f )(t; !); f (t; !) dP (!)

X

~ jL(1)(t)

Z

1j

1=q

Dq f (t; !); o~ dP (!)

X q X q 1 ~ ((L(1))(t)) k

1 q

~ + (L(1)(t))

1

k q

k=0 (F ) 1

min (F ) 1

~ f; ((L(j

~ f; ((L(

tjq ))(t)1=q

t)2q )(t))1=2q

Lq

Lq

!1=q

;

;

(17.46)

8t 2 [a; b]: And also holds 2) sup t2[a;b]

Z

1=q

Dq M (f )(t; !); f (t; !) dP (!)

X

~ kL(1)

1k1 sup

t2[a;b]

1 ~ + kL(1)k 1

(F ) 1

1=q

Dq f (t; !); o~ dP (!)

X

q X q ~ (L(1))1 k

1 q

k=0 (F ) 1

min

Z

!1=q

k q

1 1=q tj ))(t)k1 Lq ; t)2q )(t)k1=2q : 1 Lq

~ f; k(L(j

~ f; k(L(

q

(17.47)

Let q > 1 real. Then 3) Z

q

1=q

D M (f )(t; !); f (t; !) dP (!)

X

~ jL(1)(t) +2

1

1 q

1j

Z

X 1

~ (L(1)(t))

min (F ) 1

(F ) 1

1=q

q

D(f (t; !); o~) dP (!) 1 q

~ (L(1)(t) + 1)1=q ~ f; ((L(j tjq ))(t))1=q

~ f; ((L(

t)2q )(t))1=2q

Lq

Lq

;

; ;

(17.48)

334

17. Fuzzy Random Korovkin Theorems and Inequalities

8t 2 [a; b]: And also holds 4) Z

sup t2[a;b]

1=q

Dq M (f )(t; !); f (t; !) P (d!)

X

~ kL(1) 1 q

+ 21

Z

1k1 sup

t2[a;b]

(F ) 1

min (F ) 1

X

1 q

1 ~ kL(1)k 1

1=q

Dq f (t; !); o~ P (d!)

~ kL(1) + 1k1=q 1 q ~ f; k(L(j tj ))(t)k1=q 1

~ f; k(L(

t)2q )(t)k1=2q 1

Lq

Lq

;

:

(17.49)

When q 2 N f1g then inequality (17.46) is sharper than (17.48) and (17.47) sharper than (17.49). Proof. By Theorems 17.32 and 17.36. We give Corollary 17.39. All here as in Proposition 17.29 and Z D2 f (t; !); o~ dP (!) < 1; 8t 2 [a; b]: X

Then 1) Z

1=2

D2 M (f )(t; !); f (t; !) P (d!)

X

~ jL(1)(t)

1j

Z

(17.50) 1=2

D2 f (t; !); o~ P (d!)

X

1=2 ~ 1=2 ~ ~ + (L(1)(t)) L(1)(t) + 2(L(1)(t)) +1 (F ) 1

~ f; ((L((

t)2 ))(t))1=2

L2

1=2

; 8t 2 [a; b]:

and 2) sup t2[a;b]

Z

1=2

D2 M (f )(t; !); f (t; !) P (d!)

X

~ kL(1)

1k1 sup

t2[a;b]

Z

1=2

D2 f (t; !); o~ P (d!)

X

1=2 ~ 1=2 ~ ~ + kL(1)k + 1k1=2 1 kL(1) + 2(L(1)) 1 (F ) 2 1=2 ~ f; k(L(( t) ))(t)k1 q : 1

L

(17.51)

17.3 Main Results

335

Proof. By Theorem 17.32, inequalities (17.27) and (17.29). All inequalities presented in this chapter are of Shisha–Mond type (see [96]) in the fuzzy-random sense. We will derive next some Fuzzy-Random Korovkin Theorems regarding the spaces of functions Z U Kq ([a; b]) := f 2 CF R ([a; b]) : Dq f (t; !); o~ dP (!) < 1; 8t 2 [a; b] ; X

where 1

q < 1. We observe that if 1 Kq ([a; b])

k < 1 such that k

q then

Kk ([a; b]):

For the above purpose we need to put together the following assumptions and settings. Assumption 17.40. Let (X; B; P ) be a probability space, [a; b] R, f 2 U CF R ([a; b]). Let fLn gn2N be a sequence of fuzzy positive linear operators from CF ([a; b]) into itself for which there exists a corresponding sequence ~ n2N from C([a; b]) into itself such that of positive linear operators fLg (r)

(Ln g)

~ n (g (r) ); =L

(17.52)

respectively, 8r 2 [0; 1], 8n 2 N, 8g 2 CF ([a; b]). We then consider the seU quence of positive fuzzy random linear operators fMn gn2N from CF R ([a; b]) into CF R ([a; b]) de…ned by Mn (f )(t; !) := Ln (f ( ; !))(t);

(17.53)

U 8(t; !) 2 [a; b] X, 8n 2 N, 8f 2 CF R ([a; b]). Here I is the fuzzy random unit operator, i.e. I(f )(t; !) = f (t; !), 8(t; !) 2 [a; b] X. We assume also ~ n (1)gn2N is bounded. that fL From Theorem 17.35 we have

Corollary 17.41. Here all are as in Assumption 17.40, and Z D f (t; !); o~ dP (!) < 1; 8t 2 [a; b]: X

Then 1) Z

X

D Mn (f )(t; !); f (t; !) dP (!) Z ~ n (1)(t) 1j jL D f (t; !); o~ dP (!) X

+ min

~ n (1)(t) + L (F ) 1

~ n (( f; (L

q

~ n (1)(t) ; L ~ n (1)(t) + 1 L t)2 )(t))1=2

L1

;

(17.54)

336

17. Fuzzy Random Korovkin Theorems and Inequalities

8t 2 [a; b]; 8n 2 N; and 2) Z sup D Mn (f )(t; !); f (t; !) dP (!) t2[a;b]

X

~ n (1) kL

Z

1k1 sup

t2[a;b]

~ n (1) + + min kL (F ) 1

D f (t; !); o~ dP (!)

X

q ~ n (1)k1 ; kL ~ n (1) + 1k1 L

~ n (( f; kL

t)2 )(t)k1=2 1

L1

;

(17.55)

8n 2 N: From Corollary 17.39 we obtain Corollary 17.42. Here all are as in Assumption 17.40, and Z D2 f (t; !); o~ dP (!) < 1; 8t 2 [a; b]: X

Then 1) Z

1=2

D2 Mn (f )(t; !); f (t; !) P (d!)

X

~ n (1)(t) jL

1j

Z

1=2

D2 f (t; !); o~ P (d!)

X

~ n (1)(t))1=2 L ~ n (1)(t) + 2(L ~ n (1)(t))1=2 + 1 + (L (F ) 1

~ n (( f; ((L

t)2 ))(t))1=2

L2

;

1=2

(17.56)

8t 2 [a; b]; 8n 2 N: And also holds 2) sup t2[a;b]

Z

1=2

D2 Mn (f )(t; !); f (t; !) P (d!)

X

~ n (1) kL +

8n 2 N:

1k1 sup

t2[a;b]

Z

1=2

D2 f (t; !); o~ P (d!)

X

~ n (1)k1=2 kL ~ n (1) + kL 1 (F ) ~ n (( f; k(L 1

~ n (1))1=2 + 1k1=2 2(L 1 t)2 ))(t)k1=2 1

L2

;

(17.57)

17.3 Main Results

337

Note 17.43. One sees from [96] that ~ n (( L

t)2 ) (t)

1

~ n (x2 )(t) kL

t2 k1

+2ckLn (x)(t)

(17.58)

tk1 + c2 kLn (1)(t)

1k1 ;

where c := max(jaj; jbj), 8n 2 N. Then one from the above fuzzy random Shisha–Mond type inequalities (17.55) and (17.57) derives the following basic fuzzy random Korovkin Theorems, see also [78]. Theorem 17.44. Here all are as in Assumption 17.40. Furthermore assume that u u u ~ n (1) ! ~ n (id) ! ~ n (id2 ) ! L 1; L id; L id2 ;

as n ! 1, where u means uniformly and id is the identity map. Then Z lim D Mn (f )(t; !); f (t; !) dP (!) = 0; 8f 2 K1 ([a; b]): n!1

X

1;t

(17.59)

I.e. Mn (f )(t; !)

“1-mean” ! f (t; !); n!1

uniformly, 8f 2 K1 ([a; b]), that is uniformly Mn on K1 ([a; b]).

(17.60)

“1-mean” ! I, as n ! 1,

Proof. Using (17.55), (17.58) and Proposition 17.9(ii). We continue with the second basic fuzzy random Korovkin theorem. Theorem 17.45. Here all are as in Assumption 17.40. Furthermore suppose that u u u ~ n (1) ! ~ n (id) ! ~ n (id2 ) ! L 1; L id; L id2 ; as n ! 1:

Then lim

n!1

Z

D2 Mn (f )(t; !); f (t; !) P (d!)

X

= 0; 1;t

I.e. Mn (f )(t; !)

8f 2 K2 ([a; b]): (17.61)

“2-mean” ! f (t; !);

uniformly, 8f 2 K2 ([a; b]), that is uniformly Mn on K2 ([a; b]).

(17.62)

“2-mean” ! I, as n ! 1,

Proof. From (17.57), (17.58) and Proposition 17.9(ii). The related general fuzzy random Korovkin theorem follows. Theorem 17.46. Here all are as in Assumption 17.40, q > 2. Furthermore we assume that

338

17. Fuzzy Random Korovkin Theorems and Inequalities

~ n (1) (i) L and

u ! 1, n!1

(ii) ~ n (j lim k(L

n!1

tjq ))(t)k1 = 0;

(17.63)

or ~n( (ii)0 lim k(L n!1

t)2q )(t)k1 = 0.

Then lim

n!1

Z

Dq Mn (f )(t; !); f (t; !) P (d!)

X

1;t

I.e. Mn (f )(t; !)

= 0; 8f 2 Kq ([a; b]): (17.64)

“q-mean” ! f (t; !);

uniformly, 8f 2 Kq ([a; b]), that is uniformly Mn on Kq ([a; b]).

(17.65)

“q-mean” ! I, as n ! 1,

Proof. By (17.30) or (17.43) and Proposition 17.9(ii). In fact (ii)0 implies (ii). So one can use (ii) or (ii)0 as long as it is easier to be veri…ed. ~ The case mt = L(1)(t) = 1, 8t 2 [a; b] is a very important and common one. Then all results of the chapter simplify a lot as follows. Proposition 17.47. All here as in Proposition 17.29 and mt = 1, 8t 2 [a; b]. Then Z D M (f )(t; !); f (t; !) D f (s; !); f (t; !) t (ds); 8(t; !) 2 [a; b] X; [a;b]

(17.66)

where

t

is as in (17.10).

Proof. We notice that the B-measurable function Z Z (17:15) D M (f )(t; !); f (t; !) = D f (s; !) t (ds); [a;b]

(by Theorem 17.14(2))

Z

f (t; !) t (ds)

[a;b]

D f (s; !); f (t; !)

[a;b]

Thus we obtain Theorem 17.48. All here as in Proposition 17.47. Then

t (ds):

!

17.3 Main Results

339

1) Z

D M (f )(t; !); f (t; !) dP (!)

2

(F ) 1

~ f; (L((

t)2 )(t))1=2

L1

X

; 8t 2 [a; b]; (17.67)

and 2) sup t2[a;b]

Z

D M (f )(t; !); f (t; !) dP (!)

2

(F ) 1

X

~ f; kL((

t)2 )(t)k1=2 1

L1

:

(17.68)

Proof. By integrating (17.66), then it follows, in a simpler way, as the proof of Theorem 17.30. Also we have Proposition 17.49. All here as in Proposition 17.47, q > 1. Then Z

1=q

q

D(M (f )(t; !); f (t; !) dP (!)

X

Z

1+

[a;b]

js

tj h

!1=q

q

d t (s)

(F ) 1 (f; h)Lq ;

(17.69)

h > 0; 8t 2 [a; b]:

Proof. Using (17.66) in exactly the same but simpler manner as in the proof of Proposition 17.31. We present Theorem 17.50. All here as in Proposition 17.47, q > 1. Then 1) Z

1=q

Dq M (f )(t; !); f (t; !) P (d!)

(17.70)

X (F ) 1

2

~ f; ((L(j

tjq ))(t))1=q

Lq

; 8t 2 [a; b];

and 2) sup t2[a;b]

Z

1=q

Dq M (f )(t; !); f (t; !) P (d!)

X

2

(F ) 1

~ f; k(L(j

tjq ))(t)k1=q 1

Lq

:

(17.71)

Proof. We use (17.69) and it follows similarly as the proof of Theorem 17.32.

340

17. Fuzzy Random Korovkin Theorems and Inequalities

We also give Theorem 17.51. Here all as in Proposition 17.29, mt = 1, 8t 2 [a; b] and q > 1. Then 1) Z

1=q

Dq M (f )(t; !); f (t; !) dP (!)

X

2

(F ) 1

~ f; ((L(

t)2q )(t))1=2q

Lq

;

(17.72)

8t 2 [a; b]; and 2) sup t2[a;b]

Z

1=q

Dq M (f )(t; !); f (t; !) dP (!)

X

2

(F ) 1

~ f; k(L(

t)2q )(t)k1=2q 1

Lq

:

(17.73)

Proof. Similar to the proof of Theorem 17.36. We derive Theorem 17.52. Here all as in Proposition 17.29, mt = 1, 8t 2 [a; b] and q > 1. Then 1) Z

1=q

Dq M (f )(t; !); f (t; !) dP (!)

(17.74)

X (F ) 1

2 min

(F ) 1

~ f; ((L(j

~ f; ((L(

tjq ))(t))1=q

t)2q )(t))1=2q

Lq

Lq

;

; 8t 2 [a; b];

and 2) sup t2[a;b]

Z

1=q

Dq M (f )(t; !); f (t; !) dP (!)

X

2 min

(F ) 1 (F ) 1

~ f; k(L(j

~ f; k(L(

tjq ))(t)k1=q 1

t)2q )(t)k1=2q 1

Lq

Lq

:

; (17.75)

Proof. From Theorems 17.50 and 17.51. We have Corollary 17.53. All here as in Proposition 17.29, mt = 1, 8t 2 [a; b]. Then

17.3 Main Results

341

1) Z

1=2

D2 M (f )(t; !); f (t; !) P (d!)

X

2

(F ) 1

~ f; ((L((

Z

D2 M (f )(t; !); f (t; !) P (d!)

t)2 ))(t))1=2

L2

; 8t 2 [a; b]; (17.76)

and 2) sup t2[a;b]

1=2

X (F ) 1

2

~ f; k(L((

t)2 ))(t)k1=2 1

L2

:

(17.77)

Proof. By Theorem 17.50, q = 2. Corollary 17.54. Here all as in Assumption 17.40, mt = 1, 8t 2 [a; b]. Then 1) Z D Mn (f )(t; !); f (t; !) dP (!) X

2

(F ) 1

~ n (( f; (L

t)2 )(t))1=2

L1

;

(17.78)

8t 2 [a; b]; 8n 2 N; and 2) sup t2[a;b]

Z

D Mn (f )(t; !); f (t; !) dP (!)

X

2

(F ) 1

~ n (( f; k(L

t)2 ))(t)k1=2 1

L1

;

8n 2 N: (17.79)

Proof. By Theorem 17.48. Corollary 17.55. Here all as in Assumption 17.40, mt = 1, 8t 2 [a; b]. Then 1) Z

1=2

D2 Mn (f )(t; !); f (t; !) P (d!)

X

2 8t 2 [a; b]; 8n 2 N; and

(F ) 1

~ n (( f; ((L

t)2 ))(t))1=2

L2

;

(17.80)

342

17. Fuzzy Random Korovkin Theorems and Inequalities

2) sup t2[a;b]

Z

1=2

D2 Mn (f )(t; !); f (t; !) P (d!)

X

2

(F ) 1

~ n (( f; k(L

t)2 ))(t)k1=2 1

L2

;

(17.81)

8n 2 N: Proof. By Theorem 17.50, q = 2. We give now the following fuzzy random Korovkin Theorems for the case ~ of L(1)(t) = 0, 8t 2 [a; b]. Theorem 17.56. Here all are as in Assumption 17.40. Furthermore suppose that ~ n (1)(t) = 1; L

u u ~ n (id2 ) ! ~ n (id) ! id2 ; as n ! 1: 8t 2 [a; b]; L id; L

Then lim

n!1

Z

D Mn (f )(t; !); f (t; !) dP (!)

X

1;t

I.e. Mn (f )(t; !)

U = 0; 8f 2 CF R ([a; b]):

(17.82)

“1-mean” ! f (t; !);

U uniformly, 8f 2 CF R ([a; b]), that is uniformly Mn U on CF R ([a; b]).

(17.83)

“1-mean” ! I, as n ! 1,

Proof. Using (17.79) and (17.58) and Proposition 17.9(ii). We continue with Theorem 17.57. Here all are as in Assumption 17.40. Furthermore assume that u u ~ n (1)(t) = 1; 8t 2 [a; b]; L ~ n (id) ! ~ n (id2 ) ! L id; L id2 ; as n ! 1:

Then lim

n!1

Z

D2 Mn (f )(t; !); f (t; !) P (d!)

X

1;t

I.e. Mn (f )(t; !)

U = 0; 8f 2 CF R ([a; b]):

“2-mean” ! f (t; !);

(17.84) (17.85)

“2-mean” U uniformly, 8f 2 CF ! I, as n ! 1, R ([a; b]), that is uniformly Mn “2-mean” “1-mean” U on CF R ([a; b]). Notice here that Mn ! I implies Mn ! I, uniformly, i.e. Theorem 17.56.

17.4 Application

343

Proof. Using (17.81) and (17.58) and Proposition 17.9(ii). Finally we present ~ n (1)(t) = 1, Theorem 17.58. Here all are as in Assumption 17.40, L 8t 2 [a; b] and q > 2. We assume further that ~ n (j (i) lim k(L n!1

tjq ))(t)k1 = 0,

or

(ii)0 ’

~n( lim k(L

n!1

t)2q )(t)k1 = 0:

(17.86)

Then lim

n!1

Z

Dq Mn (f )(t; !); f (t; !) P (d!)

X

U 8f 2 CF R ([a; b]):

= 0; 1;t

I.e. Mn (f )(t; !)

(17.87)

“q-mean” ! f (t; !);

U uniformly, 8f 2 CF R ([a; b]) that is uniformly Mn U on CF R ([a; b]).

(17.88)

“q-mean” ! I, as n ! 1,

Proof. By (17.71) or (17.73) and Propositoin 17.9(ii). “q-mean” Remark 17.59. 1) Notice here from (17.87), that Mn ! I implies “k-mean” U Mn ! I, uniformly for any 1 k q < 1 on CF R ([a; b]). 2) In the case of mt = 1, 8t 2 [a; b], all presented results here did not require the condition Z Dq f (t; !); o~ P (d!) < 1; 8t 2 [a; b]; 1 q < 1; X

as they did the earlier ones for general mt 0. 3) One can do related research for other domains other than [a; b], e.g. [0; +1), multivariate domains in Rk , k > 1 and K compact convex subset of a metric space. Of course not all results can pass through there.

17.4 Application We consider here the fuzzy random Bernstein polynomials Bn(F ) (f )(x; !) =

n X

k=0

n k x (1 k

x)n

k

f

k ;! ; n

344

17. Fuzzy Random Korovkin Theorems and Inequalities

U 8x 2 [0; 1], 8! 2 X, 8f 2 CF R ([0; 1]), 8n 2 N, see (17.18). We apply …rst here (17.79) for

Mn (f )(t; !) = Bn(F ) (f )(t; !); 8(t; !) 2 [a; b]

X;

~ n = Bn the real Bernstein operator and L Bn (g)(x) =

n X n k x (1 x)n k

k

k n

g

k=0

;

8g 2 C([0; 1]); 8x 2 [0; 1]; 8n 2 N:

Clearly Bn ((

t)2 )(t) =

Bn ((

t)2 )(t)

t(1

t) n

Hence 1=2 1

;

t 2 [0; 1]:

1 p ; 2 n

8n 2 N:

Notice also that (Bn (1))(t) = 1, 8t 2 [0; 1]. (F ) Clearly here Bn (f )(t; !) ful…ll Assumption 17.40. Thus by (17.79) we obtain Z sup D Bn(F ) (f )(t; !); f (t; !) dP (!) t2[0;1]

X

2

(F ) 1

1 f; p 2 n

;

(17.89)

L1

U 8f 2 CF R ([0; 1]); 8n 2 N: Similarly, from (17.81) we obtain

sup t2[0;1]

Z

1=2

D2 Bn(F ) (f )(t; !); f (t; !) dP (!)

X (F ) 1

2

1 f; p 2 n

; L2

U 8f 2 CF R ([0; 1]); (17.90)

8n 2 N: Finally, from (17.75) for q > 2 we obtain sup t2[0;1]

Z

1=q

Dq Bn(F ) (f )(t; !); f (t; !) dP (!)

X

2 min (F ) 1 U 8f 2 CF R ([0; 1]); 8n 2 N:

(F ) 1

f; k(Bn (j

f; k(Bn (

tjq ))(t)k1=q 1

t)2q )(t)k1=2q 1

Lq

; ;

Lq

; (17.91)

17.4 Application

345

In particular, if f is additionally of Lipschitz type, i.e. Z D f (x; !); f (y; !) P (d!) jx yj; > 0; 8x; y 2 [0; 1]; (17.92) X

then

(F ) 1 (f;

and (F ) 1

)L1

1 f; p 2 n

L1

; p ; 2 n

> 0; 8n 2 N:

(17.93) (17.94)

Hence sup t2[0;1]

Z

X

D Bn(F ) (f )(t; !); f (t; !) dP (!)

p ; 8n 2 N; n

(17.95)

U 8f 2 CF R ([0; 1]) which is of Lipschitz type (17.92). Inequality (17.95) improves the corresponding inequality from (17.8), since over there we only get Z 3 p ; 8n 2 N: (17.96) sup D Bn(F ) (f )(x; !); f (x; !) P (d!) 2 n x2[0;1] X

18 FUZZY-RANDOM NEURAL NETWORK APPROXIMATION OPERATORS, UNIVARIATE CASE

In this chapter we study the rate of pointwise convergence in the q-mean to the Fuzzy-Random unit operator of very precise univariate Fuzzy-Random neural network operators of Cardaliaguet– Euvrard and “Squashing” types. These Fuzzy-Random operators arise in a natural and common way among Fuzzy-Random neural networks. These rates are given through ProbabilisticJackson type inequalities involving the Fuzzy-Random modulus of continuity of the engaged Fuzzy-Random function or its Fuzzy derivatives. Also several interesting results in FuzzyRandom Analysis are given of independent merit, which are used then in the proofs of the main results of the chapter. This chapter follows [17].

18.1 Introduction Let (X; B; P ) be a probability space. Consider the set of all fuzzy-random variables LF (X; B; P ). Let f : R ! LF (X; B; P ) be a fuzzy-random function or fuzzy-stochastic process. Here for t 2 R, s 2 X we denote (f (t))(s) = f (t; s) and actually we have f : R X ! RF , where RF is the set of fuzzy real numbers. Let 1 q < +1. Here we consider only fuzzy-random functions f which are (q-mean) uniformly continous over R. For each n 2 N, the fuzzy-random neural network we deal with has the following structure: G.A. Anastassiou: Fuzzy Mathematics: Approximation The., STUDFUZZ 251, pp. 347–372. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

348

18. Fuzzy-Random Neural Network Operators, Univariate case

It is a three-layer feed forward network with one hidden layer. It has one input unit and one output unit. The hidden layer has (2n2 + 1) processing units. To each pair of connecting units (input to each processing unit) we assign the same weight n1 , 0 < < 1. The threshold values nk are one for each processing unit k. The activation function b (or S) is the same for each processing unit. The Fuzzy-Random weights associated with the 1 output unit are f nk ; s ( ) n , one for each processing unit k, where IR R1 1 I = b(x)dx (or I = S(x)dx), “ ” denotes the scalar Fuzzy 1 1 multiplication. The above precisely described Fuzzy-Random neural networks induce some completely described Fuzzy-Random neural network operators of Cardaliaguet–Euvrard and “Squashing" types. We study here thoroughly the Fuzzy-Random pointwise convergence (in q-mean) of these operators to the unit operator. See Theorems 18.21, 18.24, 18.27, 18.29. This is done with rates through Probabilistic-Jackson type inequalities involving Fuzzy-Random moduli of continuity of the engaged Fuzzy- Random function and its Fuzzy derivatives. On the way to establish these main results we produce some independent and interesting results for Fuzzy-Random Analysis. The real ordinary theory of the above mentioned operators was presented earlier in [6], [9] and [47]. And the fuzzy case was treated in [11], see also here Chapters 15, 16. Of course this chapter is strongly motivated from there and is a continuation. The monumental revolutionizing work of L. Zadeh [103] is the foundation of this chapter, as well as another strong motivation. Fuzziness in Computer Science and Engineering seems one of the main trends today. Also Fuzziness has penetrated many areas of Mathematics and Statistics. The approach here is quantitative and new on the topic, started in [6], [9] and continued in [11]. It determines precisely the rates of convergence through natural very tight inequalities using the measurement of smoothness of the engaged Fuzzy-Random functions. As in Remark 4.4 ([31]) one can show easily that a sequence of operators of the form Ln (f )(x) :=

n X

f (xkn )

wn;k (x);

k=0

n 2 N;

(18.1)

P ( denotes the fuzzy summation) where f : R ! RF , xkn 2 R, wn;k (x) real valued weights, are linear over R, i.e., Ln (

f

g)(x) =

Ln (f )(x)

Ln (g)(x);

(18.2)

8 ; 2 R, any x 2 R; f; g : R ! RF . (Proof based on Lemma 1.3(iv).) We need

18.1 Introduction

349

Lemma 18.1 ([11]). Let f : R ! RF continuous, r 2 N. Then the following integrals Z sr 1 Z sr 2 Z sr 1 f (sr )dsr dsr 1 ; (F R) f (sr )dsr ; (F R) a

a

a

: : : ; (F R)

Z

Z

s

Z

s1

Z

2

sr

1

f (sr )dsr dsr

ds1 ;

1

a

a

a

a

sr

(18.3)

are continuous functions in sr 1 ; sr 2 ; : : : ; s, respectively. Here sr 1 ; sr 2 ; : : : ; s a and all are real numbers. Additionally we have Lemma 18.2. Let f : [a; b] ! RF have an existing fuzzy derivative f 0 at c 2 [a; b]. Then f is fuzzy continuous at c. Proof. By the assumption we have that lim D

h!0+

and lim+ D

f (c + h) h f (c)

h!0

f (c)

f (c h

h)

; f 0 (c)

=0

(18.4)

; f 0 (c)

= 0:

(18.5)

Furthermore here is assumed that the H-di¤erences f (c + h) f (c h) exist for small h : 0 < h < b a. Let z := f (c + h) f (c), then f (c + h) = f (c) z. Hence D(f (c + h); f (c))

f (c), f (c)

= D(f (c) z; f (c)) = D(z; o~) = D(f (c + h) f (c); o~) f (c + h) f (c) = hD ; o~ : h

(18.6)

But D

f (c + h) h

f (c)

; o~

D(f 0 (c); o~)

f (c + h) h

D

f (c)

; f 0 (c) : (18.7)

Hence D(f (c + h); f (c))

h D(f 0 (c); o~) + D

f (c + h) h

f (c)

; f 0 (c)

:

Consequently lim D(f (c + h); f (c))

(18.8)

h!0+

lim h

h!0+

D(f 0 (c); o~) + lim+ D h!0

f (c + h) h

f (c)

; f 0 (c)

= 0:

350

18. Fuzzy-Random Neural Network Operators, Univariate case

Therefore lim D(f (c + h); f (c)) = 0:

h!0+

That is, f is right continuous at c. One can prove f is left continuous at c. We have proved that f is fuzzy continuous at c 2 [a; b]. We need the Fuzzy Taylor formula

Theorem 18.3 ([11]). Let T := [x0 ; x0 + ] R, with > 0. We suppose that f (i) : T ! RF are di¤ erentiable for all i = 0; 1; : : : ; n 1, for any x 2 T . (I.e., there exist in RF the H-di¤ erences f (i) (x + h) f (i) (x), f (i) (x) f (i) (x h), i = 0; 1; : : : ; n 1 for small h : 0 < h < . Furthermore there exist f (i+1) (x) 2 RF such that the limits in D-distance exist and f (i+1) (x) = lim+ h!0

f (i) (x + h) h

f (i) (x)

= lim+

f (i) (x)

h!0

f (i) (x h

h)

;

for all i = 0; 1; : : : ; n 1.) Also we suppose that f (i) , i = 0; 1; : : : ; n are continuous on T in the fuzzy sense. Then for s a; s; a 2 T we obtain f 0 (a)

f (s) = f (a) f (n

1)

(a)

(s

a)

f 00 (a)

(s a)n 1 (n 1)!

(s

a)2 2!

Rn (a; s);

(18.9)

where Rn (a; s) := (F R)

Z

s

a

Z

s1

a

Z

sn

1

f (n) (sn )dsn dsn

1

: : : ds1 :

a

(18.10)

Here Rn (a; s) 2 CF (T ) as a function of s. Note. This formula is invalid when s < a, as it is totally based on Corollary 1.12.

18.2 Basic properties We need De…nition 18.4 (see also [66], De…nition 13.16, p. 654). Let (X; B; P ) be a probability space. A fuzzy-random variable is a B-measurable mapping g : X ! RF (i.e., for any open set U RF , in the topology of RF generated by the metric D, we have g

1

(U ) = fs 2 X; g(s) 2 U g 2 B):

(18.11)

18.2 Basic properties

351

The set of all fuzzy-random variables is denoted by LF (X; B; P ). Let gn ; g 2 “q-mean” ! LF (X; B; P ), n 2 N and 0 < q < +1. We say gn (s) g(s) if n ! +1 Z lim (D(gn (s); g(s)))q P (ds) = 0: (18.12) n!+1

X

Remark 18.5 (see [66], p. 654). If f; g 2 LF (X; B; P ), let us denote F : X ! R+ [f0g by F (s) = D(f (x); g(s)), s 2 X. Here, F is B-measurable, because F = G H, where G(u; v) = D(u; v) is continuous on RF RF , and H : X ! RF RF , H(s) = (f (s); g(s)), s 2 X, is B-measurable. This shows that the above convergence in q-mean makes sense. De…nition 18.6 (see [66], p. 654, De…nition 13.17). Let (T; T ) be a topological space. A mapping f : T ! LF (X; B; P ) will be called fuzzy-random function (or fuzzy-stochastic process) on T . We denote f (t)(s) = f (t; s), t 2 T , s 2 X. Remark 18.7 (see [66], p. 655). Any usual fuzzy real function f : T ! RF can be identi…ed with the degenerate fuzzy-random function f (t; s) = f (t), 8t 2 T , s 2 X. Remark 18.8 (see [66], p. 655). Fuzzy-random functions that coincide with probability one for each t 2 T will be considered equivalent. Remark 18.9 (see [66], p. 655). Let f; g : T ! LF (X; B; P ). Then f and k f are de…ned pointwise, i.e., (f (k

g)(t; s) f )(t; s)

g

= f (t; s) g(t; s); = k f (t; s); t 2 T; s 2 X:

De…nition 18.10 (see also De…nition 13.18, pp. 655–656, [66]). For a fuzzyrandom function f : R ! LF (X; B; P ) we de…ne the (…rst) fuzzy-random modulus of continuity ( Z 1=q

(F ) 1 (f;

)Lq

=

Dq (f (x; s); f (y; s))P (ds)

sup

;

X

x; y 2 R; jx 0 < ; 1 q < 1: De…nition 18.11. Here 1

yj

g;

q < +1. Let f : R ! LF (X; B; P ) be a fuzzy

random function. We call f a (q-mean) uniformly continuous fuzzy random function over R, i¤ 8" > 0 9 > 0: whenever jx yj ; x; y 2 R, implies that Z q (D(f (x; s); f (y; s))) P (ds) ": X

352

18. Fuzzy-Random Neural Network Operators, Univariate case

Uq We denote it as f 2 CF R (R). (F ) 1 (f;

Uq Proposition 18.12. Let f 2 CF R (R). Then

)Lq < 1, any

Proof. Let "0 > 0 be arbitrary but …xed. Then there exists 0 implies Z q (D(f (x; s); f (y; s))) P (ds) "0 < 1:

0

> 0.

> 0: jx yj

X

(F ) 1 (f; 0 )Lq

1=q

That is, "0 < 1. Let now > 0 arbitrary, x; y 2 R such that jx yj . Choose n 2 N: n 0 and set xi := x + ni (y x), 0 i n. Then D(f (x; s); f (y; s))

D(f (x; s); f (x1 ; s)) + D(f (x1 ; s); f (x2 ; s)) + + D(f (xn 1 ; s); f (y; s)):

Consequently Z

1=q q

(D(f (x; s); f (y; s))) P (ds)

X

Z

1=q q

(D(f (x; s); f (x1 ; s))) P (ds)

+

X

+ n since jxi xi+1 j = 1=q n"0 < 1.

1 n jx

Z

1=q

(D(f (xn

X (F ) 1 (f; 0 )Lq

yj

1 n

q 1 ; s); f (y; s))) 1=q

n"0 0,

0

i

P (ds)

< 1; n. Therefore

(F ) 1 (f;

)Lq

Proposition 18.13. Let f; g : R ! LF (X; B; P ) be fuzzy random functions. It holds (i)

(F ) 1 (f;

)Lq is nonnegative and nondecreasing in

(F )

(ii) lim ! 1 (f; )Lq = #0

(iii)

(F ) 1 (f; 1

+

(iv)

(F ) 1 (f; n

)Lq

(v)

(vi)

2 )Lq

n

(F ) 1 (f; 0)Lq

Uq = 0, i¤ f 2 CF R (R).

(F ) 1 (f; 1 )Lq (F ) 1 (f;

(F ) 1 (f;

(F )

(F ) 1 (f

(F )

)Lq ,

> 0.

+

(F ) 1 (f; 2 )Lq ,

1; 2

> 0.

> 0, n 2 N.

)Lq d e 1 (f; )Lq ( + 1) where d e is the ceiling of the number. g; )Lq 1 (f; )Lq + a fuzzy random function.

(F ) 1 (g;

(F ) 1 (f;

)Lq ,

)Lq ,

> 0,

> 0,

> 0. Here f

g is

18.2 Basic properties

(vii)

(F ) 1 (f;

353

Uq )Lq is continuous on R+ , for f 2 CF R (R).

Proof. Obvious. Also we have Proposition 18.14. (i) Let Y (t; !) be a real valued stochastic process such that Y is continuous in t 2 [a; b]. Then Y is jointly measurable in (t; !). (ii) Further assume that the expectation (EjY j)(t) 2 C([a; b]), or more Rb generally a (EjY j)(t)dt makes sense and is …nite. Then ! Z Z b

b

Y (t; !)dt

E

(EY )(t)dt:

=

a

a

Rb Proof. Clearly a Y (t; !)dt is a real valued random variable. Also by the assumption on EjY j we obtain that Z b (EjY j)(t)dt < 1: a

Here we assume that ! 2 S, S is a probability space. Also here Y is continuous in t, and measurable in ! and takes values in R. Clearly Y is a “Caratheodory function", see [2], p. 156, De…nition 20.14. Hence by Theorem 20.15, p. 156, [2] we get that Y is (jointly) measurable on [a; b] S. In the last we took into account the following: let f : X Y ! R be measurable on X Y, and g : Y X ! R be such that g(y; x) := f (x; y). Then g is measurable on Y X . From the above we derive that Y (t; !) is integrable on [a; b] S. The claim now is proved by applying Fubini’s theorem. According to [62], p. 94 we have the following De…nition 18.15. Let (Y; T ) be a topological space, with its -algebra of Borel sets B := B(Y; T ) generated by T . If (X; S) is a measurable space, a function f : X ! Y is called measurable i¤ f 1 (B) 2 S for all B 2 B. By Theorem 4.1.6 of [62], p. 89 f as above is measurable i¤ f

1

(C) 2 S

for all C 2 T :

We would need Theorem 18.16 (see [62], p. 95). Let (X; S) be a measurable space and (Y; d) be a metric space. Let fn be measurable functions from X into Y such that for all x 2 X, fn (x) ! f (x) in Y . Then f is measurable. I.e., lim fn = f is measurable. n!1 We need also Proposition 18.17. Let f; g be fuzzy random variables from S into RF . Then

354

18. Fuzzy-Random Neural Network Operators, Univariate case

(i) Let c 2 R, then c (ii) f

f is a fuzzy random variable.

g is a fuzzy random variable.

Proof. Obvious. Finally we need Proposition 18.18. Let f : [a; b] ! LF (X; B; P ) be a fuzzy-random function. We assume that f (t; s) is fuzzy continuous in t 2 [a; b], s 2 X. Then Rb (F R) a f (t; s)dt exists and is a fuzzy-random variable. Proof. It comes easily from: De…nition 1.9 and Corollary 13.2 of p. 644, [66] about Fuzzy-Riemann integral, Proposition 18.17, and Theorem 18.16.

18.3 Main results We need (see also [6], [11], [47]) De…nition 18.19. A function b : R ! R is said to be bell-shaped if b

belongs to L1 and its integral is nonzero, if it is nondecreasing on ( 1; a) and nonincreasing on [a; +1), where a belongs to R. In particular b(x) is a nonnegative number and at a b takes a global maximum; it is the center of the bell-shaped function. A bell-shaped function is said to be centered if its center is zero. The function b(x) may have jump discontinuities. In this chapter we consider only centered bell-shaped functions of compact RT support [ T; T ], T > 0. Denote I := T b(t)dt. Notice that I > 0. Example 18.20.

(1) b(x) can be the characteristic function on [ 1; 1]. (2) b(x) can be the hat function on [ 1; 1], i.e., 8 1 x 0; < 1 + x; 1 x; 0 < x 1; b(x) = : 0; elsewhere. Uq Here we consider functions f 2 CF R (R).

In this chapter we study among others in q-mean the pointwise convergence with rates over the real line, to the fuzzy-random unit operator of the fuzzy-random univariate Cardaliaguet–Euvrard neural network operators, 2

(Fn (f ))(x; s) := fn (x; s) :=

n X

k= n2

f

k ;s n

b n1

x In

k n

; (18.13)

18.3 Main results

355

where 0 < < 1, x 2 R, s 2 X and n 2 N. The above are linear operators over R. These operators in the ordinary real case were thoroughly studied in [9], [47] and in the fuzzy sense were studied in [11]. So as in [6], [11], for convenience we consider that n max(T + jxj, T 1= ). This implies that n2 nx T n nx + T n n2 , and card(k) 1. Furthermore, it holds that card(k) ! +1, as n ! +1. Set b := b(0), the maximum of b(x). Denote by [ ] the integral part of a number. Next we give the …rst main result. Theorem 18.21. Let x 2 R, s 2 X, T > 0, and 1R q < +1. Here n 2 N is such that n max(T +jxj, T 1= ). Assume that X Dq (f (x; s); o~)P (ds) < +1. Then Z 1=q Dq (fn (x; s); f (x; s))P (ds) X

[nx+T n ]

X

b n1

k=dnx T n e

+

b I

2T +

1 n

x In

k n

(F ) 1

f;

Z

1

1=q

Dq (f (x; s); o~)P (ds)

X

T n1

: Lq

(18.14) Proof. All parts of inequality (18.14) make sense because of the notions and basic results developed in Section 18.2. As in [6] we have that [nx+T n ]

X

b n1

k n

x In

k=dnx T n e

We see that

0

b I

2T +

[nx+T n ]

X

D(fn (x; s); f (x; s)) = D @ b n 0

D@

k n

x In

f

k=dnx T n e

!

b n1

k ;s n

[nx+T n ]

X

f (x; s)

b n

1

k=dnx T n e

0

+ D @f (x; s)

(18.15)

; f (x; s)

[nx+T n ]

X

:

k ;s n

f

k=dnx T n e

1

1 n

[nx+T n ]

X

k=dnx T n e

x In k n

x In b n

1

x In

k n

;

1 A

k n

; f (x; s)

1

1A

356

18. Fuzzy-Random Neural Network Operators, Univariate case

(by Lemma 1.2) [nx+T n ]

X

b n1

k=dnx T n e [nx+T n ]

X

+

k n

x In

b n1

1 D(f (x; s); o~) k n

x In

k=dnx T n e

k ; s ; f (x; s) : n

D f

That is, [nx+T n ]

X

D(fn (x; s); f (x; s))

b n1

k=dnx T n e [nx+T n ]

X

+

b n1

1 D(f (s; x); o~) k n

x In

k=dnx T n e

Therefore Z

k n

x In

k ; s ; f (x; s) : n

D f

1=q

Dq (fn (x; s); f (x; s))P (ds)

X

[nx+T n ]

X

b n1

k=dnx T n e [nx+T n ]

X

+

x In

b n1

X

b n1

k=dnx T n e [nx+T n ]

X

+

x In

b n1

[nx+T n ]

X

b n1

k=dnx T n e

0

+@

[nx+T n ]

X

b n1

k=dnx T n e [nx+T n ]

(by (18.15)) Z

x In

X

X

k=dnx T n e

Dq (f (x; s); o~)P (ds)

1=q

Dq (f (x; s); o~)P (ds)

k n

Z

1

1=q

k ; s ; f (x; s) P (ds) n

Dq f

X

1=q

Dq (f (x; s); o~)P (ds)

X

k n

(F ) 1

k n

x In b n1

Z

k n

x In

k=dnx T n e

Z

1

X

x In

k=dnx T n e [nx+T n ]

k n

1

f; Z

k n

x Lq 1=q

Dq (f (x; s); o~)P (ds)

X

k n

x In 1=q

+

1 A

k n

b I

(F ) 1

f;

T n1

Lq

1

2T +

1 n

(F ) 1

f;

T n1

: Lq

18.3 Main results

357

Remark 18.22. By Lemma 2.1 of [9], p. 64 we get that [nx+T n ]

X

lim

n!+1

b n1

k=dnx T n e

Uq any x 2 R. Let f 2 CF R (R), then

k n

x In

lim

n!+1

(F ) 1

1 = 0;

f; n1T

Lq

= 0, by Propo-

sition 18.13(ii). Therefore from (18.4) we obtain Z lim Dq (fn (x; s); f (x; s))P (ds) = 0: n!+1

X

“q-mean” ! f (x; s), pointwise in x 2 R with rates. n ! +1 We need also (see also [6], [47])

That is, fn (x; s)

De…nition 18.23. Let the nonnegative function S : R ! R, S has compact support [ T; T ], T > 0, and is nondecreasing there and it can be continuous only on either ( 1; T ] or [ T; T ]. S can have jump discontinuities. We call S the “squashing function". Suppose that Z T I := S(t)dt > 0: T

Clearly max S(x) = S(T ):

x2[ T;T ]

Here we consider again Uq f 2 CF R (R):

For x 2 R, s 2 X we de…ne the “univariate fuzzy-random squashing operator ” 2

(Gn (f ))(x; s) :=

n X

k ;s n

f

k= n2

S n1

x I n

k n

;

(18.16)

0 < < 1 and n 2 N: n max(T + jxj; T 1= ). The above neural network operators are also linear operators in R. These operators in the ordinary real case were thoroughly studied in [6], [47], and in the fuzzy sense were studied also in [11]. It is clear to see again that [nx+T n ]

(Gn (f ))(x; s) =

X

k=dnx T n e

f

k ;s n

S n1

x I n

k n

:

(18.17)

358

18. Fuzzy-Random Neural Network Operators, Univariate case

Here we study the pointwise convergence in x 2 R with rates of “q-mean” (Gn (f ))(x; s) ! f (x; s), 1 q < +1, as n ! +1. We present Theorem 18.24. Let x 2 R, s 2 X, T > 0 and 1 such that n max(T + jxj, T 1= ). Suppose that Z

q < +1. Let n 2 N

Dq (f (x; s); o~)P (ds) < +1:

X

Then Z

1=q

Dq ((Gn f )(x; s); f (x; s))P (ds)

X [nx+T n ]

X

S n1

k=dnx T n e

+

S(T ) I

2T +

k n

x I n

Z

1

1=q

Dq (f (x; s); o~)P (ds)

X

1 n

(F ) 1

f;

T

:

n1

Lq

(18.18)

Proof. All parts of inequality (18.18) make sense because of the notions and basic results presented in Section 18.2. Notice that [nx+T n ]

X

1

(2T n + 1):

(18.19)

k=dnx T n e

We observe that D((Gn (f ))(x; s); f (x; s)) 0 [nx+T n ] X k = D@ f ;s n k=dnx T n e 0 [nx+T n ] X k D@ f ;s n

S n

S n1

k=dnx T n e

[nx+T n

X

f (x; s)

S n1

k=dnx T n e

0

+ D @f (x; s)

k=dnx T n e

S n

x I n

k n

x I n 1

k n

k n

x I n

[nx+T n ]

X

1

1

A

x I n

k n

1

; f (x; s)A ;

; f (x; s)

1

1A

18.3 Main results [nx+T n ]

X

(by Lemma 1.2)

S n1

k=dnx T n e [nx+T n

X

+

S n1

k=dnx T n e

k n

x I n

k n

x I n

359

1 D(f (x; s); o~)

k ; s ; f (x; s) : n

D f

That is, [nx+T n ]

X

D((Gn (f ))(x; s); f (x; s))

S n1

k=dnx T n e

k n

x I n

1

D(f (x; s); o~) [nx+T n ]

X

+

S n1

k=dnx T n e

x I n

k n

k ; s ; f (x; s) : n

D f Hence Z

1=q

Dq ((Gn (f ))(x; s); f (x; s))P (ds)

X [nx+T n ]

X

S n1

k=dnx T n e [nx+T n ]

X

+

x I n

S n1

k=dnx T n e

Z

Dq f

X

[nx+T n ]

X

S n1

[nx+T n ]

X

X

S n1

k=dnx T n e

0

+@

[nx+T n ]

X

k=dnx T n e

S n1

1=q

Dq (f (x; s); o~)P (ds)

k n

k n

1

1=q

Z

1=q

Dq (f (x; s); o~)P (ds)

X

k n

x I n

x I n

Z

X

x I n

x I n

S n1

k=dnx T n e [nx+T n ]

1

k ; s ; f (x; s) P (ds) n

k=dnx T n e

+

k n

(F ) 1

k n

x I n

1

f;

Z

k n

x Lq

1=q

Dq (f (x; s); o~)P (ds)

X

k n

1 A

(F ) 1

f;

T n1

Lq

360

18. Fuzzy-Random Neural Network Operators, Univariate case [nx+T n ]

X

(by (18.19))

S n1

k=dnx T n e

Z

k n

x I n

1

1=q

Dq (f (x; s); o~)P (ds)

X

S(T ) + I

2T +

1 n

(F ) 1

f;

T n1

: Lq

Remark 18.25. From Lemma 2.2, p. 79, [9], we have that [nx+T n ]

X

lim

n!+1

S n1

k=dnx T n e

any x 2 R. Uq Let f 2 CF R (R), then

lim

n!+1

k n

x I n

(F ) 1

f; n1T

1 = 0;

Lq

= 0, by Proposition

18.13(ii). Thus from (18.18) we obtain Z Dq ((Gn f )(x; s); f (x; s))P (ds) = 0: lim n!+1

X

“q-mean” ! f (x; s), pointwise in x 2 R with rates. n ! +1 Next we study further the fuzzy random operators Fn , Gn in 1-mean, their pointwise convergence with rates to the fuzzy-random unit operator again. However, this time the domain of operators is the fuzzy-random di¤erentiable functions. That fact is re‡ected in the derived inequalities.

That is, (Gn f )(x; s)

Assumptions 18.26. Let x 2 R, s 2 X; where (X; B; P ) is a probability space, and n 2 N such that n

max(T + jxj; T

1=

);

T > 0; 0 <

< 1:

Let f : R ! LF (X; B; P ) be a fuzzy random function, such that the fuzzy derivative in x f (j) : R ! LF (X; B; P );

j = 1; : : : ; N;

exist and are fuzzy random functions. (I.e., f (j) (x; s), for all j = 0; 1; : : : ; N is B-measurable in s.) We suppose that f (N ) (x; s) is fuzzy continuous in x, and D(f (N ) (x; s); o~) M , 8(x; s) 2 R X, where M > 0. We also assume that Z D(f (j) (x; s); o~)P (ds) < +1; all j = 0; 1; : : : ; N: X

18.3 Main results

361

We …nally suppose that U1 f (j) 2 CF R (R);

all j = 0; 1; : : : ; N:

We give Theorem 18.27. All here as in Assumptions 18.26. Then Z

[nx+T n ]

X

D(fn (x; s); f (x; s))P (ds)

X

D(f (x; s); o~)P (ds) +

X (F )

+ 1 8
f;

j=1

(F ) 1

b I

+ L1

Z

2j T j j!n(1 )j

N

+

T n1

f (j) ;

T n1 1 2T + n

(F ) 1

k n

x In

k=dnx T n e

Z

:

b n1

1

f;

L1

D(f (j) (x; s); o~)P (ds)+

(18.20)

X

T n1

L1

N

2 T b 1 2T + n N !InN (1 ) Z D(f (N ) (x; s); o~)P (ds) + 3

(F ) 1

f (N ) ;

X

T n1

: L1

Remark 18.28. By Lemma 2.1 of [9], p. 64 we get that [nx+T n ]

lim

n!+1

X

b n1

k=dnx T n e

x In

k n

1 = 0;

U1 Since f (j) 2 CF R (R), j = 0; 1; : : : ; N , we have that lim

any x 2 R:

n!+1

(F ) 1

f (j) ; n1T

L1

0, by Proposition 18.13(ii). Thus from (18.20) we …nd that Z D(fn (x; s); f (x; s))P (ds) = 0: lim n!+1

That is, fn (x; s)

X

“1-mean” ! f (x; s), pointwise in x 2 R with rates. n ! +1

Proof (of Theorem 18.27). All parts of inequality (18.20) make sense and are …nite, by Assumptions 18.26 and results of Section 18.2. Because b has compact support [ T; T ] we have that [nx+T n ]

fn (x; s) =

X

k=dnx T n e

f

k ;s n

b n1

x In

k n

:

=

362

18. Fuzzy-Random Neural Network Operators, Univariate case

Clearly the terms in the sum (18.13) are non-zero i¤ n1

T

k n

x

n1

T

;

n1

k n

x

T , i¤

x 2 R:

k (j) Hence here we have x n1T , j = 0; 1; : : : ; N n . By Lemma 18.2 all f 1 are fuzzy continuous. Using the fuzzy Taylor formula (Theorem 18.3) we obtain

f

k ;s n

=

N 1 X

T

f (j) x

n1

j=0

k n

;s

x+ j!

T n1

j

RN

x

k T ; ;s ; n1 n

where

RN (F R)

Z

Z

k n

x

T n1

Z

s1

x

k T ; ;s n1 n

x

T n1

sN

1

f

(N )

T n1

x

:=

(sN ; s)dsN

!

dsN

1

!

: : : ds1

is a fuzzy random variable by Proposition 18.18, Lemma 1.15 and Lemma 18.1. Easily we derive

[nx+T n ]

X

k=dnx T n e [nx+T n ]

=

X

b n1

k ;s n

f

N 1 X

f (j) x

k=dnx T n e j=0

b n1

x In [nx+T n ] X

k=dnx T n e

b n1

x In

x In T n1

k n

;s

k n

RN k n

x

:

T k ; ;s n1 n

k n

x+ j!

T n1

j

18.3 Main results

363

That is,

fn (x; s)

[nx+T n ]

N 1 X

=

X

f

(j)

T

x

b n1

x+ j!

j

T n1

k n

x In [nx+T n ] X

RN

k=dnx T n e

b n1

;s

n1

j=0 k=dnx T n e

k n

k n

x In

T

x

;

n1

k ;s n

:

Next we estimate 0

D(fn (x; s); f (x; s)) b n

1

x In

0

+ D @f f

x

k n

!

f

b n1

k=dnx T n e

T ;s n1

1 +D f

0 N 1 X @ D

[nx+T n ]

X

x+ j!

j

T n1

b n1

f (j) x

[nx+T n ]

X

RN

k=dnx T n e

b n1

D f

k n

x In x

T n1

x !

; o~ +

x In

x In

k n

;

T ; s ; f (x; s) n1

x

j=1 k=dnx T n e

k n

T ;s n1

x

k=dnx T n e

X

;s

n1

(by Lemma 1.2)

X

[nx+T n ]

T

x

[nx+T n ]

D @fn (x; s);

T n1

;s

k n

T k ; ;s n1 n [nx+T n ]

X

b n1

k=dnx T n e

; s ; o~ + D f

x

T n1

x In

k n

1

; s ; f (x; s)

364

18. Fuzzy-Random Neural Network Operators, Univariate case N X1

[nx+T n ]

X

k n

x+ j!

j=1 k=dnx T n e

[nx+T n ]

X

b n1

[nx+T n ]

X

b n1

k=dnx T n e

D f

T

x

k n

x In

k=dnx T n e

+

b n1

k n

x In

T ; s ; o~ n1

D f (j) x +

j

T n1

D RN

k n

x In

1

; s ; o~ + D f

n1

T k ; ; s ; o~ n1 n

x

T

x

; s ; f (x; s) :

n1

Consequently we have

Z

D(fn (x; s); f (x; s))P (ds)

X [nx+T n ]

N X1

X

k n

j=1 k=dnx T n e

Z

T

D f (j) x

[nx+T n ]

X

b n1

[nx+T n ]

b n1

+

k=dnx T n e

+

X

k=dnx T n e

+

Z

X

D f

x

T n1

x In

k n

x In

k n

b n1

x In

k n

; s ; o~ P (ds)

n1

X

j

T n1

x+ j!

Z

D RN

x

X

1

Z

D f

X

; s ; f (x; s) P (ds):

T k ; ; s ; o~ P (ds) n1 n x

T ; s ; o~ P (ds) n1

18.3 Main results

365

I.e., it makes sense by Remark 18.5 and Proposition 18.17 the integral Z

D(fn (x; s); f (x; s))P (ds)

X

b I

(by (18.15)) Z

8
1 2T + n

:

D(f (j) (x; s); o~)P (ds) +

2j T j j!n(1 )j

j=1

(F ) 1

f (j) ;

X

[nx+T n ]

X

+

b n

1

x In

k=dnx T n e [nx+T n ]

X

+

b n1

Z

D RN

x

X

x In

k=dnx T n e

Z

k n

T n1

D(f (x; s); o~)P (ds) +

k n

(F ) 1

L1

T n1

;

k ; s ; o~ P (ds) n (18.21)

1 f;

X

T n1

+

(F ) 1

f;

L1

T n1

=: ( ): L1

In the following we work on Z

T k D RN x ; ; s o~ P (ds) 1 n n X Z k T T ; ; s ; f (N ) x ;s D RN x 1 1 n n n X ! N k x + n1T n P (ds) N! ! Z N k x + n1T T (N ) n + D f x ; o~ P (ds); ;s n1 N! X

the last makes sense by Remark 18.5 and Proposition 18.17(i). That is, Z

X

D RN Z

x

D RN

T k ; ; s o~ P (ds) n1 n x

X

P (ds) +

2N T N N !nN (1

Finally we estimate

T n1 )

T k ; ; s ; f (N ) x ;s n n1 Z D(f (N ) (x; s); o~)P (ds) + X

k n

(F ) 1

x + n1T N! f (N ) ;

T n1

N

! :

L1

366

18. Fuzzy-Random Neural Network Operators, Univariate case

Z

D RN

k ; ; s ; f (N ) x n

T

x

n1

X

P (ds) Z Z = D (F R) X

T n1

x

ds1 ; (F R)

Z

k n T n1

x

Z

k n

dsN

1

T n1

Z

T n1

1

T n1 1

Z

Z

(sN ; s)dsN

f (N ) x

Z

k n T n1

x

T

!

x

T n1

x

x

P (ds)) dsN dsN Z

Z

k n

x

T n1

dsN k n

x + n1T N!

2N T N N !nN (1

)

1

T n1

T n1

T

;s

n1

dsN

!

dsN

T n1

X

sN

x

1

(F ) 1

T n1

f

(N )

N (F ) 1 (F ) 1

; sN

x+

X

D RN

x

T n1 P (ds)

1

ds1

T ;s n1

dsN

n1

L1

f (N ) ;

2T f (N ) ; 1 n

k n

x+

T n1

L1

: L1

Thus, we got that Z

!

:::

!!

!

T

ds1

1

1

: : : ds1 Z

s1

x

dsN

; s dsN

n1

P (ds) (by Lemma 1.13, bounded convergence theorem, Proposition 18.14(i) and Fubini theorem) Z nk Z s1 Z sN 1 Z = D f (N ) (sN ; s); f (N ) x T n1

!

s1

x

D f (N ) (sN ; s); f (N ) x

T n1

x

f

(N )

T n1

x

X

sN

sN

1

N

x + n1T N!

: : : ds1 P (ds)

(by Lemmas 1.13, 1.15, 18.1) Z

sN

x

s1

x

Z

k n

;s

n1

s1

x

Z

T

k ; ; s ; f (N ) x n 2N T N N ! nN (1

)

T n1 (F ) 1

k n

;s f (N ) ;

2T n1

x + n1T N!

L1

N

!

!

!

18.3 Main results

2N +1 T N N !nN (1 )

(F ) 1

f (N ) ;

T n1

367

: L1

Putting things together we derive Z

D RN

T k ; ; s ; o~ P (ds) n1 n

x

X

2N +1 T N N !nN (1 ) 2N T N + N !nN (1

(F ) 1

f (N ) ;

Z

)

T n1

L1

D(f (N ) (x; s); o~)P (ds) +

(F ) 1

f (N ) ;

X

T

:

n1

L1

That is, we …nd Z

D RN

T

x

n1

X

2N T N N !nN (1

)

; Z

k ; s ; o~ P (ds) n

(18.22)

D(f (N ) (x; s); o~)P (ds) + 3

(F ) 1

X

f (N ) ;

T n1

: L1

Also it holds

[nx+T n ]

X

b n1

k=dnx T n e

(by (18.15)) +3

f

(N )

D RN

;

)

2T +

T n1

1 n :

L1

Putting things in ( ), see (18.21), etc. Finally we present

x

X

2N T N b N !InN (1 (F ) 1

Z

k n

x In

Z

X

T n1

;

k ; s o~ P (ds) n

D(f (N ) (x; s); o~)P (ds)

368

18. Fuzzy-Random Neural Network Operators, Univariate case

Theorem 18.29. All here are as in Assumptions 18.26. Then Z D((Gn f )(x; s); f (x; s))P (ds) X

[nx+T n ]

X

S n1

x I n

k=dnx T n e

Z

k n

D(f (x; s); o~)P (ds) +

1 (F ) 1

f;

X

+

(F ) 1

f;

T n1

T n1

L1

8
+

L1

X N

2 T S(T ) 1 2T + n N !I nN (1 ) Z D(f (N ) (x; s); o~)P (ds) + 3

T n1

L1

(18.23) (F ) 1

f (N ) ;

X

T n1

Remark 18.30. From Lemma 2.2, p. 79, [9], we have that [nx+T n ]

lim

n!+1

X

S n1

k=dnx T n e

x I n

k n

1 = 0;

any x 2 R:

U1 Since f (j) 2 CF R (R), j = 0; 1; : : : ; N , we have that

lim

n!+1

(F ) 1

f (j) ;

T n1

= 0: L1

Hence from (18.23) we get that Z lim D((Gn f )(x; s); f (x; s))P (ds) = 0: n!+1

X

That is, (Gn f )(x; s) pointwise in x 2 R with rates.

“1-mean" ! f (x; s); n ! +1

: L1

18.3 Main results

369

Proof (of Theorem 18.29). Basic facts here are justi…ed as in the proof of Theorem 18.27. Again we see, that the terms in the sum (18.16) are non-zero i¤ n1 x nk T , i¤ T

Thus here x have

f

k ;s n

=

T n1

N 1 X

f

k n.

(j)

k n

x

n1

T

;

n1

x 2 R:

By the fuzzy Taylor formula (Theorem 18.3) we

T

x

;s

n1

j=0

k n

x+ j!

j

T n1

RN

x

k T ; ;s ; n1 n

where

RN := (F R)

Z

Z

k n

x

x

T n1

s1 T n1

x

dsN

k T ; ;s n1 n Z sN ::: x

1

f

T n1

(N )

(sN ; s)dsN

1 ) : : :) ds1 :

Hence [nx+T n ]

X

f

k=dnx T n e [nx+T n ]

=

X

S n1

k ;s n N 1 X

k=dnx T n e j=0 k n

x + n1T j! 1 S n x nk I n [nx+T n ] X RN

k=dnx T n e

S n1

x I n

k n

f (j) x

x I n

k n

T ;s n1

j

x

:

T k ; ;s n1 n

!

370

18. Fuzzy-Random Neural Network Operators, Univariate case

That is

(Gn f )(x; s)

[nx+T n ]

N 1 X

=

X

j=0 k=dnx T n e k n

x+ j!

T ;s n1

f (j) x j

T n1

S n1

[nx+T n ]

X

RN

k=dnx T n e

S n1

k n

x I n

T

x

n1

k n

x I n ;

k ;s n

:

Next we estimate

D((Gn f )(x; s); f (x; s)) [nx+T n ]

X

D((Gn f )(x; s);

+ D @f

n1

x

k n

1 +D f 0 j

T

RN

x

S n1

k=dnx T n e

x

T n1

x I n

x I n

k n

; s ; f (x; s)

k n

;

T ;s n1

f (j) x k n

S n

T k ; ;s n1 n

x I n

k n

x I n

T ; s ; f (x; s) n1

x

X

S n1

k=dnx T n e

+D f

S n1

j=1 k=dnx T n e

[nx+T n ]

X

;s

[nx+T n ]

N 1 X

n1

[nx+T n ]

S n1

k=dnx T n e

D@

X

X

;s

n1

T ;s n1

x+ j!

[nx+T n ]

T

x

(by Lemma 1.2)

+

x

k=dnx T n e

0

f

f

T

1

1

D f

x I n x

k n

1

; o~A

T ; s ; o~ n1

1 A

18.3 Main results [nx+T n ]

N X1

X

k n

x+ j!

j=1 k=dnx T n e

[nx+T n ]

X

S n1

x I n

k n

[nx+T n ]

S n1

x I n

k n

k=dnx T n e

X

+

k=dnx T n e

+D f

S n1

k n

x I n

T ; s ; o~ n1

D f (j) x +

j

T n1

371

T

x

D RN

T k ; ; s ; o~ n1 n

x

1

D f

S n1

x I n

T ; s ; o~ n1

x

; s ; f (x; s) :

n1

Therefore we derive Z D((Gn f )(x; s); f (x; s))P (ds) X

[nx+T n ]

N X1

X

k n

x+ j!

j=1 k=dnx T n e

Z

T

D f (j) x

n1

X [nx+T n ]

X

+

S n1

k=dnx T n e [nx+T n ]

X

+

S n1

k=dnx T n e

+

Z

D f

X

j

; s ; o~ P (ds)

x I n x I n

k n

Z

D RN

(F ) 1

f (j) ;

[nx+T n ]

+

X

k=dnx T n e

x

X

k n

1

Z

D f

X

D((Gn f )(x; s); f (x; s))P (ds) 8
k n

k T ; ; s ; o~ P (ds) n1 n x

T n1

; s ; o~ P (ds)

T ; s ; f (x; s) P (ds): n1

x

X

That is Z

T n1

Z

D(f (j) (x; s); o~)P (ds)

X

T

n1

S n1

L1

x I n

k n

Z

X

D RN

x

T k ; ; s ; o~ P (ds) n1 n

372

18. Fuzzy-Random Neural Network Operators, Univariate case [nx+T n ]

X

+

S n1

k=dnx T n e

Z

k n

x I n

(F ) 1

D(f (x; s); o~)P (ds) +

(18.24)

1 f;

X

T n1

+ L1

(F ) 1

f;

T n1

=: ( ): L1

Using (18.22) we …nd [nx+T n ]

X

S n1

k=dnx T n e

(by (18.19))

x I n

Z

k n

T

x

n1

X

2N T N S(T ) N !I nN (1 ) +3

D RN

(F ) 1

f

(N )

2T + ;

Putting things in ( ), see (18.24), etc.

Z

1 n

X

T n1

: L1

;

k ; s ; o~ P (ds) n

D(f (N ) (x; s); o~)P (ds)

19 A-SUMMABILITY AND FUZZY KOROVKIN APPROXIMATION

The aim of this chapter is to present a fuzzy Korovkin-type approximation theorem by using a matrix summability method. We also study the rates of convergence of fuzzy positive linear operators. This chapter is based on [27].

19.1 Introduction The study of the classical Korovkin type approximation theory is a wellestablished area of research, which deals with the problem of approximating a function f by means of a sequence fLn (f )g of positive linear operators. Most of the classical approximation operators tend to converge to the value of the function being approximated. However, at points of discontinuity, they often converge to the average of the left and right limits of the function. In this case, the matrix summability methods of the Cesáro type are applicable to correct the lack of convergence [45]. The main purpose of using summability theory has always been to make a nonconvergent sequence "converge". Some results regarding matrix summability for positive linear operators may be found in the papers [38], [39], [94]. In this chapter using a A-summation process we study the approximation properties of sequence of positive linear operators de…ned on the space of all fuzzy continuous functions on the interval [a; b]. G.A. Anastassiou: Fuzzy Mathematics: Approximation The., STUDFUZZ 251, pp. 373–384. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

374

19. A-Summability and Fuzzy Korovkin Approximation

Anastassiou and Duman [29] introduced the fuzzy analog of A statistical convergence by using the metric D. Let ( n )n2N be a fuzzy number valued sequence and let A = (ajn ) is a non-negative regular summability matrix. Recall that the regularity conditions on a matrix A are known as Silverman-Toeplitz conditions in the functional analysis (see for details, [70]). Then, ( n )n2N is A statistically convergent to 2 RF , which is denoted by stA lim D ( n ; ) = 0 if for n!1 every " > 0 X lim ajn = 0 j!1 n:D(

n;

) "

holds. Some basic results regarding A statistical convergence for number sequences may be found in the papers [65], [77], [86]. Of course, the case of A = C1 , the Cesáro matrix of order one, immediately reduces to the statistical convergence of fuzzy valued sequences which de…ned by Nuray and Sava¸s [91]. Also, if A is replaced by the identity matrix, then we have the fuzzy convergence introduced by Matloka [85]. Let A := (An )n 1 , An = ankj be a sequence of in…nite nonk;j2N

negative real matrices. For a sequence of real numbers, x = (xj )j2N , the double sequence n Ax := f(Ax)k : k; n 2 Ng n

de…ned by (Ax)k :=

1 P

j=1

ankj xj is called the A transform of x whenever the

series converges for all k and n. A sequence x is said to be A summable to L if 1 X ankj xj = L lim k!1

j=1

uniformly in n ([42], [99]). If An = A for some matrix A, then A summability is the ordinary j k + n, (n = 1; 2; :::), matrix summability by A. If, ankj = k1 , for n and ankj = 0 otherwise, then A summability reduces to almost convergence [82].

19.2 A Fuzzy Korovkin Type Theorem In this section we prove a fuzzy Korovkin-type approximation theorem via the concept of A summation process. Let f : [a; b] ! RF be fuzzy number valued functions. Then f is said to be fuzzy continuous at x0 2 [a; b] provided that whenever xn ! x0 , then D (f (xn ) ; f (x0 )) ! 0 as n ! 1. Also, we say that f is fuzzy continuous on [a; b] if it is fuzzy continuous at every point x 2 [a; b]. The set all fuzzy continuous functions on the interval [a; b] is denoted by CF [a; b] (see, for

19.2 A Fuzzy Korovkin Type Theorem

375

instance [18]). Notice that CF [a; b] is only a cone not a vector space. Now let L : CF [a; b] ! CF [a; b] be an operator. Then L is said to be fuzzy linear if, for every 1 ; 2 2 R, f1 ; f2 2 CF [a; b], and x 2 [a; b] ; L(

1

f1

2

f2 ; x) =

1

L (f1 ; x)

2

L (f2 ; x)

holds. Also L is called fuzzy positive linear operator if it is fuzzy linear and the condition L (f ; x) L (g; x) is satis…ed for any f; g 2 CF [a; b] and all x 2 [a; b] with f (x) g (x). Throughout this chapter the symbol kf k denotes the usual supremum norm of f . A sequence fLj g of positive linear operators of C [a; b] into itself is called an A-summation process on C [a; b] if fLj (f )g is A-summable to f for every f 2 C [a; b], i.e., lim

k!1

1 X

ankj Lj (f )

f = 0;

uniformly in n;

(19.1)

j=1

where it is assumed that the series in (19.1) converges for each k; n and f . Let fLj g be sequence of positive linear operators of C [a; b] into itself such that for each k; n 2 N 1 X sup ankj kLj (1)k < 1:

(19.2)

n;k j=1

Furthermore, for k; n 2 N and f 2 C [a; b], let Bkn (f ; x) :=

1 X

ankj Lj (f )

j=1

which is well de…ned by (19.2) and belongs to C [a; b] : In this chapter, we establish a theorem of Korovkin type with respect to the convergence behavior (19.1) for a sequence of positive linear operators of CF [a; b] into itself. So our results are extensions of type (19..1) lim

k!1

1 X

ankj Lj (f )

f = 0;

uniformly in n:

j=1

If a sequence fLj g of positive linear operators of CF [a; b] into itself is called an A-summation process on CF [a; b] if fLj (f )g is A-summable to f for every f 2 CF [a; b]; i.e., 0 1 1 X lim D @ ankj Lj (f ) ; f A = 0; uniformly in n: k!1

j=1

376

19. A-Summability and Fuzzy Korovkin Approximation

Throughout the chapter we use the test functions ei (x) = xi (i = 0; 1; 2) Then Anastassiou [18] obtained the following fuzzy Korovkin theorem. Theorem 19.1. ([18]) Let fLn gn2N be a sequence of fuzzy positive linear operators from CF [a; b] into itself. Assume that there exists a corresponding sequence

Ln

of positive linear operators from C [a; b] into itself with n2N

the property

(r)

fLn (f ; x)g

= Ln f

(r)

;x

(19.3)

for all x 2 [a; b], r 2 [a; b], n 2 N and f 2 CF [a; b]. Assume further that lim

n!1

ei = 0 for each i = 0; 1; 2:

Ln (ei )

Then, for all f 2 CF [a; b], we have lim D (Ln (f ) ; f ) = 0:

n!1

Recently the A statistical analog of Theorem 19.1 has been studied by Anastassiou and Duman [29]. It will be read as follows. Theorem 19.2. ([29]) Let A = (ajn ) be a non-negative regular summability matrix and let fLn gn2N be a sequence of fuzzy positive linear operators from CF [a; b] into itself. Assume that there exists a corresponding sequence

Ln

of positive linear operators from C [a; b] into itself n2N

with the property (19.3). Assume further that stA

lim

n!1

Ln (ei )

ei = 0 for each i = 0; 1; 2:

Then, for all f 2 CF [a; b], we have stA

lim D (Ln (f ) ; f ) = 0:

n!1

We now give the following generalization by using a matrix summability method. Theorem 19.3. Let A = (An )n 1 be a sequence of in…nite non-negative real matrices such that 1 X sup ankj < 1 (19.4) n;k j=1

and fLj gj2N be a sequence of fuzzy positive linear operators from CF [a; b] into itself. Assume that there exists a corresponding sequence

Lj

of j2N

19.2 A Fuzzy Korovkin Type Theorem

377

positive linear operators from C [a; b] into itself with the property (19.3) and satisfying (19.2). Assume further that 1 X

lim

k!1

ankj Lj (ei )

ei = 0 for each i = 0; 1; 2

(19.5)

j=1

uniformly in n: Then, for all f 2 CF [a; b], we have 0 1 1 X lim D @ ankj Lj (f ) ; f A = 0 k!1

j=1

uniformly in n. Proof. Let f 2 CF [a; b], x 2 [a; b] and r 2 [0; 1]. By the hypothesis, (r) 2 C [a; b], we may write, for every " > 0, that there exists a since f (r)

(r)

number > 0 such that f (y) f (x) < " holds for every y 2 [a; b] satisfying jy xj < . Then we get for all y 2 [a; b] ; that

where M

(r)

f

(r)

:= f

(r)

(y)

f

(r)

(x)

" + 2M

2

(r) (y

x) 2

;

(19.6)

: Now using the linearity and the positivity of the

operators Lj and considering inequality (19.6), we can write 1 X

ankj Lj f

(r)

;x

f

(r)

(x)

j=1

1 X

ankj Lj

f

(r)

(y)

f

(r)

(x) ; x + f

(r)

(x)

j=1

1 X

1 X

ankj Lj (1; x)

1

j=1

ankj Lj

" + 2M

(r) (y

!

2

x) 2

;x

j=1

"+"

1 X

ankj Lj (1; x)

1 +

2M

j=1

+M

(r)

1 X

+ f

1 X

(r)

ankj Lj (1; x)

1

j=1

1 (r) X

2

ankj Lj (y

2

x) ; x

j=1

ankj Lj (1; x)

1

j=1

"+ "+M

(r)

1 X j=1

ankj Lj (1; x)

1 +

2M

1 (r) X

2

j=1

ankj Lj (y

2

x) ; x

378

19. A-Summability and Fuzzy Korovkin Approximation

= "+ "+M

1 X

(r)

ankj Lj (1; x)

2M

1 +

(r)

2

j=1

j=1

3

1 1 X X 2x ankj Lj (y; x) + x2 ankj Lj (1; x)5 j=1

j=1

"+ "+M

+

2M

(r)

2 4

2

1 X

+2 jxj

1 X

(r)

ankj Lj (1; x)

2 1 X 4 ankj Lj y 2 ; x

1

j=1

1 X

ankj Lj y 2 ; x

x2

j=1

ankj Lj (y; x)

1 X

x + x2

j=1

3

15

ankj Lj (1; x)

j=1

"+ "+M

(r)

1 X

ankj Lj (1; x)

1

j=1

+

2M

1 X

(r)

2

ankj Lj

2

2

y ;x

x

4M

+

(r) 2

j=1

+

2M

(r) 2

c

2

1 c X

ankj Lj (y; x)

x

j=1

1 X

ankj Lj (1; x)

1

j=1

"+

+

"+M

4M

(r) 2

(r)

1 c X

+

2M

(r) 2

c

2

!

ankj Lj (e1 ; x)

1 X

ankj Lj (e0 ; x)

e0

j=1

e1 +

2M

(r)

2

j=1

1 X

ankj Lj (e2 ; x)

j=1

where c := max fjaj ; jbjg : If we take

K

(r)

(

(") := max " + M

(r)

+

2c2 M 2

(r)

;

4cM 2

(r)

;

2M

(r)

2

)

;

e2

19.2 A Fuzzy Korovkin Type Theorem

379

then we get 1 X

ankj Lj f

(r)

;x

f

(r)

"+K

(x)

(r)

(")

j=1

+

1 X

ankj Lj (e1 ; x)

e1 +

1 X

8 1 <X :

ankj Lj (e0 ; x)

j=1

ankj Lj (e2 ; x)

e2

j=1

j=1

Then we observe that sup max r2[0;1]

1 X

ankj Lj f

j=1

" + sup max K

(r)

r2[0;1]

+

1 X

ankj Lj (e2 ; x)

j=1

(")

(r)

;x

8 1 <X

: j=1 9 = e2 : ;

f

(r)

9 = ;

e0

:

(x)

ankj Lj (e0 ; x)

1 X

e0 +

ankj Lj (e1 ; x)

e1

j=1

o n (r) (r) Taking K := K (") := sup max K (") ; K+ (") , we get r2[0;1]

sup sup max x2[a;b]r2[0;1]

" + K (")

+ sup

8 <

sup

:x2[a;b]

1 X

x2[a;b] j=1

1 X

1 X

ankj Lj f

(r)

;x

f

(r)

(x)

j=1

ankj Lj (e0 ; x)

j=1

ankj Lj (e2 ; x)

e2

Consequently, we get 0 1 1 X D @ ankj Lj (f ; x) ; f (x)A

9 = ;

e0 + sup

1 X

x2[a;b] j=1

ankj Lj (e1 ; x)

:

"+K

j=1

1 X

ankj Lj (e0 )

e0

j=1

+K

1 X

ankj Lj (e1 )

e1

ankj Lj (e2 )

e2 :

j=1

+K

1 X j=1

e1

380

19. A-Summability and Fuzzy Korovkin Approximation

By taking limit as k ! 1 and by using (19.5);we obtain the desired result. If we take An = I , the identity matrix and An = A, for some matrix A in Theorem 19.3, we immediately get Theorem 19.1 and Theorem 19.2, respectively. Remarks. We now present three examples of sequences of positive linear operators. The …rst one shows that approximation Theorem 19.3 works but the statistical version of Theorem 19.2 does not work. The second one shows that approximation Theorem 19.3 works but Theorem 19.1 does not work and the last one shows that statistical version of Theorem 19.2 works but Theorem 19.3 does not work. Assume that A := (An )n 1 = ankj is a sequence of in…nite mak;j2N

j k + n, (n = 1; 2; :::), and ankj = 0 trices de…ned by ankj = k1 if n otherwise, then A nsummability reduces to almost convergence. o j (i) Take (uj ) = ( 1) . Observe that u is almost convergent to zero, but it is not statistically convergent. Then consider the Fuzzy Bernsteintype polynomials as follows: (F ) Bj

j X

(f ; x) = (1 + uj )

k=0

j k x (1 k

j k

x)

k j

f

;

where f 2 CF [0; 1], x 2 [0; 1] and j 2 N. So we write j n o(r) X j k (F ) (r) Bj (f ; x) = Bj f ; x = (1 + uj ) x (1 k

j k

x)

k=0

where f

(r)

f

(r)

k j

2 C [0; 1]. Then, we get Bj (e0 ; x)

=

(1 + uj )

Bj (e1 ; x)

=

(1 + uj ) x

Bj (e2 ; x)

=

2

(1 + uj ) x +

x2

x j

!

:

Since u is almost convergent to zero we get lim

k!1

1 X

ankj Bj (ei )

ei = 0 for each i = 0; 1; 2;

j=1

uniformly in n: So, by Theorem 19.3, we obtain, for all f 2 CF [0; 1], that 0 1 1 X lim D @ ankj Bj (f ) ; f A = 0; k!1

j=1

;

19.3 Rate of Convergence

381

uniformly in n. However, since u is not statistically convergent to zero, we observe that fBj g satis…es the Theorem 19.3 but it does not satisfy the statistical version of Theorem 19.2. (ii) Let (vj ) = (1; 0; 1; 0; :::). We see that (vj ) is almost convergent to 12 , but not convergent. Then consider (F )

Lj

j X

1 + vj 2

(f ; x) =

k=0

j k x (1 k

j k

x)

k j

f

;

for f 2 CF [0; 1], x 2 [0; 1] and j 2 N: Hence fLj g satis…es the Theorem 19.3 but it does not satisfy the Theorem 19.1. (iii) Now consider the sequence of 0’s and 1’s de…ned as follows: 0; :::; 0 ; 1; :::; 1 ; 0; :::; 0; 1; :::1; 1; ::: ! 100

!

10

where the blocks of 0’s are increasing by factors of 100 and the blocks of 1’s are increasing by factors of 10. If we denote this sequence by (yj ) ; then (yj ) is not almost convergent but it is statistically convergent to zero [87]. Then we de…ne (F )

Tj

(f ; x) = (1 + yj )

j X

k=0

j k x (1 k

j k

x)

f

k j

for f 2 CF [0; 1], x 2 [0; 1] and j 2 N: We observe that fTj gj2N satis…es the statistical version of Theorem 19.2 but it does not satisfy the Theorem 19.3.

19.3 Rate of Convergence In this section we study the rates of convergence of the sequence of fuzzy positive linear operators examined in Theorem 19.3. Let f : [a; b] ! RF : Then the (…rst) fuzzy modulus of continuity of f , which is introduced by Gal [66] (see also [18]), is de…ned by (F )

!1

(f; ) :=

sup

D (f (x) ; f (y))

x;y2[a;b];jx yj

for any 0 < b a: As in [15], we have (F )

!1

n (r) (f; ) = sup max ! 1 f ; r2[0;1]

Then we give the following.

(r)

; ! 1 f+ ;

o

:

382

19. A-Summability and Fuzzy Korovkin Approximation

Theorem 19.4. Let A = (An )n real matrices such that

1

be a sequence of in…nite non-negative

1 X sup ankj < 1 n;k j=1

and fLj gj2N be a sequence of fuzzy positive linear operators from CF [a; b] into itself. Assume that there exists a corresponding sequence

Lj

of j2N

positive linear operators from C [a; b] into itself with the property (19.3) and satisfying (19.2). Suppose that Lj satisfy the following conditions:

i. lim

k!1

1 P

j=1

(F )

ii. lim ! 1 k!1

(F )

iii. lim ! 1 k!1

ankj Lj (e0 )

(f;

(f;

n k)

n k)

and ' (y) = (y

e0 = 0 uniformly in n

1 P

j=1

ankj Lj (e0 )

e0 = 0 uniformly in n

= 0 uniformly in n, where

n k

2

x) for each x 2 [a; b].

Then for all f 2 CF [a; b] we have 0 1 1 X lim D @ ankj Lj (f ) ; f A = 0

k!1

uniformly in n.

j=1

v u 1 u P =t ankj Lj (') j=1

19.3 Rate of Convergence

383

Proof. Using the linearity and the positivity of the operators Lj we may write, for each j 2 N, that 1 X

ankj Lj f

(r)

;x

f

(r)

(x)

j=1

f

(r)

1 X

(x)

ankj Lj (1; x)

1 +

1 X

(r)

ankj Lj (e0 ; x)

e0 +

M

1 X

f

(r)

(y)

f

(r)

(x) ; x

1 X

ankj Lj ! 1 f

(r)

;

(y

x)

;x

j=1

j=1

(r)

ankj Lj

j=1

j=1

M

1 X

ankj Lj (e0 ; x)

e0

j=1

1 X + ankj Lj

jy

xj

ankj Lj (e0 ; x)

e0

1+

!1 f

(r)

;

;x

j=1

M

1 X

(r)

j=1

1 X + ankj Lj

1+

2

jy

xj 2

j=1

M

1 X

(r)

ankj Lj (e0 ; x)

!

!1 f

(r)

e0 + ! 1 f

!

;

(r)

;x

1 X

;

j=1

!1 f +

(r)

ankj Lj (e0 ; x)

j=1

;

1 X

2

ankj Lj (y

2

x) ; x

j=1

M

(r)

1 X

ankj Lj (e0 ; x)

e0 + ! 1 f

(r)

1 X

;

j=1

+! 1 f

(r)

ankj Lj (e0 ; x)

e0

j=1

!1 f ;

+

(r)

;

2

1 X

ankj Lj (y

2

x) ; x

j=1

where M

(r)

1 X j=1

:= f

(r)

ankj Lj f

(r)

. Then we observe that

;x

f

(r)

(x)

M

(r)

1 X j=1

ankj Lj (e0 ; x)

e0 +

384

19. A-Summability and Fuzzy Korovkin Approximation

+ !1 f

(r)

1 X

;

ankj Lj (e0 ; x)

e0

j=1

+ !1 f

(r)

!1 f ;

(r)

+

;

1 X

2

ankj Lj ('; x) :

j=1

If we take

=

1 X

n k,

we conclude that

ankj Lj f

(r)

;x

f

(r)

M

(x)

(r)

1 X

ankj Lj (e0 ; x)

e0

j=1

j=1

+! 1 f

(r)

;

n k

1 X

ankj Lj (e0 ; x)

e0 + 2! 1 f

(r)

;

n k

:

j=1

Then we get 0 1 1 X D @ ankj Lj (f ) ; f A

M

1 X

ankj Lj (e0 )

e0

(19.7)

j=1

j=1

+

1 X

ankj Lj (e0 )

(F )

e0 ! 1

(f;

n k)

j=1

(F )

+2! 1 where M := sup r2[0;1]

(f;

n k)

r M r ; M+ :Letting k ! 1 and using the hypothesis

(i), (ii) and (iii) we get 1 0 1 X lim D @ ankj Lj (f ) ; f A = 0

k!1

j=1

uniformly in n; which completes the proof. We note that the rate of convergence of the involved operators may be obtained from (19.7).

20 A-SUMMABILITY AND FUZZY TRIGONOMETRIC KOROVKIN APPROXIMATION

The aim of this chapter is to present a fuzzy trigonometric Korovkin-type approximation theorem by using a matrix summability method. We also study the rates of convergence of fuzzy positive linear operators in trigonometric environment. This chapter is based on [28].

20.1 Introduction Approximation theory which has a close relationship with other branches of mathematics has been used in the theory of polynomial approximation and various domains of functional analysis [3], in numerical solutions of di¤erential and integral operators [78], in the studies of the interpolation operator of Hermit-Fejér [44], [45], [55] and the partial sums of Fourier series [79]. Most of the classical approximation operators tend to converge to the value of the function being approximated. However, at points of discontinuity, they often converge to the average of the left and right limits of the function. The main purpose of using summability theory has always been to make a nonconvergent sequence "converge". Some results regarding matrix summability for positive linear operators may be found in the paper [38], [39], [94]. In this chapter using a A-summation process we study the G.A. Anastassiou: Fuzzy Mathematics: Approximation The., STUDFUZZ 251, pp. 385–397. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

386

20. A-Summability and Fuzzy Trigonometric Korovkin Approximation

approximation properties of sequence of positive linear operators on the space of all 2 -periodic and continuous functions on the whole real axis.

20.2 Background Anastassiou and Duman [29] (see also [63]) introduced the fuzzy analog of A statistical convergence by using the metric D. Let ( n )n2N be a fuzzy number valued sequence and let A = (ajn ) is a non-negative regular summability matrix. Recall that the regularity conditions on a matrix A are known as Silverman-Toeplitz conditions in the functional analysis (see for details, [70]). Then, ( n )n2N is A statistically convergent to 2 RF , which is denoted by stA lim D ( n ; ) = 0 if for n!1 every " > 0 X lim ajn = 0 j!1 n:D(

n;

) "

holds. Some basic results regarding A statistical convergence for number sequences may be found in the papers [65], [77], [86]. Of course, the case of A = C1 , the Cesáro matrix of order one, immediately reduces to the statistical convergence of fuzzy valued sequences which de…ned by Nuray and Sava¸s [91]. Also, if A is replaced by the identity matrix, then we have the fuzzy convergence introduced by Matloka [85]. Let A := (An )n 1 , An = ankj be a sequence of in…nite nonk;j2N

negative real matrices. For a sequence of real numbers, x = (xj )j2N , the double sequence n

Ax := f(Ax)k : k; n 2 Ng n

de…ned by (Ax)k :=

1 P

j=1

ankj xj is called the A transform of x whenever the

series converges for all k and n. A sequence x is said to be A summable to L if lim

1 X

k!1

ankj xj = L

j=1

uniformly in n ([42], [99]). If An = A for some matrix A, then A summability is the ordinary matrix summability by A. If, ankj = k1 , for n j k + n, (n = 1; 2; :::), and ankj = 0 otherwise, then A summability reduces to almost convergence [82].

20.3 A Fuzzy Korovkin Type Theorem

387

20.3 A Fuzzy Korovkin Type Theorem In this section we prove a fuzzy trigonometric Korovkin-type approximation theorem via the concept of A summation process. Let f : R ! RF be fuzzy number valued functions. Then f is said to be fuzzy continuous at x0 2 R provided that whenever xn ! x0 , then D (f (xn ) ; f (x0 )) ! 0 as n ! 1. Also, we say that f is fuzzy continuous on R if it is fuzzy continuous at every point x 2 R. The set all fuzzy continuous functions on R is denoted by CF (R) (see, for instance, [18], (F ) [32]). Notice that CF (R) is only a cone not a vector space. By C2 (R) we mean the space of all fuzzy continuous and 2 -periodic functions on R. Also the space of all real valued continuous and 2 -periodic functions is denoted by C2 (R). Suppose that f : [a; b] ! RF be fuzzy number valued functions. Then, f is said to be fuzzy-Riemann integrable (or, FR-integrable) to I 2 RF if, for given " > 0, there exists a > 0 such that, for any partition P = f[u; v] ; g of [a; b] with the norms (P ) < , we have ! X D (v u) f ( ) ; I < ". P

In this case, we write I := (F R)

Zb

f (x) dx:

a

By Corollary 13.2 of ([66], p. 644), we conclude that if f 2 CF [a; b] (fuzzy continuous on [a; b]), then f is FR-integrable on [a; b] : Now let L : CF (R) ! CF (R) be an operator. Then L is said to be fuzzy linear if, for every 1 ; 2 2 R, f1 ; f2 2 CF (R), and x 2 R; L(

1

f1

2

f2 ; x) =

1

L (f1 ; x)

2

L (f2 ; x)

holds. Also L is called fuzzy positive linear operator if it is fuzzy linear and the condition L (f ; x) L (g; x) is satis…ed for any f; g 2 CF (R) and all x 2 R with f (x) g (x). A sequence fLj g of positive linear operators of C (R) into itself is called an A-summation process on C (R) if fLj (f )g is A-summable to f for every f 2 C (R), i.e., lim

k!1

1 X

ankj Lj (f )

f = 0;

uniformly in n;

(20.1)

j=1

where it is assumed that the series in (20.1) converges for each k; n and f vand khk denotes the usual supremum norm of h 2 C (R).

388

20. A-Summability and Fuzzy Trigonometric Korovkin Approximation

If a sequence fLj g of positive linear opeartors of CF (R) into itself is called an A-summation process on CF (R) if fLj (f )g is A-summable to f for every f 2 CF (R), i.e., 0 1 1 X lim D @ ankj Lj (f ) ; f A = 0; uniformly in n: k!1

j=1

Some uni…cation on Korovkin-type results through the use of a matrix summability method may be found in [3], [88], [89], [90], [100]. Let fLj g be a sequence of positive linear opeartors of C (R) into itself such that for each k; n 2 N 1 X sup ankj kLj (1)k < 1:

(20.2)

n;k j=1

Furthermore, for k; n 2 N and f 2 C (R), let Bkn (f ; x) :=

1 X

ankj Lj (f )

j=1

which is well de…ned by (20.2) and belong to C (R) : In this chapter, we establish a theorem of Korovkin type with respect to the convergence behavior (20.1) for a sequence of positive linear operators (F ) of C2 (R) into itself. So the results of type (20.1) are extensions of lim

k!1

1 X

ankj Lj (f )

f = 0;

uniformly in n:

j=1

Throughout the chapter the symbol kf kdenotes the usual supremum norm of f 2 CF (R) and we use the test functions fi (i = 0; 1; 2) de…ned by f0 (x) = 1; f1 (x) = cos x; f2 (x) = sin x: Then Anastassiou and Gal [32] obtained the following fuzzy Korovkin theorem. Theorem 20.1. Let fLn gn2N be a sequence of fuzzy positive linear (F ) operators de…ned on C2 (R). Suppose that there exists a corresponding sequence property

Ln

of positive linear operators de…ned on C2 (R) with the n2N (r)

fLn (f ; x)g

= Ln f

(r)

;x

(20.3)

for all x 2 [a; b], r 2 [0; 1], n 2 N and f 2 C2F (R). Assume further that lim

n!1

Ln (fi )

fi = 0 for each i = 0; 1; 2:

20.3 A Fuzzy Korovkin Type Theorem

389

Then, for all f 2 C2F (R), we have lim D (Ln (f ) ; f ) = 0:

n!1

Recently the A statistical analog of Theorem 20.1 has been studied by Anastassiou and Duman [29]. It will be read as follows. Theorem 20.2. ([29]) Let A = (ajn ) be a non-negative regular summability matrix and let fLn gn2N be a sequence of fuzzy positive linear (F ) operators de…ned on C2 (R). Suppose that there exists a corresponding sequence

Ln

of positive linear operators de…ned on C2 (R) with the n2N

property (20.3). Assume further that stA

lim

n!1 (F )

Then, for all f 2 C2

fi = 0 for each i = 0; 1; 2:

Ln (fi )

(R), we have

stA

lim D (Ln (f ) ; f ) = 0:

n!1

It is clear that if we replace the matrix A in Theorem 20.2 by the Cesáro matrix C1 , we immediately get the statistical fuzzy Korovkin result in the trigonometric case (see Corollary 2.2. of [29]). We now give the following generalization by using a A-summation process . Theorem 20.3. Let A = (An )n 1 be a sequence of in…nite non-negative real matrices such that 1 X sup ankj < 1 (20.4) n;k j=1

and let fLn gn2N be a sequence of fuzzy positive linear operators de…ned on (F )

C2

(R). Suppose that there exists a corresponding sequence

Ln

of n2N

positive linear operators de…ned on C2 (R) with the property (20.3) and satisfying (20.2). Suppose further that

lim

k!1

1 X

ankj Lj (fi )

fi = 0 for each i = 0; 1; 2:

j=1

(F )

uniformly in n. Then, for all f 2 C2 (R), we have 0 1 1 X lim D @ ankj Lj (f ) ; f A = 0 k!1

j=1

(20.5)

390

20. A-Summability and Fuzzy Trigonometric Korovkin Approximation

uniformly in n: Proof. Assume that I is a closed bounded interval with h length 2 of R.i (F ) (r) (r) (r) Let f 2 C2 (R), x 2 I and r 2 [0; 1]. Taking [f (x)] = f (x) ; f+ (x) (r)

we get f (x) 2 C2 (R). Then, we immediately see from ([78], p.7) that, for every " > 0; there exists a > 0 such that f

(r)

(y)

f

(r)

" + 2M

(x)

holds for all y 2 R and x 2 I where M

(r)

y x 2

2 (r) sin

(20.6)

2

sin

:= f

(r)

2

. Now using the linearity

and the positivity of the operators Lj and considering inequality (20.6), we can write 1 X

ankj Lj f

(r)

;x

f

(r)

(x)

(r)

(y)

f

(r)

(x) ; x + f

j=1

1 X

ankj Lj

f

(r)

1 X

(x)

ankj Lj

" + 2M

y x 2 sin2 2

2 (r) sin

j=1

"

1

j=1

j=1

1 X

ankj Lj (1; x)

1 X

ankj Lj (1; x) +

j=1

+M

(r)

1 X

1 (r) X

2M sin2

ankj Lj (1; x)

!

;x

+ f

1 X

(r)

ankj Lj (1; x)

1

j=1

y

ankj Lj sin2

x 2

2 j=1

;x

1

j=1

"+ "+M

(r)

1 X

ankj Lj (1; x)

1

j=1

+

1 (r) X

2M sin2

ankj Lj sin2

y

2 j=1

"+ "+M

(r)

1 X j=1

2

;x

(r)

ankj Lj (1; x)

0 1 X cos x @ ankj Lj (cos y; x) j=1

x

1 + 1

cos xA

M sin2

2

80 1 < X @ ankj Lj (1; x) : j=1

1

1A

20.3 A Fuzzy Korovkin Type Theorem

0

sin x @ "+

1 X

ankj Lj (sin y; x)

j=1

(r)

"+M

(r)

M + sin2 1 X

(r)

+ jcos xj

M sin2

+ jsin xj

M sin2

2

(r)

If we take K

1 X

ankj Lj f

(r)

(r)

2

(") := " + M

;x

f

(r)

1 X

ankj Lj (1; x)

1

j=1

ankj Lj (cos y; x)

cos x

ankj Lj (sin y; x)

sin x :

j=1

1 X j=1

(r)

(x)

2

!

19 = sin xA ;

(r)

+

M sin2

; then we get 2

"+K

(r)

(")

j=1

+

1 X

8 1 <X :

ankj Lj (f0 ; x)

j=1

ankj Lj (f1 ; x)

f1

j=1

1 X

+

ankj Lj (f2 ; x)

f2

j=1

Then we observe that

sup max r2[0;1]

1 X

ankj Lj f

(r)

;x

j=1

+ sup max K

(r)

(")

r2[0;1]

+

1 X

ankj Lj (f1 ; x)

f1

f

+

j=1

ankj Lj (f2 ; x)

(r)

8 1 <X :

j=1

1 X

391

f2

j=1

9 = ;

:

(x)

9 = ;

:

"

ankj Lj (f0 ; x)

f0

f0

392

20. A-Summability and Fuzzy Trigonometric Korovkin Approximation

n o (r) (r) Taking K := K (") := sup max K (") ; K+ (") ; we get r2[0;1]

1 X

sup sup max x2Rr2[0;1]

ankj Lj f

(r)

;x

f

(r)

(x)

j=1

8 1 < X ankj Lj (f0 ; x) " + K (") sup :x2R

f0

j=1

+ sup x2R

+sup

1 X

ankj Lj (f1 ; x)

f1

j=1

1 X

x2R j=1

Consequently, we obtain 0 1 1 X D @ ankj Lj (f ) ; f A

ankj Lj (f2 ; x)

"+K

f2

1 X

9 = ;

:

ankj Lj (f0 ; x)

f0

j=1

j=1

+K

1 X

ankj Lj (f1 ; x)

f1

ankj Lj (f2 ; x)

f2 :

j=1

+K

1 X j=1

By taking limit as k ! 1 and by using (20.5) we obtain the desired result. Remarks. We now present three examples of sequences of positive linear operators. The …rst one shows that approximation Theorem 20.3 works but the statistical version of Theorem 20.2 does not work. The second one shows that approximation Theorem 20.3 works but Theorem 20.1 does not work and the last one shows that statistical version of Theorem 20.2 works but Theorem 20.3 does not work. Assume that A := (An )n 1 = ankj is a sequence of in…nite matrik;j2N

ces de…ned by ankj = k1 if n j k+n, (n = 1; 2; :::), and ankj = 0 otherwise, then A summability reduces to almost convergence. Now de…ne the fuzzy Fejer operators Fj as follows: 8 9 Z < = sin2 2j (y x) 1 (F R) f (y) dy ; Fj (f ; x) = : ; j 2 sin2 y 2 x

20.3 A Fuzzy Korovkin Type Theorem

393

(F )

where j 2 N, f 2 C2 (R) and x 2 R. Then observe that the operators Fj are fuzzy positive linear. So we write Z sin2 2j (y x) 1 (r) (r) dy fFj (f ; x)g = Fj f ; x := f (y) j 2 sin2 y 2 x where f

(r)

2 C2 (R) and r 2 [0; 1]. Then, we get (see [78]) Fj (f0 ; x)

=

Fj (f1 ; x)

=

1 n

1 n

cos x

n 1 Fj (f2 ; x) = sin x: n n o j (i) Take (uj ) = ( 1) . Observe that u is almost convergent to zero, but it is not statistically convergent. Now using the sequence (uj ) and the fuzzy Fejer operators Fj ; we introduce the following fuzzy positive linear (F ) operators de…ned on the space C2 (R) : Bj (f ; x) = (1 + uj )

Fj (f ; x) ;

(F )

where j 2 N, f 2 C2 (R) and x 2 R. So, the corresponding real positive linear operators are given by Z sin2 2j (y x) 1 + uj (r) f (y) dy; Bj f ; x := j 2 sin2 y 2 x where f

(r)

2 C2 (R). Then we get, for all j 2 N and x 2 R, that Bj (f0 )

f0

Bj (f1 )

f1

Bj (f2 )

f2

= uj 1 + uj j 1 + uj uj + : j uj +

Since u is almost convergent to zero we get n+k

lim

k!1

1X Bj (fi ) k j=n

fi = 0 for each i = 0; 1; 2 (F )

uniformly in n: So, by Theorem 20.3, we obtain, for all f 2 C2 0 1 1 X lim D @ ankj Bj (f ) ; f A = 0 k!1

j=1

(R), that

394

20. A-Summability and Fuzzy Trigonometric Korovkin Approximation

uniformly in n. However, since u is not statistically convergent to zero, we observe that fBj g satis…es the Theorem 20.3 but it does not satisfy the statistical version of Theorem 20.2. (ii) Let (vj ) = (1; 0; 1; 0; :::). We see that (vj ) is almost convergent to 21 , but not convergent (in the ordinary sense). Then consider 1 + vj 2

Lj (f ; x) =

Fj (f ; x) ;

(F )

for f 2 C2 (R), x 2 R and j 2 N: Hence fLj g satis…es the Theorem 20.3 but it does not satisfy the Theorem 20.1. (iii) Now consider the sequence of 0’s and 1’s de…ned as follows: 0; :::; 0 ; 1; :::; 1 ; 0; :::; 0; 1; :::; 1; 1; ::: ! 100

!

10

where the blocks of 0’s are increasing by factors of 100 and the blocks of 1’s are increasing by factors of 10. If we denote this sequence by (yj ) ; then (yj ) is not almost convergent but it is statistically convergent to zero ([87]). Then we de…ne Tj (f ; x) = (1 + yj ) Fj (f ; x) (F )

for f 2 C2 (R), x 2 R and j 2 N: We observe that fTj gj2N satis…es the statistical version of Theorem 20.2 but it does not satisfy Theorem 20.3.

20.4 Rate of Convergence In this section we study the rates of convergence of the sequence of fuzzy positive linear operators examined in Theorem 20.3. (F ) Let f 2 C2 (R) : Then the (…rst) fuzzy modulus of continuity of f , which is introduced by Gal [66] (see also [18], [32]), is de…ned by (F )

!1

(f; ) :=

sup

D (f (x) ; f (y))

x;y2R;jx yj

for any > 0: (r) It is easy to see that, for any c > 0 and all f 2 C2 (R) (F )

!1

f

(r)

;c

(F )

(1 + [c]) ! 1

f

(r)

;

where [c] is de…ned to be greatest integer less than or equal to c. (F ) Lemma 20.4. [18] Let f 2 C2 (R). then it holds: n o (F ) (r) (r) ! 1 (f; ) = sup max ! 1 f ; ; ! 1 f+ ; ; r2[0;1]

20.4 Rate of Convergence

for any > 0. Then we have the following. Theorem 20.5. Let A = (An )n real matrices such that

1

395

be a sequence of in…nite non-negative

1 X sup ankj < 1 n;k j=1

and fLj gj2N be a sequence of fuzzy positive linear operators de…ned on (F )

C2

(R). Assume that there exists a corresponding sequence

Lj

of j2N

positive linear operators de…ned on C2 (R) with the property (20.3). Suppose that Lj satisfy the following conditions:

i. lim

k!1

1 P

j=1

(F )

(f;

n k)

(F )

(f;

n k)

ii. lim ! 1 k!1

iii. lim ! 1 k!1

where

n k

ankj Lj (f0 )

f0 = 0 uniformly in n;

= 0 uniformly in n,

1 P

j=1

ankj Lj (f0 )

f0 = 0 uniformly in n,

v u 1 u P = t ankj Lj (') and ' (y) = sin2 j=1

(F )

Then for all f 2 C2

y x 2

(R) ; we have

0 1 1 X lim D @ ankj Lj (f ) ; f A = 0

k!1

uniformly in n.

j=1

for each x 2 R.

396

20. A-Summability and Fuzzy Trigonometric Korovkin Approximation

Proof. Using the linearity and the positivity of the operators Lj and using property (20.4) by [97], we may write, for each j 2 N, > 0; that 1 X

ankj Lj f

(r)

;x

f

(r)

(x)

j=1

f

(r)

1 X

(x)

ankj Lj (1; x)

1 +

1 X

(r)

ankj Lj

f

(r)

(y)

f

(r)

(x) ; x

j=1

j=1

M

1 X

ankj Lj (f0 ; x)

f0

j=1

1 X + ankj Lj

2

1+

sin2

y

j=1

M

1 X

(r)

ankj Lj (f0 ; x)

x

!1 f

2

f0 + ! 1 f

(r)

1 X

;

j=1

!1 f

(r)

;

1 X

y

ankj Lj sin2

x

1 X

(r)

ankj Lj (f0 ; x)

f0 + ! 1 f

(r)

!1 f

(r)

;

1 X

ankj Lj sin2

y

x 2

j=1

(r)

1 X

ankj Lj (f0 ; x)

f0

j=1

2

where M

ankj Lj (f0 ; x)

1 X

;

j=1

+

;x

;x

2

j=1

M

;

j=1

2

+

(r)

:= f

(r)

ankj Lj f

; x + !1 f

(r)

;

. Taking supremum over x 2 R, we easily see that

(r)

;x

f

(r)

(x)

j=1

M

(r)

1 X

ankj Lj (f0 ; x)

f0 + ! 1 f

(r)

1 X

;

j=1

!1 f

(r)

;

1 X

ankj Lj sin2

j=1

Now putting

:=

f0

j=1

2

+

ankj Lj (f0 ; x)

n k,

we conclude that

y

x 2

;x

+ !1 f

(r)

;

:

20.4 Rate of Convergence

1 X

ankj Lj f

(r)

;x

f

(r)

397

(x)

j=1

M

(r)

1 X

ankj Lj (f0 ; x)

f0 + ! 1 f

(r)

;

n k

1 X

ankj Lj (f0 ; x)

f0

j=1

j=1

+2! 1 f

(r)

n k

;

:

Then we get from Lemma 20.4 0 1 1 X D @ ankj Lj (f ) ; f A

(20.7)

j=1

1 X

M

ankj Lj (f0 )

f0

j=1

(F )

+! 1

(f;

n k)

1 X

ankj Lj (f0 )

(F )

f0 + 2! 1

(f;

n k) ;

(20.1)

j=1

o n (r) (r) . If we take limit as k ! 1 on the where M := sup max M+ ; M r2[0;1]

both sides of inequality (20.7) and use the hypotesis (i) ; (ii) and (iii), we immediately see that 0 1 1 X lim D @ ankj Lj (f ) ; f A = 0: k!1

j=1

Inequality (20.7) gives the above convergence with rates.

21 UNIFORM REAL AND FUZZY ESTIMATES FOR DISTANCES BETWEEN WAVELET TYPE OPERATORS AT REAL AND FUZZY ENVIRONMENT

The basic fuzzy wavelet type operators Ak ; Bk ; Ck ; Dk ; k 2 Z were studied in [14], [16], see also Chapters 12, 16, for their pointwise and uniform convergence with rates to the fuzzy unit operator. Also they were studied in [23], see also Chapter 13, in terms of estimating their fuzzy di¤erences and giving their pointwise convergence with rates to zero. For prior related and similar study of convergence to the unit of real analogs of these wavelet type operators see [9], Section II. Here in Section 21.1 we present the complete study of …nding uniform estimates for the distances between the real Wavelet type operators Ak ; Bk ; Ck ; Dk ; k 2 Z: Their di¤erences converge to zero with rates. This is done via elegant tight Jackson type inequalities involving the modulus of continuity of the higher order derivative of the engaged real function. Based on these real analysis results in Section 21.2 we establish the corresponding fuzzy results regarding uniform estimates for the fuzzy di¤erences between the fuzzy wavelet type operators. These fuzzy diferences converge to zero with rates give via fuzzy Jackson type tight inequalities. The last inequalities involve the fuzzy modulus of continuity of the higher order fuzzy derivative of the engaged fuzzy function. The de…ning all these operators real scaling function is not assumed to be orthogonal and is of compact support.

G.A. Anastassiou: Fuzzy Mathematics: Approximation The., STUDFUZZ 251, pp. 399–429. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

400

21. Real and Fuzzy Estimates for Distances between Wavelet type Operators

21.1 Estimates for Distances of Real Wavelet type Operators The real wavelet type operators Ak ; Bk ; Ck ; Dk ; k 2 Z we study here converge to the unit operator I; and in that respect were studied extensively in [9], Section II. We need De…nition 21.1. Let f : R ! R be continuous function. We de…ne the …rst modulus of continuity of f by ! 1 (f; ) =

sup jf (x)

f (y)j ; > 0:

x;y jx yj

We give Theorem 21.2. Let f 2 C N (R) ; N 1; x 2 R; and k 2 Z: Let ' be a bounded function of compact support [ a; a] ; a > 0 such that 1 P ' (x j) = 1; all x 2 R: Suppose ' 0: Put j= 1

(Bk f ) (x) (Dk f ) (x)

1 X

=

j= 1 1 X

=

j 2k

f

kj

' 2k x

(f ) ' 2k x

j ;

j ;

j= 1

where kj

(f ) =

n X

wr f

r=0

n 2 N; wr E1k (x)

0;

n P

j r + k k 2 2 n

;

wr = 1: Then

r=0

=

(Dk f ) (x) n P

N

N X Bk f (i) (x) 2ki ni i! i=1

(Bk f ) (x)

wr r ! 1 f

(N )

r=1

2kN nN N !

;

n X r=1

wr ri

!

r 2k n

:

(21.1)

Remark. (i) Given that f (N ) is continuous and bounded or uniformly continuous we have that ! 1 f (N ) ; 2krn < 1; and E1k (x) ! 0; x 2 R; as k ! 1:

21.1 Estimates for Distances of Real Wavelet type Operators

401

(ii) One also has kE1k k1

Dk f

=

n P

N X Bk f (i)

Bk f N

n X

2ki ni i! i=1

wr r ! 1 f

(N )

;

r=1

wr ri

r=1

!

1

r 2k n

:

2kN nN N !

Under the assumptions of (i) we get also kE1k k1 ! 0; as k ! 1: (iii) By (21.1) we get j(Dk f ) (x)

(Bk f ) (x)j

+ Given that f (i) f (i) 1 ; and j(Bk f ) (x)

1

n X

N X Bk f (i) (x) 2ki ni i! i=1 n P

r=1

wr r

i

r=1

!

wr rN ! 1 f (N ) ; 2krn :

2kN nN N !

(21.2)

< 1; for i = 1; : : : ; N; we obtain Bk f (i) (x) N X f (i) 1 2ki ni i! i=1

(Dk f ) (x)j

+

n P

r=1

n X

wr r

r=1

i

!

wr rN ! 1 f (N ) ; 2krn :

2kN nN N !

(21.3)

Clearly we get j(Bk f ) (x)

N X f (i) 1 ! 1 f (N ) ; 21k + 2ki i! 2kN N ! i=1

(Dk f ) (x)j

=: T1

and kBk f

Dk f k1

T1 :

(21.4)

So as k ! 1; we have kBk f Dk f k1 ! 0: Proof of Theorem 21.2. Because f 2 C N (R) ; N n X

wr f

r=0

n X + wr r=0

Z

j r + k k 2 2 n (j=2

j=2k

k

)+ kr 2 n

=f

f

(N )

(t)

j 2k

+

N X f (i) i=1

f

(N )

k

j=2

j 2k

i!

n X r=0

j 2k

1 we have wr

ri 2ki ni

+ 2krn t (N 1)!

N

1

dt:

402

21. Real and Fuzzy Estimates for Distances between Wavelet type Operators

Hence we get 1 X

kj

(f ) ' 2k x

j= 1 N X

+

1 P

j 2k

f (i)

n 1 X X + wr ' 2k x r=0

j 2k

f

j= 1

j= 1

i=1

1 X

j =

' 2k x

2ki i!ni Z (j=2k )+

j

j

' 2k x

n X

wr r

r=0

r 2k n

f (N ) (t)

i

j

!

f (N ) j=2k

j=2k

j= 1

j 2k

N

+ 2krn t (N 1)!

1

dt:

So, we observe that (Dk f ) (x)

N X Bk f (i) (x) 2ki ni i! i=1

(Bk f ) (x)

where R1

=

n X

wr

r=1

1 X

' 2k x

(j=2k )+

r 2k n

wr ri

r=0

f (N ) (t)

!

= R1 ;

f (N ) j=2k

j=2k

j= 1

j 2k

Z

j

n X

+ 2krn t (N 1)!

N

1

dt:

Set jr

=

Z

(j=2k )+

r 2k n

f

(N )

(t)

f

(N )

j=2

j 2k

k

+ 2krn t (N 1)!

j=2k

So that

n X

jR1 j Next we see that Z (j=2k )+

r 2k n

jr

wr

r=1

1 X

' 2k x

j

(j=2k )+

r 2k n

f (N ) (t)

f (N ) j=2k

!1 f

(N )

; t

k

(j=2 )

j=2k

! 1 f (N ) ; = !1 f

(N )

r 2k n

r ; k 2 n

Z

1

dt :

jr :

j= 1 j 2k

j=2k

Z

N

(j=2k )+

j=2k N r 2k n

N!

:

r 2k n

j 2k

j 2k

+ 2krn t (N 1)! + 2krn t (N 1)!

+ 2krn t (N 1)!

N

1

dt

N

1

dt N

1

dt

21.1 Estimates for Distances of Real Wavelet type Operators

That is ! 1 f (N ) ;

jr

N r 2k n

r 2k n

N!

403

:

So we have found that n P

r=1

jR1 j

wr rN ! 1 f (N ) ; 2krn :

2kN nN N !

The proof of the theorem is complete. We continue with Theorem 21.4. Let f 2 C N (R) ; N 1; x 2 R; and k 2 Z: Let ' be a bounded function of compact support [ a; a] ; a > 0 such that 1 P ' (x j) = 1; all x 2 R: Suppose ' 0: Put j= 1

(Bk f ) (x)

=

(Ck f ) (x)

=

1 X

j 2k

f

j= 1 1 X

kj

' 2k x

j ;

(f ) ' 2k x

j ;

Z

t+

j= 1

where k kj (f ) = 2

Z

k

2

(j+1)

f (t) dt = 2k

2

kj

2

k

f

0

j 2k

dt:

Then E2k (x)

=

(Ck f ) (x)

(Bk f ) (x)

! 1 f (N ) ; 21k : 2kN (N + 1)!

N X Bk f (i) (x) 2ki (i + 1)! i=1

(21.5)

Remark 21.5. (i) Given that f (N ) is continuous and bounded or uniformly continuous we have that ! 1 f (N ) ; 21k < 1; and E2k (x) ! 0; x 2 R; as k ! 1: (ii) One also has kE2k k1

=

Ck f

Bk f

! 1 f (N ) ; 21k : 2kN (N + 1)!

N X Bk f (i) ki 2 (i + 1)! i=1

1

404

21. Real and Fuzzy Estimates for Distances between Wavelet type Operators

Under the assumptions of (i) we get also kE2k k1 ! 0; as k ! 1: (iii) By (21.5) we get j(Bk f ) (x)

N X ! 1 f (N ) ; 21k Bk f (i) (x) + kN : ki 2 (i + 1)! 2 (N + 1)! i=1

(Ck f ) (x)j

Given that f (i) j(Bk f ) (x)

(21.6)

< 1; for i = 1; : : : ; N; we obtain

1

N X

f (i) 1 ! 1 f (N ) ; 21k + =: T2 ; 2ki (i + 1)! 2kN (N + 1)! i=1

(Ck f ) (x)j

and kBk f

Ck f k1

T2 :

(21.7)

So as k ! 1; we get kBk f Ck f k1 ! 0: Proof of Theorem 21.4. Because f 2 C N (R) ; N f

j 2k

t+

j 2k

= f

+

Z

+

N X f (i) i=1

j 2k

i!

1 we have

ti

t+(j=2k )

(N )

f

(s)

f

(N )

j=2

N

t + 2jk (N

k

j=2k

s 1)!

1

ds:

Hence we get

kj

(f ) = 2

k

Z

2

k

f

t+

0

+2k

Z

Z

k

2

j 2k

j 2k

dt = f

+

t+(j=2k )

f (N ) (s)

t + 2jk (N

f (N ) j=2k

j=2k

0

N X f (i) 2jk 2ki (i + 1)! i=1 N

s 1)!

1

ds dt:

Hence we get 1 X

kj

(f ) ' 2k x

j= 1

1 X

j= 1

1 X

j =

j 2k

f

j= 1 N 1 X X f (i) i=1 j= 1

k

' 2 x

j 2

k

Z

2

k

0

t + 2jk (N

j 2k 2ki

Z

' 2k x (i + 1)!

' 2k x j

j +

+

t+(j=2k )

f (N ) (s)

j=2k N

s 1)!

1

!

ds dt:

!

f (N ) j=2k

21.1 Estimates for Distances of Real Wavelet type Operators

405

So, we see that (Ck (f )) (x)

N X Bk f (i) (x) = R2 ; 2ki (i + 1)! i=1

(Bk (f )) (x)

where 1 X

R2 = Z

2

Z

k

' 2k x

j 2k

j= 1

k

f

(N )

(s)

f

(N )

j=2

k

j=2k

0

N

t + 2jk (N

t+(j=2 )

!

1

s 1)!

ds dt:

Set Z

j (t) =

t+(j=2k )

f (N ) (s)

j=2k

So that

1 X

jR2 j

' 2k x

j 2k

Z

j

f

(N )

! 1 f (N ) ; s

j=2

t + 2jk (N

(j=2k )

j=2k

;t

Z

t+(j=2k )

t + 2jk (N

j=2k

= ! 1 f (N ) ; t

(t) dt:

t + 2jk (N

k

t+(j=2k )

!1 f

ds :

k

2

j=2k

(N )

1

s 1)!

0

j= 1

Next we observe that Z t+(j=2k ) f (N ) (s) j (t) Z

N

t + 2jk (N

f (N ) j=2k

N

N

s 1)!

N

s 1)!

1

s 1)!

ds

tN : N!

I.e. we get j

(t)

! 1 f (N ) ; t

tN ; N!

and 2k

Z

2

k

j

2k

(t) dt

0

Z

2

k

0

k

2 1 ! 1 f (N ) ; k N! 2 = =

tN dt N!

! 1 f (N ) ; t Z

2

k

tN dt

0

2k 1 ! 1 f (N ) ; k 2 (N + 1)! 2 1 1 ! 1 f (N ) ; k kN 2 (N + 1)! 2

k(N +1)

:

1

ds 1

ds

406

21. Real and Fuzzy Estimates for Distances between Wavelet type Operators

That is we obtain Z 2 2k

k

1 1 ! 1 f (N ) ; k 2kN (N + 1)! 2

(t) dt

j

0

It is clear that

:

! 1 f (N ) ; 21k : 2kN (N + 1)!

jR2 j

The proof of the theorem is now …nished. We continue with Theorem 21.6. Let f 2 C N (R) ; N 1; x 2 R; and k 2 Z: Let ' be a bounded function of compact support [ a; a] ; a > 0 such that 1 X ' (x j) = 1; all x 2 R: Suppose ' 0: Put j= 1

1 X

(Ck f ) (x) =

kj

(f ) ' 2k x

j ;

j= 1

where k kj (f ) = 2

Z

2

2

k

(j+1)

f (t) dt = 2k kj

Z

2

k

f

t+

0

and

1 X

(Dk f ) (x) =

kj

(f ) ' 2k x

j 2k

dt;

j ;

j= 1

where kj

(f ) =

n X

j r + k k 2 2 n

wr f

r=0

n 2 N; wr E3k (x)

0;

n X

;

wr = 1: Then

r=0

=

(Ck f ) (x) 1 n X r=0

i+1

r n wr

(Dk f ) (x) i+1

( 1) h

r N +1 n

2kN

+ 1

(N + 1)!

N X Dk f (i) (x) 2ki (i + 1)! i=1

r n

i+1

r N +1 n

(21.8) i

!

! 1 f (N ) ;

1 2k

:

21.1 Estimates for Distances of Real Wavelet type Operators

407

Remark 21.7 (i) Given that f (N ) is continuous and bounded or uniformly continuous we have that ! 1 f (N ) ; 21k < 1; and E3k (x) ! 0; x 2 R; as k ! 1: (ii) One also has kE3k k1

=

Ck f

N X

Dk f

i=1

1 n X

i+1

r n wr

r=0

Dk f (i) 2ki (i + 1)! r n

i+1

( 1) h

r N +1 n

i+1

r N +1 n

+ 1

2kN (N + 1)!

i

!1

! 1 f (N ) ;

1 2k

:

Under the assumption of (i) we get also kE3k k1 ! 0; as k ! 1: (iii) By (21.8) we get j(Ck f ) (x)

+

Dk f (i) (x) ki 2 1 (i + 1)!

i=1

n X r=0

N X

(Dk f ) (x)j wr

h

r N +1 n

2kN

r N +1 n

+ 1

(N + 1)!

i

!

! 1 f (N ) ;

1 2k

:

(21.9)

Clearly then j(Ck f ) (x)

N X

(Dk f ) (x)j

! 1 f (N ) ; 21k Dk f (i) (x) + : (21.10) 2ki 1 (i + 1)! 2kN 1 (N + 1)!

i=1

Given that f (i)

1

< 1; for i = 1; : : : ; N; we get Dk f (i) (x)

f (i)

1

;

and j(Ck f ) (x)

(Dk f ) (x)j

N X i=1

2ki

f (i) 1 ! 1 f (N ) ; 21k + =: T3 1 (i + 1)! 2kN 1 (N + 1)! (21.11)

and k(Ck f ) (x) So as k ! 1; we get kCk f

(Dk f ) (x)k1 Dk f k1 ! 0:

T3 :

(21.12)

408

21. Real and Fuzzy Estimates for Distances between Wavelet type Operators

Proof of Theorem 21.6. Because f 2 C N (R) ; N f

+

t+

Z

j 2k

j r + k 2k 2 n

=f

+

(j=2k )+

j 2k

r 2k n

i

r

t

2k n N

t + 2jk (N

j r + k 2k 2 n

f (N )

r 2k n

+ i!

i=1

t+(j=2k)

f (N ) (s)

N X f (i)

1 we have

1

s 1)!

ds:

Hence we get Z

k kj (f ) = 2

k

2

f

t+

0

+

N X n X

wr f

j 2k

(i)

i!

i=1 r=0

Z

0

2

k

Z

r 2k n

+

n X j dt = wr f 2k r=0 Z 2 k r 2k t k 2 n 0

t+(j=2k )

f

(j=2k )+

(N )

(s)

f

r 2k n

i

dt +

n X

wr 2k

r=0

j 2k

t+ (N

r j + k k 2 2 n

(N )

j r + k 2k 2 n

N

s 1)!

1

!

ds dt:

That is we have 1 X

k kj (f ) ' 2 x

1 X

j =

1 X

N X j=

kj

1

r n

1 1 X

j= 1

Z

2

j

k

0

f (i) ' 2k x

j

2ki (i + 1)!

i=1

+

(f ) ' 2k x

j= 1

j= 1

+

kj

i+1

r n

i+1

( 1)

' 2k x

n X

j

i+1

wr 2k

r=0

Z

t+(j=2k )

f (N ) (s)

f (N )

(j=2k )+ kr 2 n N

t + 2jk (N

s 1)!

1

j r + k k 2 2 n

!

ds dt:

Consequently we get (Ck f ) (x) (Dk f ) (x)

N X Dk f (i) (x) 2ki (i + 1)! i=1

1

r n

i+1

i+1

( 1)

r n

i+1

= R3 ;

21.1 Estimates for Distances of Real Wavelet type Operators

409

where

R3 = Z

2

k

Z

' 2k x

f

(s)

n X

wr 2k

r=0

t+(j=2 ) (N )

j

j= 1

k

(j=2k )+

0

1 X

f

r 2k n

N

t + 2jk (N

j r + k 2k 2 n

(N )

!

1

s 1)!

ds dt:

Set

jr (t) =

Z

t+(j=2k )

f (N ) (s)

(j=2k )+

r 2k n

N

t + 2jk (N

j r + k 2k 2 n

f (N )

1

s 1)!

ds :

So that

jR3 j

1 X

k

' 2 x

j

n X

k

wr 2

2

k

jr

(t) dt:

0

r=0

j= 1

Z

Next we observe that (i) Case of 2krn t: So we have

jr

(t)

Z

t+(j=2k )

f (N ) (s)

(j=2k )+

Z

r 2k n

t+(j=2k )

(j=2k )+

!1 f

!1 f

r 2k n

(N )

(N )

!1 f

;t

;t

! 1 f (N ) ;

1 2k

r k 2 n t

(N )

j 2k

; s

Z

t+(j=2k )

(j=2k )+

r 2k n

r k 2 n

t + 2jk (N

t + 2jk (N

s 1)!

N

N r 2k n

N! t 2krn N!

N

:

So we obtain

jr

(t)

!1 f

(N )

1 ; k 2

t

N r 2k n

N!

N

t + 2jk (N

j r + k k 2 2 n

f (N )

:

s 1)! N

s 1)! 1

ds

1

ds

1

ds

410

21. Real and Fuzzy Estimates for Distances between Wavelet type Operators r 2k n

(ii) Case of

jr

(t)

=

Z

t: We have

(j=2k )+

r 2k n

f (N ) (s)

j r + k 2k 2 n

f (N )

t+(j=2k )

s Z

t + 2jk (N 1)!

(j=2k )+

r 2k n

N

1

ds

f (N ) (s)

t+(j=2k )

Z (j=2k )+ kr 2 n

!1 f

(N )

j r + k 2k 2 n

;

t+(j=2k )

!1 f

(N )

r ; k 2 n

t

r 2k n

t

= ! 1 f (N ) ; !1 f

(N )

1 ; k 2

t + 2jk (N 1)!

t+(j=2k ) r 2k n

t

t + 2jk (N 1)!

s

s

Z (j=2k )+ kr s 2 n

t + 2jk (N 1)!

s

j r + k 2k 2 n

f (N )

N

N

N

1

ds 1

ds

1

ds

N

N!

r 2k n

t

N

:

N!

So we derive jr (t)

r 2k n

1 2k

! 1 f (N ) ;

N

t

:

N!

Therefore we have found jr (t)

! 1 f (N ) ; 21k N!

N

r

t

:

2k n

Furthermore we see that Z

2

k

jr

! 1 f (N ) ; 21k N!

(t) dt

0

=

! 1 f (N ) ; 21k "Z rN ! 2k n r k 2 n 0

Z

2

k

t

r 2k n

dt +

Z

0

N

t "

=

! 1 f (N ) ; 21k (N + 1)!

=

! 1 f (N ) ; 21k 1 k(N (N + 1)! 2 +1)

r 2k n

2

dt

k

t

r k 2 n

1 2k

r 2k n

r 2k n

N +1

+ r n

N

N +1

+ 1

N

dt

#

N +1

r n

#

N +1

:

21.1 Estimates for Distances of Real Wavelet type Operators

411

Thus k

2

Z

2

k

jr

(t) dt

0

! 1 f (N ) ; 21k 2kN (N + 1)!

N +1

r n

+ 1

r n

N +1

:

Finally we derive that jR3 j

! 1 f (N ) ; 21k 2kN (N + 1)!

!

n X

N +1

r n

wr

r=0

+ 1

r n

N +1

;

proving the theorem. We continue with Theorem 21.8. Let f 2 C N (R) ; N 1; x 2 R and k 2 Z; also f (i) 1 < 1; i = 1; : : : ; N: Let ' be a bounded function of compact 1 X support [ a; a] ; a > 0 such that ' (x j) = 1 all x 2 R: Suppose j= 1

'

0 and ' is Lebesgue measurable (then 'kj (x) := 2k=2 ' 2k x f; 'kj =

Z

1

j

R1

1

' (x) dx = 1): De…ne

all k; j 2 Z;

f (t) 'kj (t) dt;

1

and (Ak f ) (x)

=

=

1 X

f; 'kj 'kj (x)

j= 1 1 X

j= 1

Z

also de…ne (Bk f ) (x) =

1

u ' (u 2k

f

1

1 X

f

j= 1

j 2k

j) du ' 2k x

' 2k x

j ;

j :

Then j(Ak f ) (x)

(Bk f ) (x)j

kAk f N X i=1

x 2 R: So as k ! 1; we get kAk f

B k f k1

(i)

(21.13)

f aN a 1 i a + kN ! 1 f (N ) ; k ; ki 2 i! 2 N! 2

Bk f k1 ! 0:

412

21. Real and Fuzzy Estimates for Distances between Wavelet type Operators 1 X

Proof. By Notice that (Ak f ) (x)

' 2k x

j = 1 we have that

j= 1

(Bk f ) (x)

1 X

=

j= 1 1 X

' (u

1 X

j) du = 1:

j 2k

f

j= 1

Z

1

f

u ' (u 2k

j 2k

' 2k x

f

u ' (u 2k

1

f

j= 1 Z 1 1 X

=

1

f; 'kj 'kj (x)

j= 1 1 X

=

R1

1

j= 1

' 2k x

j) du ' 2k x

j

j

j) du

j 2k

f

k

' 2 x j Z 1 1 X

=

f

u 2k

f

j 2k

' (u

j) du

f

u 2k

f

j 2k

' (u

j) du

1

j= 1

' 2k x

j :

That is (Ak f ) (x)

(Bk f ) (x)

Z

1 X

=

j= 1

1 1

' 2k x

j :

By supp ' [ a; a] we have that ' (u j) is nonzero when a; that is when j a u j + a: Hence

1 X

j= 1

Z

(Ak f ) (x) j+a

u 2k

f

j a

f

a

(Bk f ) (x) = j 2k

' (u

j) du ' 2k x

Next we see that f

u 2k

j 2k

f

=

N X f (i) i=1

j 2k

i!

i

(u

j) 2ki

+ R4 ;

where R4 =

Z

u=2k

j=2k

f

(N )

(t)

u j

f

(N )

j 2k

u 2k

(N

t

N

1)!

1

dt:

j :

j

21.1 Estimates for Distances of Real Wavelet type Operators

(i) Case of j

413

u: We have Z

jR4 j

u=2k (N )

f

(t)

f

j=2k

Z

u=2k

j=2k

= ! 1 f (N ) ; (ii) Case of j jR4 j =

(u

Z

j) 2k

(u

N

t

(N u 2k

j 2k

! 1 f (N ) ; t

! 1 f (N ) ;

u 2k

j 2k

(N )

1

dt

1)! N

t

(N

j=2k

N

t

u 2k

dt

1)!

(N

u=2k

1

1

dt

1)!

N

(u j) : 2kN N !

j) 2k

u: We have Z

j=2k

f (N ) (t)

f (N )

(N )

(N )

u=2k

Z

j=2k

f

(t)

f

u=2k

Z

j=2k

!1 f

(N )

u=2k

! 1 f (N ) ; = ! 1 f (N ) ;

j

u 2k

j

u=2k

u 2k

(N

(N (N

1)!

u N 2k

t (N

dt

1

1)!

u N 2k

t

1

1)!

u N 2k

t

j 2k

j t ; 2k Z j=2k

u N 2k

t

j 2k

dt

1

dt

1

dt

1)!

N

(j u) : 2kN N !

So we have proved that jR4 j

! 1 f (N ) ;

ju

jj 2k

N

ju jj 2kN N !

! 1 f (N ) ;

a 2k

aN ; 2kN N !

i.e. jR4 j

aN a ! 1 f (N ) ; k : kN 2 N! 2

Furthermore we observe that Z j+a u j f k f 2 2k j a Z j+a N X f (i) j=2k = (u 2ki i! j a i=1

' (u i

j) ' (u

j) du j) du +

Z

j+a

j a

R4 ' (u

j) du:

414

21. Real and Fuzzy Estimates for Distances between Wavelet type Operators

Therefore Z j+a f j a

u 2k

j 2k

f

' (u

N X f (i) 1 i a + 2ki i! i=1

j) du

a aN ! 1 f (N ) ; k : kN 2 N! 2

The last proves the theorem. We continue with Theorem 21.9. Let f 2 C N (R) ; N 1; x 2 R and k 2 Z; also f (i) 1 < 1; i = 1; : : : ; N: Let ' be a bounded function of compact 1 X support [ a; a] ; a > 0 such that ' (x j) = 1 all x 2 R: Suppose

'

j= 1

0 and ' is Lebesgue measurable (then

R1

1

' (x) dx = 1): De…ne

'kj (x) := 2k=2 ' 2k x j all k; j 2 Z; Z 1 f; 'kj = f (t) 'kj (t) dt; 1

and (Ak f ) (x)

=

=

1 X

j= 1 1 X j= 1

f; 'kj 'kj (x) Z

Also de…ne (Dk f ) (x) =

1

f

1 1 X

u ' (u 2k

kj

j) du ' 2k x

(f ) ' 2k x

j :

j ;

j= 1

where kj

(f ) =

n X

wr f

r=0

n 2 N; wr

0;

n X

j r + k 2k 2 n

;

wr = 1: Then

r=0

j(Ak f ) (x)

(Dk f ) (x)j

kAk f

N X i=1

Dk f k1

(21.14)

f (i) 1 i (a + 1) + 2ki i! N

(a + 1) (a + 1) ! 1 f (N ) ; kN N !2 2k

:

21.1 Estimates for Distances of Real Wavelet type Operators

So as k ! 1; we get kAk f Dk f k1 ! 0: 1 X R1 Proof. By ' 2k x j = 1 we have that 1 ' (u Notice that

(Ak f ) (x)

415

j) du = 1:

j= 1

(Dk f ) (x)

=

=

1 X

f; 'kj 'kj (x)

j= 1 1 X

j= 1 1 X

1 X

kj

(f ) ' 2k x

j= 1

2k=2 f; 'kj Z

1

kj

(f ) ' 2k x

j

u ' (u j) du k 2 1 j= 1 ! n X j r wr f + k ' 2k x j 2k 2 n r=0 " n Z 1 1 X X u = wr f k ' (u j) du 2 1 j= 1 r=0 =

f

r j + k ' 2k x j k 2 2 n " n Z 1 1 X X u = wr f k 2 1 j= 1 r=0 f

' (u

j) du)] ' 2k x

f

j r + k k 2 2 n

j :

That is, by the compact support of ' we have

1 X

j= 1

"

(Ak f ) (x) n X

wr

Z

j+a

j a

r=0

(Dk f ) (x) =

u f k 2

f

' 2k x

r j + k 2k 2 n

' (u

j) du

#

j :

Next we see that u 2k

f

j r + k 2k 2 n

f

=

N X f (i) i=1

j 2k

+ i!

r 2k n

u

r i n

j 2ki

+ R5 ;

where R5 =

Z

u 2k j 2k

+

f (N ) (t) r 2k n

f (N )

j r + k k 2 2 n

u 2k

(N

t

N

1)!

1

dt:

j

416

21. Real and Fuzzy Estimates for Distances between Wavelet type Operators r n

(i) Case j +

u: We have easily that ! 1 f (N ) ; 21k u j N !2kN

jR5 j (ii) Case j + Z

jR5 j =

Z

r n

u

N

r n

j

:

u: We have that

j 2k

+

r 2k n

j r + k k 2 2 n

f (N )

u 2k j 2k

r n

+

r 2k n

f

j r + k k 2 2 n

(N )

u 2k

! 1 f (N ) ; 21k j + N !2kN

r n

u

f j+

(N )

r n

u N 2k

t

f (N ) (t)

(N

(t)

1)!

u N 2k

t (N

1

1)!

dt

1

dt

N

u

:

So we have proved that jR5 j

! 1 f (N ) ; 21k j + N !2kN

r n

u

j+

i.e.

r n

N

u

! 1 f (N ) ; a+1 N 2k (a + 1) ; kN N !2

N

jR5 j

(a + 1) a+1 ! 1 f (N ) ; k N !2kN 2

Furthermore we observe that Z j+a r u j + k f k f k 2 2 2 n j a Z N j+a X f (i) 2jk + 2krn r i u j ' (u ki i!2 n j a i=1

' (u j) du +

:

j) du = Z

j+a

j a

R5 ' (u

j) du:

Therefore Z

j+a

j a

f

u 2k

f

r j + k 2k 2 n

' (u

j) du

N N X f (i) 1 (a + 1) (a + 1) i (a + 1) + ! 1 f (N ) ; ki i! kN 2 N !2 2k i=1

:

The last proves the theorem. We continue with Theorem 21.10. Let f 2 C N (R) ; N 1; x 2 R and k 2 Z; also f (i) 1 < 1; i = 1; : : : ; N: Let ' be a bounded function of compact

21.1 Estimates for Distances of Real Wavelet type Operators

support

1 X

[ a; a] ; a > 0 such that

' (x

j= 1

'

0 and ' is Lebesgue measurable (then 'kj (x) := 2k=2 ' 2k x

f; 'kj =

Z

1

j) = 1 all x 2 R: Suppose

R1

j

417

1

' (x) dx = 1): De…ne

all k; j 2 Z;

f (t) 'kj (t) dt;

1

and (Ak f ) (x)

=

=

1 X

f; 'kj 'kj (x)

j= 1 1 X j= 1

Z

1

f

1

u ' (u 2k

j) du ' 2k x

j :

Also de…ne (Ck f ) (x) =

1 X

kj

(f ) ' 2k x

j ;

j= 1

where k kj (f ) = 2

Z

2

2

k

(j+1)

f (t) dt = 2k kj

Z

2

k

f

t+

0

j 2k

dt:

Then

N X i=1

j(Ak f ) (x)

(Ck f ) (x)j

(i)

kAk f

N

Ck f k1

f (a + 1) (a + 1) i 1 (a + 1) + kN ! 1 f (N ) ; i!2ki 2 N! 2k

So as k ! 1; we get kAk f Ck f k1 ! 0: 1 X R1 Proof. By ' 2k x j = 1 we have that 1 ' (u Notice that

(Ak f ) (x)

(21.15) :

j) du = 1:

j= 1

(Ck f ) (x) =

1 X

j= 1

f; 'kj 'kj (x)

1 X

j= 1

kj

(f ) ' 2k x

j

418

21. Real and Fuzzy Estimates for Distances between Wavelet type Operators 1 X

=

2k=2 f; 'kj

j= 1 1 X

=

j= 1 1 X

=

j= 1

"

Z

(f ) ' 2k x

1

u f k ' (u 2 1

Z

k

2

2

Z

k

0

' 2k x j " Z 1 X = 2k

2

' 2k x j " Z 1 X = 2k

2

Z

k

0

j= 1

j) du

2

j Z

k

u f k ' (u 2 1

1

f

2

k

dt ' 2k x

Z

k

2

f

0

u f k ' (u 2 1 u f k 2

!

j t+ k 2

j) du dt

1

1

k

2

0

1

Z

k

0

j= 1

' 2k x

kj

j) du

j t+ k 2

f

dt

#

j) du dt

#

j t+ k 2

f

' (u

j t+ k 2

j :

That is, by the compact support of ' we have

1 X

j= 1

"

(Ak f ) (x) k

2

Z

2

k

0

Z

j+a

(Ck f ) (x) =

u f k 2

j a

j t+ k 2

f

' 2k x

' (u

j) du dt

#

j :

Next we see that

f

u 2k

f

R6 =

Z

t+

j 2k

=

N X f (i) t + i! i=1

j 2k

u 2k

t

j 2k

i

N

1

+ R6 ;

where

(i) Case t +

u 2k

t+

j 2k

jR6 j

f

(N )

(s)

f

(N )

j 2k

u : 2k

u 2k

j t+ k 2

(N

s

ds:

1)!

We have easily that

!1 f

(N )

;

u 2k

t

j 2k

u 2k

t N!

j N 2k

:

j

dt

#

21.1 Estimates for Distances of Real Wavelet type Operators

(ii) Case t +

j 2k

Z

jR6 j =

Z

u : 2k t+

We have that

j 2k

f

j 2k

f

(N )

u 2k

! 1 f (N ) ; t +

N

j t+ k 2

(N )

u 2k

t+

419

f

s 2uk (N 1)!

(s)

N

j t+ k 2

f

(N )

t+

u 2k

j 2k

(N )

s 2uk (N 1)!

(s)

u N 2k

j 2k

1

ds

1

ds

:

N!

So we have proved that

! 1 f (N ) ; t +

jR6 j

! 1 f (N ) ;

(j

u)

(j u) 2k

t+

2k

(a + 1) 2k

N

N! N

(a + 1) : 2kN N !

I.e. we found that N

jR6 j

(a + 1) (a + 1) ! 1 f (N ) ; kN 2 N! 2k

Furthermore we observe that Z j+a u j f k f t+ k 2 2 j a Z N i j j+a (i) Xf t + 2k (u j) t ' (u i! 2k j a i=1

:

' (u

j) du =

j) du +

Z

j+a

j a

R6 ' (u

j) du:

Therefore Z

j+a

j a

f

u 2k

f

t+

j 2k

' (u

j) du

N N X f (i) 1 (a + 1) (a + 1) i (a + 1) + ! 1 f (N ) ; ki kN N ! i!2 2 2k i=1

:

The last proves the theorem. We give Theorem 21.11. Let f 2 C N (R) ; N 1; x 2 R and k 2 Z: Let ' be a bounded function of compact support [ a; a] ; a > 0 such that

420

21. Real and Fuzzy Estimates for Distances between Wavelet type Operators

1 X

' (x

j= 1

0: Assume further f (i)

j) = 1 all x 2 R: Suppose '

1; i = 1; : : : ; N: Put

1 X

(Bk f ) (x) =

j 2k

f

j= 1 1 X

(Ck f ) (x) =

kj

' 2k x

j ;

(f ) ' 2k x

j ;

1

<

j= 1

where

k kj (f ) = 2

Z

2

k

f

t+

0

and

1 X

(Dk f ) (x) =

kj

j 2k

dt;

(f ) ' 2k x

j ;

j= 1

where kj (f ) =

n X

j r + k 2k 2 n

wr f

r=0

n 2 N; wr

0;

n X

wr = 1: Then

r=0

(i)

j(Bk f ) (x)

(Dk f ) (x)j

kBk f

j(Bk f ) (x)

(Ck f ) (x)j

kBk f

N X i=1

(iii) j(Ck f ) (x)

Dk f k1

(21.16)

N X f (i) 1 ! 1 f (N ) ; 21k + ; 2ki i! 2kN N ! i=1

(ii)

(Dk f ) (x)j

N X

So as k ! 1; we get Dk f k1 ! 0; kBk f

Ck f k1

(i)

(21.17) (N )

1 2k

f !1 f ; 1 + kN ; (i + 1)! 2 (N + 1)!

2ki

kCk f i=1

kBk f

;

2ki

Dk f k1

(21.18)

f (i) 1 ! 1 f (N ) ; 21k + : 1 (i + 1)! 2kN 1 (N + 1)!

Ck f k1 ! 0; kCk f

Dk f k1 ! 0:

Proof. By Theorems 21.2, 21.4, 21.6 and especially use of (21.4), (21.7) and (21.12).

21.2 Estimates for Distances of Fuzzy Wavelet type Operators

421

21.2 Estimates for Distances of Fuzzy Wavelet type Operators We need some background which follows. (r) (r) Remark 21.12 ([15]). Here r 2 [0; 1] ; xi ; yi 2 R; i = 1; : : : ; m 2 N: Suppose that (r)

(r)

sup max xi ; yi

2 R; for i = 1; : : : ; m:

r2[0;1]

Then one sees easily that sup max r2[0;1]

m X

(r) xi ;

i=1

m X

(r) yi

i=1

!

m X

(r)

(r)

sup max xi ; yi

:

i=1 r2[0;1]

De…nition 21.13. Let f : R ! RF ; we de…ne the (…rst) fuzzy modulus of continuity of f by (F )

!1

(f; ) =

sup D (f (x) ; f (y)) ; > 0: x;y2R jx yj

U We de…ne CF (R) the space of uniformly continuous functions from R ! RF ; also C (R; RF ) the space of fuzzy continuous functions on R: (F )

U (R) : Then ! 1 Proposition 21.14 ([15]). Let f 2 CF > 0: (F )

Proposition 21.15 ([15]). It holds lim ! 1 f2

U CF

!0

(R) :

(f; ) < 1; any

(F )

(f; ) = ! 1

(f; 0) = 0; i¤

Proposition 21.16 ([15]). Let f : R ! RF be a fuzzy real number val(r) (F ) (r) are …nite; ued function. Assume that ! 1 (f; ) ; ! 1 f ; ; ! 1 f+ ; for > 0; all r 2 [0; 1] : Then n o (r) (r) (F ) : ! 1 (f; ) = sup max ! 1 f ; ; ! 1 f+ ; r2[0;1]

U Note. It is clear from Propositions 21.15, 21.16 that if f 2 CF (R) ; then f 2 CU (R) (uniformly continuous on R). We denote by C N (R; RF ) ; N 1; the space of all N times continuously fuzzy di¤erentiable functions from R into RF : (r)

We mention

422

21. Real and Fuzzy Estimates for Distances between Wavelet type Operators

Theorem 21.17 ([71]). Let f : [a; b] R ! RF be H fuzzy di¤erentiable. Let t 2 [a; b] ; 0 r 1: Clearly h i r (r) (r) [f (t)] = (f (t)) ; (f (t))+ R: (r)

Then (f (t))

are di¤erentiable and r

(r)

I.e. (f 0 )

= f

(r)

0

0

(r)

[f 0 (t)] =

(f (t))

(r)

; (f (t))+

0

:

; 8r 2 [0; 1]:

Remark 21.18 ([15]). Let f 2 C N (R; RF ) ; N rem 21.17 we obtain f

(i)

(t)

r

(r)

(r)

(i)

; (f (t))+

(f (t))

=

1: Then by Theo-

(i)

; for i =

0; 1; 2; : : : ; N; and in particular we have that f (i)

(r)

= f

(r)

(i)

;

for any r 2 [0; 1] : (r)

Note 21.19. (i) Let f : R ! RF fuzzy continuous, then f :R!R are continuous, 8r 2 [0; 1] : (ii) Let f 2 C N (R; RF ) ; N 1: Then by Theorem 21.17, we have (r) f 2 C N (R) ; for any r 2 [0; 1] : We need also NB (R) := ff : R ! RF j such that De…nition 21.20. Denote by CF all fuzzy derivatives f (i) : R ! RF ; i = 0; 1; : : : ; N exist and are fuzzy continuous and furthermore D f (i) ; e 0 < 1; for i = 1; : : : ; N g; N 1: Notice here that f (i) ; e 0

D

=

sup max

f (i)

(r)

;

r2[0;1]

=

sup max

f (r)

(i)

1

;

r2[0;1]

f (i)

1

f (r)

(r) + (i) +

1

; i = 1; : : : ; N: 1

Notice also that

D

f (i) ; e 0 < 1; implies

f (i)

(r) 1

< 1; i = 1; : : : ; N; 8r 2 [0; 1] :

21.2 Estimates for Distances of Fuzzy Wavelet type Operators

423

We mention Theorem 21.21 ([67]). Let f : [a; b] ! RF be fuzzy continuous. Then Rb (F R) a f (x) dx exists and belongs to RF ; furthermore it holds # r "Z # " Z b

b

f (x) dx

(F R)

=

(r)

(f )

a

a

(r)

(x) dx; (f )+ (x) dx ;

8r 2 [0; 1] : (r) Clearly f : [a; b] ! R are continuous functions. In this section we study the fuzzy corresponding analogs of real wavelet type operators Ak ; Bk ; Ck ; Dk ; k 2 Z; of …rst section. For simplicity we keep the same notation at the fuzzy level. So, depending on the context we understand accordingly whether our operator is real of fuzzy, that is whether is operating on real valued functions or on fuzzy valued functions. We present the next main fuzzy wavelet type result. NB (R) ; N 1; x 2 R; and k 2 Z: Let ' be Theorem 21.22. Let f 2 CF a bounded real valued function of compact support [ a; a] ; a > 0 such 1 P that ' (x j) = 1; all x 2 R: Suppose ' 0: Put j= 1

1 X

f

(F R)

Z

(Bk f ) (x) =

j 2k

j= 1

(Ck f ) (x) =

1 X

k

2

2

k

f

0

j= 1

and (Dk f ) (x) =

1 X

kj

(f )

' 2k x

j t+ k 2

j ;

dt

' 2k x

!

' 2k x

j ;

j ;

j= 1

where kj

n 2 N; wre Then (i)

0;

n P

r e=0

(f ) =

wre = 1:

D ((Bk f ) (x) ; (Dk f ) (x))

n X

r e=0

wre

f

j re + k k 2 2 n

;

D (Bk f; Dk f ) N D X i=1

f (i) ; e 0

2ki i!

(21.19) (F )

+

!1

f (N ) ; 21k ; 2kN N !

424

21. Real and Fuzzy Estimates for Distances between Wavelet type Operators

(ii) D ((Bk f ) (x) ; (Ck f ) (x))

D (Bk f; Ck f ) f (i) ; e 0

N D X i=1

(21.20) (F )

+

2ki (i + 1)!

!1 f (N ) ; 21k ; 2kN (N + 1)!

and (iii) D ((Ck f ) (x) ; (Dk f ) (x))

D (Ck f; Dk f ) N X i=1

D

f

2ki

1

(i)

;e 0

(21.21)

(i + 1)!

+

(F ) !1 f (N ) ; 21k : 2kN 1 (N + 1)!

Note. We see that D f (N ) (x) ; e 0 + D f (N ) (y) ; e 0

D f (N ) (x) ; f (N ) (y)

f (N ) ; e 0 < 1:

2D (F )

Thus ! 1 f; 21k < 1; 8k 2 Z: Consequently as k ! 1 we obtain

D (Bk f; Dk f ) ; D (Bk f; Ck f ) ; D (Ck f; Dk f ) ! 0 with rates. Proof. (i) We observe the following D ((Bk f ) (x) ; (Dk f ) (x))

=

sup max r2[0;1]

=

n

(r)

(r)

((Bk f ) (x))

((Dk f ) (x))

(r)

(r)

((Bk f ) (x))+ ((Dk f ) (x))+ n (r) sup max Bk f (x)

o

Dk f

r2[0;1]

(r)

Bk f+

sup max r2[0;1] (r)

Bk f+

n

(x) Bk f

(r)

Dk f+ (r)

Dk f

(r)

Dk f+

1

o

(x) (r)

o 1

(r)

;

;

(x) ;

21.2 Estimates for Distances of Fuzzy Wavelet type Operators

(21:16)

sup max r2[0;1]

!1 +

f

(r)

i=1

8 > N <X

sup max

> : i=1

r2[0;1]

(r) f (i) + 1 2ki i!

N X i=1

=

1

2kN N !

f (i)

1

1

!1

f (N )

!1 +

(r)

; 21k

;

> ;

(r)

f (i)

; f (N )

(r)

;

r2[0;1]

;

9 > =

1 2k

+

1

f (i) ; e 0 +

> > ;

2kN N !

(r)

f (i)

9 > > =

; 21k

2kN N !

(r) f (N ) + 2kN N !

+

(N )

(r)

f+

+

(r)

2ki i!

1

; !1

2ki i!

(i)

2ki i!

(i)

sup max ! 1

N X 1 D ki i! 2 i=1

(r)

; 21k

f+

N X 1 sup max ki i! 2 r2[0;1] i=1

+

f

> > : i=1

2kN N !

N X

=

(N )

(r)

8 > > N <X

425

1 2k

1

f (N )

; !1

1 1 (F ) ! f (N ) ; k 2kN N ! 1 2

(r) +

;

1 2k

;

proving the theorem’s (21.19). (ii) We observe the following D ((Bk f ) (x) ; (Ck f ) (x))

=

sup max r2[0;1]

=

n

(r)

(r)

((Bk f ) (x))

((Ck f ) (x))

(r)

(r)

((Bk f ) (x))+ ((Ck f ) (x))+ n (r) sup max Bk f (x)

o

Ck f

r2[0;1]

(r)

Bk f+

sup max r2[0;1]

n

(x) Bk f

(r)

Ck f+ (r)

Ck f

(x) (r)

o 1

(r)

;

;

(x) ;

426

21. Real and Fuzzy Estimates for Distances between Wavelet type Operators

o (r) Ck f+ 1 8 (i) (r) > > f N <X (21:17) 1 sup max + ki (i + 1)! > 2 r2[0;1] > : i=1 (r)

Bk f+

!1

f

(r)

(N )

; 21k

;

2kN (N + 1)! (r)

N X i=1

=

sup max r2[0;1]

N X i=1

N X i=1

+ =

f (i)

1

f (i)

(r) + 1

+

(r) +

i=1

2ki

1 D (i + 1)!

; 21k

> ;

(r)

(r)

f (i)

;

+

1

f (i) ; e 0 +

;

9 = ; 21k >

f (i)

1 sup max ! 1 2kN (N + 1)! r2[0;1]

N X

> > ;

2kN (N + 1)!

2kN (N + 1)!

1 sup max (i + 1)! r2[0;1]

(r)

f (N )

+

f (N )

!1

9 > > =

; 21k

2kN (N + 1)!

!1

1

ki > : i=1 2 (i + 1)!

f+

+

(r)

(N )

(r)

!1

2ki (i + 1)!

8 > N <X

2ki (i + 1)!

2ki

(i)

f+

(r)

f (N )

2kN

;

1 2k

1

f (N )

; !1

1 1 (F ) !1 f (N ) ; k (N + 1)! 2

(r) +

1 2k

;

;

proving the theorem’s (21.20). (iii) We observe the next D ((Ck f ) (x) ; (Dk f ) (x))

=

sup max r2[0;1]

=

n

(r)

(r)

((Ck f ) (x))

((Dk f ) (x))

(r)

(r)

((Ck f ) (x))+ ((Dk f ) (x))+ n (r) sup max Ck f (x)

o

Dk f

r2[0;1]

(r)

Ck f+

(x)

(r)

Dk f+

(x)

o

(r)

;

(x) ;

21.2 Estimates for Distances of Fuzzy Wavelet type Operators

sup max r2[0;1]

n

Ck f

(r)

Dk f

(r) 1

427

;

o (r) Dk f+ 1 8 (i) (r) > > f N <X (21:18) 1 sup max + ki 1 (i + 1)! > 2 r2[0;1] > i=1 : (r)

Ck f+

!1

f

2kN

(N )

(r) 1

i=1

=

sup max r2[0;1]

2ki

1

N X

2ki

1

i=1

i=1

+

2kN

!1 =

f (i)

N X

N X i=1

1

;

(N + 1)! (i)

(r)

N X

; 21k

f+ 2ki

8 > N <X

1

1

f (i)

ki > : i=1 2 (r) + 1

(i + 1)!

1

1

+

1

1 sup max (i + 1)! r2[0;1]

2kN

(r) +

2ki 1

(r) +

;

1

9 > > =

; 21k

> > ;

(N + 1)!

(r)

; 21k

;

(N + 1)!

9 = ; 21k >

(N + 1)! > ; (r)

f (i)

(r)

f (i)

;

+

1

1 sup max ! 1 (N + 1)! r2[0;1]

f (N )

1

f (N )

!1

f (N )

2kN

f+

2kN

(r)

(i + 1)!

!1

+

+

(i + 1)!

(N )

(r)

!1

f (N )

(r)

;

1 2k

1

;

1 2k

1 D (i + 1)!

f (i) ; e 0 +

2kN

1

1 1 (F ) !1 f (N ) ; k (N + 1)! 2

proving the theorem’s (21.21). Above we need that ([16])

(r)

= Bk f

(r)

;

(r)

= Ck f

(r)

; and

(r)

= Dk f

(r)

;

(Bk f ) (Ck f ) (Dk f )

;

428

21. Real and Fuzzy Estimates for Distances between Wavelet type Operators

8r 2 [0; 1]: Denote by Cb (R; RF ) the space of bounded fuzzy continuous functions on R with respect to metric D: We …nish with the following fuzzy wavelet type main result NB Theorem 21.23. Let f 2 CF (R) \ Cb (R; RF ) ; N 1; x 2 R; and k 2 Z: Let the scaling function ' (x) a real valued function with supp ' (x) [ a; a] ; 0 < a < +1; ' is continuous on [ a; a] ; ' (x) 0; such 1 R1 P ' (x j) = 1 on R (then 1 ' (x) dx = 1): that j= 1

De…ne

'kj (t) := 2k=2 ' 2k t Z f; 'kj := (F R)

j

all k; j 2 Z; t 2 R

j+a 2k

f (t)

j a 2k

'kj (t) dt;

and the fuzzy wavelet type operator (Ak f ) (x) =

1 X

f; 'kj

'kj (x) ; x 2 R:

j= 1

The fuzzy wavelet type operators Bk ; Ck ; Dk are as in Theorem 21.22. Then (i) D ((Ak f ) (x) ; (Bk f ) (x)) N D X i=1

f (i) ; e 0

2ki i!

D (Ak f; Bk f ) aN

ai +

(F )

2kN N !

!1

f (N ) ;

a ; 2k

(21.22)

(ii) D ((Ak f ) (x) ; (Ck f ) (x)) N D X i=1

f (i) ; e 0

i!2ki

D (Ak f; Ck f )

N

i

(a + 1) +

(a + 1) (a + 1) (F ) !1 f (N ) ; kN 2 N! 2k

;

(21.23)

and (iii)

D ((Ak f ) (x) ; (Dk f ) (x)) N D X i=1

f (i) ; e 0

2ki i!

D (Ak f; Dk f )

(21.24)

N

i

(a + 1) +

(a + 1) (a + 1) (F ) ! f (N ) ; N !2kN 1 2k

:

21.2 Estimates for Distances of Fuzzy Wavelet type Operators

429

Notice that D (Ak f; Bk f ) ; D (Ak f; Ck f ) ; D (Ak f; Dk f ) ! 0 as k ! 1 with rates. Proof. Similar to the proof of Theorem 21.22. Also notice here (see also (r) (r) [16]) that (Ak f ) = Ak f ; 8r 2 [0; 1] : It is based on (21.13), (21.14) and (21.15). U Note 21.24. In [14] we proved, see also Chapter 12, for f 2 CF (R) as k ! 1; we get uniformly that Ak ; Bk ; Ck ; Dk ! I unit operator with rates in the D metric. In the case of Ak we need also f be fuzzy bounded.

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List of Symbols

RF ; 5 [ ]r ; 5 D; 5 ;5 ;5 oP e; 6 ;6 D ;5 f 0; 7 Rb (F R) a f (x)dx; 8 C[0; 1]; 16 C n ([a; b]; RF ); 51 CF [0; 1]; 75 (F ) Bn (f ); 75 (F ) ! 1 (f ; ); 76 C2F (R); 77 Jn (f ); 77 ! B1 g; n1 [ 1 ; 1 ] ; 91 4 4

C2 (R); 110 U CF (X); 116 BU CF (X) ; 116 B CF (X) ; 116

440

List of Symbols

id; 144 id2 ; 144 B CF (R+ ) ; 146 Cb (R; RF ) ; 192 C (R; R R Fs ) ; 192 (F H) t f (x)dx; 267 CF (R) ; 265 NB CF (R) ; 275 LF (X; B; P ); 310 Uq CF R ([a; b]) ; 311 U CF R ([a; b]) ; 315 E; 353 A = (An )n 1 ; 374 An = (ankj )k;j2N ; 374 Ax, 374 Fj (f ; x); 392 Ak ; Bk ; Ck ; Dk ; 399 '; 400 C N (R); 400 f0g ; 6 (RF ; D); 5

Index

2 -periodic function, 76 A-statistical, 374 A-summability, 374 A-summable, 374 attained inequality, 70 Banach space, 16 bell-shaped function, 271 Bernstein-type fuzzy polynomial, 75 B-measurable mapping, 310 Bojanic-DeVore operator, 93 Borel measure, 135 compact set, 135 compact support, 399 convex, 179 equicontinuous functions, 162 expectation, 353 fuzzy Szasz-Mirakjan operator, 148 fuzzy Baskakov-type operator, 148 fuzzy bounded, 129 fuzzy continuous, 129 fuzzy convolution type operator, 237 fuzzy Fejer operators, 392 fuzzy global smoothness, 115

442

Index

fuzzy Jackson operator, 119 Fuzzy Korovkin Theorem, 125 fuzzy linear operator, 132 fuzzy logic, 3 Fuzzy mean value theorem, 30 fuzzy modulus of continuity, 100 fuzzy neural network operator, 263 fuzzy order, 5 fuzzy polynomials, 75 fuzzy positive linear operator, 99 fuzzy random Bernstein Polynomial, 343 fuzzy random function, 347 fuzzy random Korovkin Theorem, 309 fuzzy random modulus of continuity, 351 fuzzy random variable, 351 fuzzy real number, 5 Fuzzy Riesz Representation Theorem, 135 fuzzy set, 1 fuzzy squashing operator, 282 Fuzzy Taylor formula, 350 fuzzy trigonometric approximation, 75 Fuzzy trigonometric Korovkin Theorem, 112 fuzzy trigonometric polynomial, 76 fuzzy trigonometric Shisha-Mond inequality, 111 fuzzy uniformly continuous, 116 fuzzy wavelet type operator, 119 fuzzy weak convergence, 139 fuzzy-random neural network operator, 348 Fuzzy-Riemann integral, 241 geometrc moment theory, 140 global smoothness preservation, 115 H-derivative, 7 H-di¤erence, 7 Henstock integral, 8 integrable function, 134 joint measurable function, 353 Korovkin Theorem, 101 Langrange interpolation polynomial, 10 Lebesgue measurable, 238 Lipschitz mapping, 96 Lp convergence, 282 matrix summability method, 376 matrix, 96 measurable function, 134 metric space, 5

Index

multivariate fuzzy chain rule, 33 multivariate Fuzzy Taylor formula, 48 Ostrowski type inequality, 69 positive fuzzy random linear operator, 316 positive linear operator, 316 probability measure, 2 q-mean uniformly continuous fuzzy random function, 311 q-mean, 311 real normed vector space, 11 scaling function, 12 Shisha and Mond inequality, 101 -algebra of Borel sets, 353 Silverman-Toeplitz conditions, 374 Star-shaped set, 137 Tonelli-Fubini theorem, 320 topological space, 351 uniform estimate, 399 univariate fuzzy chain rule, 31 usual modulus of continuity, 110 wavelet type operator, 12 Zadeh, 1

443