Weighted Approximation with Varying Weights

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, ZOrich F. Takens, Groningen

1569

Vilmos Totik

WeightedApproximation with VaryingWeight

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest

Author Vilmos Totik Bolyai Institute University of Szeged Aradi v. tere 1 6720 Szeged, Hungary and Department of Mathematics University of South Florida Tampa, FL 33620, USA

Mathematics Subject Classification (1991): 41A10, 41A17, 41A25, 26Cxx, 31A10, 31A99, 41A21, 41A44, 42C05, 45E05

ISBN 3-540-57705-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57705-X Springer-Verlag New York Berlin Heidelberg Library of Congress Cataloging-in-Publication Data. Totik, V. Weighted approximation with varying weights/Vilmos Totik. p. cm. - (Lecture notes in mathematics; 1569) Includes bibliographical references and index. ISBN 3-540-57705-X (Berlin: softcover: acid-free). - ISBN 0-387-57705-X (New York: acid-free) 1. Approximation theory. 2. Polynomials. I. Title. II. Series: Lecture notes in mathematics (Springer-Verlag); 1569. QA3.L28 no. 1569 [QA221] 510 sdc20 [511'.42] 93-49416 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, re-use of illustrations, recitation, broadcasting, reproduction on microfilms 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-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1994 Printed in Germany SPIN: 10078788

46/3140-543210 - Printed on acid-free paper

Contents 1

I

Introduction

Freud weights

7

2

Short proof for the approximation

problem for Freud weights

3

Strong asymptotics 3.1 T h e upper estimate . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The lower estimate . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 T h e / 2 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 10 14 17 19

II

Approximation with general weights

21

4

A general approximation theorem 4.1 Statement of the main results . . . . . . . . . . . . . . . . . . . . 4.2 Examples and historical notes . . . . . . . . . . . . . . . . . . . .

21 21 23

5

Preliminaries to the proofs

25

6

P r o o f o f T h e o r e m s 4.1, 4.2 a n d 4.3

32

7

C o n s t r u c t i o n o f E x a m p l e s 4.5 a n d 4.6 7.1 Example 4.5 . . . . . . . . . . . . . . . 7.2 Example 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

III 8

9

Varying weights Uniform approximation weights

by weighted polynomials with varying

in g e o m e t r i c m e a n s

Applications

by

49 57 59 60 64 7O

7'9

11 F a s t d e c r e a s i n g p o l y n o m i a l s 12 A p p r o x i m a t i o n

38 38 44

49

Modification of the method The lower estimate . . . . . . . . . . . . . . . . . . . . . . . . . . The upper estimate . . . . . . . . . . . . . . . . . . . . . . . . . . The asymptotic estimate . . . . . . . . . . . . . . . . . . . . . . .

10 A p p r o x i m a t i o n

IV

: ..............

W(anx)Pn(x)

79 85

vi 13 E x t r e m a l problems with varying weights

91

14 A s y m p t o t i c p r o p e r t i e s of o r t h o g o n a l polynomials with varying weights 95 15 Freud weights revisited

103

16 M u l t i p o i n t Pad~ a p p r o x i m a t i o n

106

17 Concluding remarks

109

References

111

Index

115

1

Introduction

In this work we are going to discuss polynomial approximation with weighted polynomials of the form w" Pn, where w is some fixed weight and the degree of Pn is at most n. We emphasize that the exponent of the weight w n changes with n, so this is a different (and in some sense more difficult) type of approximation than what is usually called weighted approximation. In fact, in the present case the polynomial P , must balance exponential oscillations in w". To have a basis for discussion let us consider first an important special case. Let w(x) = exp(-clxl~), c > 0 be a so called Freud weight. H. N. Mhaskar and E. B. Saff [34] considered weighted polynomials of the form w'~P,~, where the degree of P,, is at most n. They found that the norm of these weighted polynomials live on a compact set ,5,~, i.e. for every such weighted polynomial we have IIw"P.IIR

= IIw"P.lls~,

furthermore, w"P./llwnP.IIs~ tends to zero outside Sw. They also explicitly determined Sw: (1.1)

1/a~-l/ct ^,l/a ~-l/a]

S~o = [-7~

~

,,~

~

J

where 1

ro :=

f0

v o-1

~

1

~

dv = r ( 7 ) r ( 7 ) / ( 2 r ( 7 +

1

)),

(see Section 3 below). One of the most challenging problems of the eighties in the theory of orthogonal polynomials was Freud's conjecture (see Section 3) about the asymptotic behavior of the recurrence coefficients for orthogonal polynomials with respect to the weights w. The solution came in three papers [16], [29] and [27] by D. S. Lubinsky, A. Knopfmacher, P. Nevai, S. N. Mhaskar and E. B. Serf. The most difficult part of the proof was the following approximation theorem ([29]). T h e o r e m 1.1 If w~(t) = e x p ( - 7 , ~ l t l ' ~ ) , c~ > 1 is a Freud weight normalized so that Swo = [-1, 1], then for every continuous f which vanishes outside ( - 1 , 1) there are polynomials Pn of degree at most n, n = 1 , 2 , . . . such that w~P,~ uniformly tends to f on the whole real line.

Let us mention that it follows from what we have said about w~,P,~ tending to zero outside [-1, 1], if f can be uniformly approximated by wn~Pn, then it must vanish outside [-1, 1]. In the next section we shall present a rather elementary and direct proof for Theorem 1.1. Then, in Section 3, we shall derive a short proof for the strong asymptotic result of Lubinsky and Salt for an extremal problem associated with Freud weights. With this we will provide a self contained and short proof for the most important result of the monograph [28]. In Section 4.1 we shall considerably generalize Theorem 1.1 and solve the analogous approximation problem for a large family of weights. In earlier works the approximation problem was mostly considered for concrete weights such as

2

Introduction

Section 1

Freud, Jacobi or Laguerre weights. The generalization given in Theorem 4.2 is the first general result in the subject and is far stronger than the presently existing results (e.g. it allows St~ to lie on different intervals). It also solves several open conjectures. However, the new and relatively simple method is perhaps the most important contribution of the present paper (Lubinsky and Saff themselves generalized Theorem 1.1 in a different direction, see e.g. [28] and Section 12). We shall first restrict our attention to the important special case given in Theorem 1.1 in order to get a simple proof for the above mentioned asymptotics (and hence for the so called Freud conjecture) and in order not to complicate our method with the technical details that are needed in the proof of Theorem 4.2 (see Section 5). In the third part of this work we shall present a modification of the method. This will allow us to consider varying weights in the stronger sense, that we shall allow even wn to vary with n. Recently a lot of attention has been paid to such varying weights which are connected to some interesting applications to be discussed in Chapter IV. In essence our approximation problem can be reformulated as follows: how well can we discretize logarithmic potentials, i.e. replace them by a potential of a discrete measure which are the sums of n (n = 1, 2 , . . . ) equal point masses (see the discussion below for the relevant concepts). The usual procedure is the following: divide the support into n + 1 equal parts with respect to the measure and place masses 1/n at these division points. This approach has proven to be sufficient and useful in many problem. However, the process introduces singularities on the support which has to be avoided in finer problems. Our method in its simplest form is a modification of the previous idea. We also divide the support into n equal parts with respect to the measure, but we use the weight points of these parts instead of their endpoints for placing the mass points to, then we vertically shift this discrete measure by an amount L,,/n where Ln -* cr is appropriately chosen. This modification will result in a dramatic increase in the speed of approximation. In the rest of this introduction we shall briefly outline the results from the theory of weighted potentials that we will need in the paper. We shall use logarithmic potentials of Borel measures. If/~ is a finite Borel measure with compact support, then its logarithmic potential is defined as its convolution with the logarithmic kernel: Ug(z) =

1 log ~ - ~

d/t(t).

Let ~E be a closed subset of the real line. For simplicity we shall assume that " Z is regular with respect to the Dirichlet problem in C \ R " , by which we mean that every point x0 of ~ satisfies Wiener's condition: if := {x c r [2

<

-

< 2-"},

Introduction

Section 1

then

oo

(1.2)

Z

3

n

log 1/cap(E,) = oo

n-----1

(see the following discussion for the definition of the logarithmic capacity). In particular, this is true if E consists of finitely many (finite or infinite) intervals. This regularity condition is not too essential in our considerations, but it simplifies some of our proofs. A weight function w on ~ is said to be admissible if it satisfies the following three conditions

(1.3)

(i)

w is continuous;

(ii) (iii)

E0 := {x 6 E iw(x) > 0} has positive capacity; if ~ is unbounded, then ]xiw(x ) --. 0 as Ix I ~ oo, z 6 ~.

We are interested in approximation of continuous functions by weighted polynomials of the form wnp,. To understand the behavior of such polynomials we have to recall a few facts from [34] and [35] about the solution of an extremal problem in the presence of a weight (often called external field). We define Q = Q~ by

w(x) =: exp(-Q(x)).

(1.4)

Then Q : ~ --* ( - o o , cx)] is continuous everywhere where w is positive, that is where Q is finite. Let AJ(~) be the set of all positive unit Borel measures/~ with supp(/~) C_ ~, and define the weighted energy integral (1.5)

I~(/~)

:=

.f/log[lz-tlw(z)w(t)]-:dp(z)d~(t)

The classical case corresponds to choosing H to be compact and w = 1 on H: If /~ is a Borel measures with compact support on R, then its logarithmic energy is defined as

I(.) := / U"(z)d.(z) = / / lOglz l--tld.(t)d~,(z)9 If K is a compact set, then its logarithmic capacity cap(K) is defined by the formula 1 (1.6) log cap(K) := inf {I(/~)I/~ 6 .~4(K)}. Now the capacity of an arbitrary Borel set B is defined as the supremum of the capacities of compact subsets of B, and a property is said to hold quasieverywhere on a set A if it holds at every point of A with the exception of points of a set of capacity zero.

4

Introduction

Section 1

The equilibrium measure (see [51] or [17]) wK of K is the unique probability measure wK minimizing the energy integrals in (1.6). Its potential has the following properties: (1.7)

1 V ~K(z) S log cap(K)

(1.8)

V~K(z) = log

1 cap(K)

for

z E C,

for quasi-every

z e g.

If K is regular (which means that its complement C \ K is regular with respect to the Dirichlet problem), then we have equality for every z in (1.8). Returning to the general case of weighted energies, the next theorem was essentially proved in [34] and [35]. T h e o r e m A Let w be an admissible weight on the set ~, and let

(1.9)

V~ := inf{I~(~)I~ ~ M ( ~ ) ) .

Then the following properties are true. (a) Vw is finite. (b) There exists a unique t~,o E Ad(E) such that ~,.(/,,.)= v,..

Moreover, I~w has finite logarithmic energy. (c) Sw := supp(/~) is compact, is contained in So (c.f. property (ii) above), and has positive capacity. (d) The inequality Ui'~(z) >_ -q(z)

+ v~ - f Od,,~ =: -~(z) + F~

holds on ~. (e) The inequality U "~ (z) <_ - Q ( z ) + F~ holds for all z E Sw. (f) In particular, for every z E &o, u .~(~) = -Q(~) + F~.

The proof is an adaptation of the classical Frostman method. In fact, in [34] and [35] property (d) was proved to hold for quasi-every z E ~. But the regularity of Z implies that then the set of points where

/~og

Tr~_,i..o(t) + ~(.) > v. -

/

~ . . --:~.

Introduction

Section 1

5

holds is dense at every point of ~ in the fine topology (see [12, Chapter 10] or [17, Chapter III]), hence the inequality in question is true at every z 9 ~ by the continuity of Q (where it is finite) and the continuity of logarithmic potentials in the fine topology. The measure juw is called the equilibrium or eztremal measure associated with w. Above we have used the abbreviation :=

- f Odu

for this important quantity. We cite another theorem of H. N. Mhaskar and E. B. Saff [34, Theorem 2.1], which says that the supremum norm of weighted polynomials wnpn lives on Sw. Let us agree that whenever we write Pn, then it is understood that the degree of Pn is at most n. T h e o r e m B Let w be an admissible weight on ~ C_ It. If Pn is a polynomial of degree at most n and

(1.10)

Iw(z)npn(z)l <_ M

for

z 9 &,

then for all z 9 C (1.11)

[Pn(z)I < M e x p (n(-UU~(z) + F~o)).

Furthermore, (1.10) implies (1.12)

[w(z)npn(z)[ < M

for

z 9 ~.

This theorem asserts that every weighted polynomial must assume its maximum modulus on St0. Soon we shall see that S~ is the smallest set with this property. Theorem B is an immediate consequence of the principle of domination (see the proof of Lemma 5.1 in Section 5).

Part I

Freud weights In the first part of the paper we shall consider exponential type (also called Freud) weights. We shall illustrate our method on them. The other purpose of this part is to give a self-contained and relatively short proof for the strong asymptotic results of Lubinsky and Saff [28].

2

S h o r t p r o o f for t h e a p p r o x i m a t i o n for Freud weights

problem

In this section we give a short and simple proof for Theorem 1.1. Let Q(x) = 7,lxl ", so that w,(x) = w(x) = exp(-Q(x)). First we simplify the problem. I. Obviously, it is enough to consider f ' s that are positive in ( - 1 , 1) and less than, say, 1. Furthermore, we know that it is sufficient to approximate on, say, [-2, 2], because w'~P, tends to zero outside [-3/2, 3/2] (see Theorems A and B from the introduction and the formula (3.7) in Section 3). II. It is enough to approximate by the absolute values of weighted polynomials. In fact, if wnlP,~l uniformly tends to v/-]', then w2'~lPnl2 uniformly tends to f , and here IPnl 2 is already a real polynomial. This shows our claim when the degree n is even. For odd degree one can get the statement by approximating f / w with even degree polynomials and then by multiplying through by w. III. It is enough to show the following: for every e > 0 and L > 0 there is a continuous function gL and for every large n polynomials Qn of degree at most n such that with J~ := [-1 + e, 1 - e] (2.1)

w"(x)lQ.(x)l = exp(gL(x)+ RL(x)),

x e J.,

where the remainder term Rz(x) satisfies IRL(~)I _< C,/L uniformly for x E J, with some C~ ~ 1 independent of L, and for every x E [-3, 3] (2.2)

~"(~)IQ,(~)I_

Dn3,

where D = DL,e is a constant independent of n. In fact, suppose this is true, and apply it to w ~ instead of w with some ,k > 1. The corresponding extremal support is [-0~,0~] with ~ = ~ - l / a tending to 1 together with ~, hence, by choosing )~ > 1 close to 1 and then applying the statement above to a smaller e if necessary, we can see that there are polynomials Q[,~/~] of degree at most In~h] such that with some gL and RL as above

w"(x)lQ[,/~](~)l = exp(gL(x) - (n - )~[n/),])Q(x) + RL(~)),

9 ~ L,

Freudweights

8

and

w"(x)lq[,,p,](x)l

Section 2

x E [-2, 2].

< D , n 3,

Since 0 <_ n - A[n/A] _< A, and the family of function {gL -- sQ [ 0 <_ s <_ i} (considered on [-1 + e, 1 - ~]) is compact, for every large n there are polynomials S,~-[n/A] of degree at most n - [n/A] such that - f(x)exp(--gL(x) 4- (n -- ~[n/~])Q(x))l < exp(--gL(z) + (n - A[n/A])Q(x))/L,

9 e J2,,

< f ( , ) e x p ( - - g L ( x ) + (n - A [ n / A ] ) Q ( x ) ) , and (2.3)

_< n -4,

x 6 J2e \ Je,

x 6 [-2, 2] \ Je.

Now we set Pn = Q[n/~]S,-[n/~], which has degree at most n. If ~ > 0 is given, then choose first e > 0 so that the maximum of f outside J2~ is smaller than ~, then chooose A > 1 as above, and finally choose L large enough to have C e l l < ~. Then our estimates show that for sufficiently large n the difference [wnlPnl- fl is at most 3~ on [-2,2], and this is what we need to prove. IV. Thus, we only have to verify (2.1) and (2.2). Let us consider the so called Ullman distribution #to given by its density function

Ot~$I Ua-I

(2.4)

v(t) =

I

d..

It is well-known (see the computation in Section 3, especially (3.6) and (3.7)) that w(z) and e x p ( U ~ ( z ) ) differ on [-1, 1] only in a multiplicative constant, and elsewehere the weight w(x) is smaller than exp(U I'~ (x)) times this constant. Hence it is enough to show (2.1) and (2.2) with w = we replaced by e x p ( U " ' ) . In doing so we are going to use the standard discretization technique for logarithmic potentials (c.f. [42] and [28]) with some modifications, but exactly these modifications permit good approximation. Let v be the density of the Ullman distribution pto (see (2.4)), and let us divide [-1, 1] by the points - 1 = to < tt < ... < tn = 1 into n intervals Ij, j = 0,1,...,n1 with pto(Ij) = 1In. Since v is continuous and positive in ( - 1 , 1), there are two constants c, C (depending on e) such that if I 1 N J~2 # 0, then c/n < [b[ < C/n. Let 1

=

d,,(,)

be the weight point of the restriction of pt0 to Ij, and set

Qn(t) = H ( t - i L / n - (j). J We claim that this choice will satisfy (2.1) and (2.2) (with w replaced by exp(U"~)).

Short proof for the approximation problem for Freud weights

Section ~

9

First of all let us consider the partial derivative of U g" (z) at z = x + iy with respect to y:

(2.5)

ou.-(z)

~_ Y 1 (x - t) 2 + y2

__

Oy

v(t)dt

.-.-}

,~v(~)

as y --, 0 - 0 uniformly for x E J~ by the properties of the Poisson kernel. This, and the mean value theorem implies that (2.6)

Um'(x) - Um~(x - i L / n ) - rLv(x-----~)+ ~ (

uniformly in x E Jr The same argument shows that L

(2.7)

IU,~(~) - u z - ( x - iL/n)[ = O ( n )

uniformly for x E R.

Actually, (2.5) and (2.6) uniformly hold on R because v is continuous (even at + l ) and vanishes outside [-1, 1]. We shall use this fact in Section 3, but for the present purposes we keep the above formulation because in Section 4.1 we shall consider weights the density of which is not necessarily continuous around the endpoints, and it will be easier to point out the necessary changes if we work with (2.6) and (2.7). Let pn(t) = p ~ ( t - i L / n ) , i.e. we are defining pn on the interval [-1, 1] + i L / n , which is obtained by shifting [-1, 1] upwards on the plane by the amount L / n . Then the preceding two estimates tell us how far apart the two potentials U z- and U z" can be on [-1, 1] and on R. Next we estimate for x E J~, x E Ijo

(zs)

Ilog IQ.(x)l + nU~"(x)l (log Ix - iL / n - t I - log Ix - iL / n - ~j l) dpw( t) j =0

lj

Here the integrand is log 1 + x - ' ~ ' ~ ' - ~ j

=~log

1+ x-ET"n-~j

'

Since the absolute value of

r x- iL/n-~j' is at most 1/2 for large L (check this separately for [~j - t[ < C / n and for the opposite case which can only occur if I t f3 ge~ = 0 and hence Ix - ~j[ > e/2 while IIj[ < e2), it easily follows that then the last expression can be written in the form / 1 =(5a-tl~ _iL/n_5 i +0

l~-tl~ 2~,

10

Freud weights

Section 3

and since the integral of the first term on/3' against dpw(t) is zero because of the choice of ~j, we have to deal only with the second term. For it we have the upper estimate

0 ((L/n) 2 -~(C/n)2 ~ ( j - jo ) / n ) 2 ) if Ij n J o ~- 0 and

otherwise (recall that x E ge), hence we can continue (2.8) as

<_Ci k=0

C2

L2+c2k2+ClmaxlIj[ 3

~

C~

IIilc-~<-z-

linJ~2=~

if n is sufficiently large. Now log IQ.(x)[ +

nUU"(x) = (log IQ.(x)l + nUU"(x)) + (nUU'~(x) - nUU"(x)),

and here, by the preceding estimate, the first term is at most C J L in absolute value, while by (2.6) the second term is 7rv(x)i + o(i)uniformly in x E J~ as n --* c~. This gives (2.1) (recall that we are working with exp(U u~) instead of w). The proof of (2.2) is standard: using the monotonicity of the logarithmic function we have for example for x E Ij0, j0 < j < n - 1 the inequality log Ix - iLIn - ~1 < n ft i+' log Ix - iLIn - t[dp=(t), and adding these and the analogous inequalities for j < j0 together one can easily deduce the estimate log IQ,(x)l + nU'"(x)

(2.9) jo+l

+

Z n /=j0-1

/

1 log i x _ i L / n _ t l

<_31og6 +

d~(t) < 31og6n/L

for every x e [--3, 3]. This and (2.7) prove (2.2).

3

|

Strong asymptotics

The theorems of this section are not new, they can be found in the monograph [28] by D. S. Lubinsky and E. B. Soft. We closely follow many steps from [28], but we substitute the approximation part of the proof with the simple method of Section 2 which allows us to make shortcuts and simplifications, thereby

Strong asymptotics

Section 3

11

significantly reducing the length of the original proof (which is scattered through about 100 pages). First we shall consider the L 2 extremal problem and then the L p one at the end of the section. In order to have a complete proof we add a few standard calculations that may help the reader. Let w(~) = w~(x) = e -~~ a > 1 be a Freud weight on R normalized so that Sw = [-1, 1] (this normalization is made for convenience, any other positive constant can replace 3'~ on the right; for the explicit form of the constant ")'a see (3.5) below), and consider the orthonormal polynomials with respect to w~:

+ ... defined by the orthogonality relation

f Vn( ; x)Vm( ;

Z)W2(X) dx = 6.,m.

When a -- 2 these are the classical Hermite polynomials, for other a's G. Freud started to investigate their properties. Let l'In denote the set of polynomials of degree n and leading coefficients 1, i.e.

+...}.

nn =

The leading coefficient 7-(w) of the orthonormal polynomials Pn are closely related to a weighted extremal (minimum) problem, namely (3.1)

1

(W)

~--- inf

P. 1"I.

/ PrOw 2 2,

and it is one of the most important quantities related to pn. In fact, their behavior determines the behavior of the Pn'S which can also be seen by the fact, that in the recurrence formula

xpn(w; Z) = An+lp,+I(w; z) + Anpn-I(w; z) the recurrence coefficients are given by

A,, = %-a(w)/Tn(w). In [6] G. Freud made two conjectures: one on the asymptotics of the largest zeros of the Pn's and another one on the recurrence coefficients. E. A Rahmanov [42] solved the first conjecture, but the second one, which claimed that (3.2)

l i m n-l/'~An = ~,

was open for some while, until it was settled in a series of papers [16], [29] and [27] by D. S. Lubinsky, A. Knopfmacher, P. Nevai, S. N. Mhaskar and E. B. Salt

Freud weights

12

Section 3

(Freud himself verified the conjecture for a = 2, 4, 6, and for even integers it was settled by A. Magnus [32]). This was a typical conjecture that was obviously bound to be true (already Freud new that the terms on the left are in between two positive constants, and if the limit exists then it has to be 1/2; and there was no reason why the limit should not exist), but its proof required genuinely new tools. Shortly after settling Freud's conjecture, D. S. Lubinsky and E. B. Saff [28] proved the following strong (as opposed to (3.2), which is called ratio) asymptotics for the 7n(w)'s themselves, which is probably one of the all-time best results in the theory of orthogonal polynomials: lim 7n(w)rrl/22-ne-n/C'n(n+U2)/a= 1.

(3.3)

n "-'~ O 0

Below we shall present a relatively short proof for (3.3) that utilizes the approximation technique in Section 2. The original proof is scattered through the monograph [28] and is quite long. Let us start with the Ullman distribution given by its density

v(t)=~O~fit 1I ~ Udcut - I

(3.4)

on [-1, 1]. For its potential we have (writing instead of the measure its density as a parameter in U) by switching the order of integration -V"(x)

/o'

=

~u ~ - ~ 1 or

f,o,,._,, ,,~

dt du.

The expression after u a-1 is nothing else than the negative of the equilibrium potential of the interval [-u, u], hence it equals log u - log 2 if Ixl < u and log ]x + v f ~ - u 2] - log2 if Izl > u. Thus, for - 1 < x < 1 we can continue the above equality as =

=

-log2+

/:

au~-llogu du+ I

-log2-i+lzl 01

~

+

/o

1

/0'"

~u~-lloglx+

v~-u~l du

)

o~va-'log(l+V~-v2) dv .

Integration by parts yields that the last integral is

=

~ot

v '~-1

1+ ~

1r

v~

v~d~

=

fol

v ~-1

l v q - ~v-~d~ -

1

~'

where w e used the identity v 2 = (I - vrf - v2)(I + ~/1 - v=). Since

(3.5)

~ := fo~ ~ v~-~

we finally get for z E [-1, 1]

d~ = r ~( 7 ) r ( ~1) / ( 2 r ( 7o~ + 1 )),

Strongasymptotics

Section 3

13

1 U ' ( x ) = - 7 , 1 x l ~ + log2 + - . a

(3.6)

Let now {xI > 1, x E It. By symmetry we can assume x > 1. Exactly as above

U~(x) = log2 -

~u a-1 log(x + V / ~ - u2)du,

and by differentiation we get

(u~(x))' = -

~01~2~-~-~0~I~-1 ~11 O~Ua-1 ~ 2d~ = -~7~lxl~-~ + /~ ~ / T - ~ du.

This tells us first of all that outside [-1, 1] we have on R (3.7)

U"(x) > -~,l~l"

1

+ log2 + - ,

and so by (3.6) and Lemma 5.1 to be proven in Section 5 we can conclude that

d , . ( O = ,(Odt and (3.8)

F,o = log 2 + l / a .

It also follows that for Ixl e (1,2)

(u"(x))' + o,~,x "-1 ~ Ilxl- 1l 1/2, where ~ indicates that the ratio of the two sides lies in between two absolute constants (in the range of the arguments indicated), and so

(3.9)

(u",.(x) - F.) + ~,~lxl" "-' Ilxl- 1l ~/2

for Ix{ E (I,2), and (3.10)

(u.-(x) - F.) + ~.{x{ ~ ~ {xl"

when {x} > 2. This and Theorem B of the introduction easily imply the following inequality of D. S. Lubinsky [25]: for Pn = 1 + n -7/12

F

(3.11) degSUp. _<. .

~1/o

dx/J[j ,o

dx

)

= 1 + o(1) as n --~ oo (for completeness we shall give a short proof for (3.11) at the end of this section). The awkward looking p~/" in the limits of integration on the left is just to match the proof below, we shall only need that it is larger than 1 + cn -7/12 with some c > 0. After these preliminaries let us return to (3.3). Let ~o(x) = 1 - x 2. We need the following formula of S. N. Bernstein (see [2, pp. 250-254] or [28, p. 111]):

14

Freud weights

Section 3

Let R2q be a polynomial of degree 2q, positive on ( - 1 , 1) with possibly simple zeros at +1. Then for n > q

(inf

: n'l/12-n exp

L1

~112

)

In what follows let us abbreviate the geometric mean appearing on the right as i.e.

G[V]=exp(~f' l~ -77 dx)

(3.13)

9

3.1

The upper

estimate

Let Pn = 1 - n -2/3, and let us carry out the substitution z = p~/<~nl/ay in the integrals in (3.1), and then restrict the integrals to [-1, 1]. We get (3.14)

- - - - L 1 >n(2"+l)/~p (2n+1)/~ 7n(W) 2 --

inf [le-'r<'P"2ril~l<'P~(x) v.El-i,,d-1

dx.

We are going to show with the method of Section 2 that there are polynomials Hn of degree at most n such that if (3.15)

h.(~) = e - ~ . ~ " l ' l ~

- ~2)-1/4,

then

(3.16) and

hn(x)> 1

for xE[-1,1]

lim G[h,,] =

(3.17)

n'--~" O0

1.

Then we will have by (3.14) and (3.16) 1 %(w)'----'~ _> n(2n+z)/~p(2n+l)/a

inf

L1

P.~II.

~~ I7 2 z lHnl 2 " '

and so by Bernstein's formula

(3.18) But here

1 %(w) 2 >_n(in+l)l~,p~2,,+z)i<,~r2_2 n (G[~ll411H,,I]) 2.

~Zl4(z)llH,,(~)l

and (3.19)

7~

= e-'y<,p,,"l~l<'/h,.,(x)

1 [1 ixia 1 j _ l ~/~--_-z2 d~ = --,a

which, together with (3.17) imply that p~n/,~ times the geometric mean on the right hand side of (3.18) has the form (1 -F o(1))e -2'~1`~ exp ((2n/a)((1 - Pn) 4- log pn)) -- (1 -t- o(1))e -2nla,

15

Strong asymptotics

Section 3

and this proves that lim sup 7n (w) 7r1122-ne-nl~n(n+ll2)la ~ 1. n--~ OO

Thus, everything boils down to the existence of polynomials Hn with properties (3.16) and (3.17). We follow the proof in Section 2, but now we need somewhat finer analysis around the endpoints. By symmetry, we can restrict our attention to the left endpoint -1. For the Ullman distribution (3.4) it immediately follows that

v(t),~(1-t2) I/2

as t~+l,

and this property alone implies for the Ik's and ~k's of Section 2 that for 0 <

k < n/2 (_~)2/a (3.20)

1

1 + ~k "~

,

Ilkl N (k + 1)1~an2/a'

and analogous estimates hold for k > n/2. Hence for j r j0 dist(~j,/jo) ,.~ 1

1

n~/3 E

k~[/o,j]

(k+ 1)1/3 "~

Ij2/3 - j2o/3I n:/s

Since for x E Ijo, the absolute value of the the j-th, j ~ j0, term in the sum in (2.8) is at most dist(~j,/jo)

'

(see the argument after (2.8)), it follows from the preceding estimate that (3.21)

E nlf I (loglx-iL/n-tl-loglx-iL/n-~jl)dl~(t) J#jo

<_ C

J (

E

1

)2/(ij2/z__j02/a,~ 2 = O(1).

(j + 1),/an~/a

_

n2/a

]

j#jo

Thus, it has left to estimate the j0-th term in (28). Its absolute value is obviously bounded by

llog(L/n)2+[I'~ (L/n) 2

1 - ( 2 l~

(1+ 1---0(( n ~ 2 / 3 ~ 1 1 L2 \j--o--o-~] ] ] < zl~ l - x2

if L is sufficiently large (recall that x E Ijo, hence
16

F r e u d weights

Section 3

Taking into account (2.6), which, as we have seen in Section 2, uniformly holds on It, we get that the polynomials Q , from Section 2 satisfy (3.22)

CX/1

- x 2 _<

e-~nl~l"lQn(x)le"F~ <

- -

C

/[--

Z2

uniformly in x 6 [-1, 1] and n with some constant C (recall (3.8) and (3.6)), and (3.23) lim e - ~ n l x l " l Q n ( x ) l e nF~ = e t'v(x) n-=* OO

uniformly on compact subsets of ( - 1 , 1). Next we improve the estimate (3.22) for x lying close to +1. The proof of (3.21) and (3.22) easily yields that there is a c > 0 such that (3.24) e-'Y-'q~l=lQn(x)le nF" >_ e

for all n and 0 < 1 -I~1 < c n - 2 / 3 .

In fact, recall that, say, II01 ~ (1 + ~0) ~ n -2/s, and so (c.f. (3.21) and the argument after (2.8) in Section 2) n ~

(log Ix - i L / n

log Ix - i L / n

- tI-

-

~0l) d/~w(t)

o

1

/

Ir163

2

= n / z O(1)d/z~(t) _< C' o

provided 0 < 1 - Ix[ <_ e n - 2 / ~ and c is sufficiently small. For later purposes let us record here that the proof gives also the following: (3.25)

1

~ < exp(nUt"(x))lQn(x)l < C

uniformly on R \ [-1, 1]. Now let us remove the two zeros from Q m which have the smallest and the largest real parts, respectively. On the interval [-1 + c n - 2 / 3 , 1 - c n - 2 / 3 ] this introduces a factor ,,, c l / ( 1 - x2) -1, hence for the so modified polynomial Q* we get from (3.22), (3.23) and (3.24) that (3.26)

1 e_.y.nlxl ~ ~ < IQ~(x)lenF" -<

C (i -x2)3/2

holds with some constant C uniformly in x 6 [-1, 1] and n, and eL=v(z) (3.27)

li--mooe - ~ " n l x l " l Q ~ ( x ) l e n r ~

uniformly on compact subsets of ( - 1 , 1).

-

1 - x2

Strong asymptotics

Section 3

17

After these preparatory steps let us return to (3.16) and (3.17). We apply the preceding estimates for Q[*np,], which has degree at most [n - n 1/3] to conclude

1

(3.28)

C

and

_ eL~v(x)

(3.29)

nlim e -~"[p"n]I~I"IQ~...I(x)Ie["""1F~

1 - x2

uniformly on compact subsets of ( - 1 , 1). Then it is easy to find polynomials Rn-[np.] of degree at most n-[npn] >_ n 1/3 such that with Hn = Q[np.]R.-[np.] both properties (3.16) and (3.17) are satisfied (use exactly as in Section 2 the fact that the family of functions { - s Q [ 0 < s < 1} considered on [-1, 1] is compact). The point is that in the definition of hn in (3.15) the factor (1 - x2) -1/4 appears, which only improves the lower estimate in (3.28), and so (3.16) is easy to achieve. To achieve (3.17) at the same time is a simple approximation procedure if we use the upper estimate in (3.28) and the asymptotic relation (3.29).

|

3.2

The

lower

estimate

The proof of the lower estimate is very similar to the above argument. In fact, let now pn = 1 + n -7/12, and let us carry out the substitution x = pln/anl/ay in the integrals in (3.1). The result is 1

_(2n+l)/._(2.+x)/~ inf [ ~ e-'r~P"2'q~:l"P~(x) dx. e. P.eII. J-~

7 . ( w ) 2 - ,,

Since now we want to prove a lower estimate, we cannot restrict the integral to [-1, 1], rather we need the so called infinite-finite range inequality (3.11) which tells us that the part of the integrals of weighted polynomals away form the extremal support is negligible. In the above integral not the weight w 2n but (wP") 2n appears, and the corresponding extremal measure has support ,Sw,, = [-pnX/'~,pnl/'~]. Thus, if we apply (3.11) then we can conclude that, by restricting the integrals to [-1, 1] we introduce only a constant that tends to zero, i.e. (3.30)

1 7,(w)2

-

(1 + o(1))n(~n+l)/ap (2n+1)/c'

inf

PnEl'In

f l e-~'P"2"l~l" P~(x) dx. 1

Actually, the infinite-finite range inequality (3.11) has to be applied to the 1/~] intervals [--pnlZa,pn l[a] and [-1, 1] rather than to [-1, 1] and rt - P -1/~ ,vn j,

18

Freud weights

Section

3

which only means a linear transformation not introducing any new constant 1/c~ in the ratios in question. In general, this linear transformation x --~ p , y introduces in our formulae only a constant that tends to 1 as n --+ oo, hence in what follows we shall use it without explicit mentioning. Let us now consider the weight wP~ for which ~ p ~ = [_p.~21O,,p.~2/c~]. By (3.22), (3.23) and (3.25) there are polynomials Q[n/p,] of degree at most [n/pn] such that with w, = wP~ (3.31)

C c(p'~ 4/~' - x2) 1/2 <_ e-'Y~P""I~I~IQtn/..](x)Ie'~/""F'. < (p~4/~ _ x2)1/2

uniformly in

x E [-pn21~,pn 21~] and 1

(3.32)

< exp

n,

(n_._UU,~.) (z) IQt./p.](x)l _< c \Pn

uniformly in n and z g [-p~2/~(1 -

cn-2/3), p~2/~(1 - cn-2/3)],

limo,:, e -'r~p~['~/p"]I~'W iQt./,~.](x)letn/~.]F~,.

(3.33)

and

= eL~'"<=')

uniformly on compact subsets of ( - 1 , 1). On applying (3.9) we can conclude from (3.32) that log \(e -'r`'p"nlr[`' II'n~ L. ". l P n J,tz~le(nlp.)F,.,, (1 - z2) 1/4) dz k 11

fp

~/1

;"/'~ <1~1_<1

-

x 2

= o (n(.~ - 1)3/2(.o _ 1)1/2 + (.. _ 1)1/2log

1

)

p, - 1

= O(n -1/6)

= o(1)

by the choice of the pn'S, which estimate is used in (3.36) below. From (3.31) and (3.33) it easily follows that we can multiply this Q[n/p,,] by a suitable Rn-l-[n/p,] of degree at most n - 1 - [n/pn] > In 5/12 to get a H , - I with the following properties: Hn-1 does not vanish on ( - 1 , 1), if

(3.34)

h.(x) = e-~P""l~l~lHn_i(x)(1

then (3.35)

hn(x)

and (3.36)

<_ 1 lim n---~ O0

for

- x2)1/2l(1 - x2) -1/4,

x E [-1, 1],

G[hn] = 1.

From here the proof is the same as in the case of the upper estimate: set R2n(X) - IHn-l(X)[2(1 - x 2)

Section 3

19

Strong asymptotics

into Bernstein's formula, use (3.35) and (3.36) instead of (3.16) and (3.17), and reverse the corresponding inequalities. In the end we obtain

linminf Tn(w)Trl/22-ne-n/an(n+1/2)/a > 1, and the proof is complete.

|

We have promised a proof for (3.11). First we need a crude Nikolskii-type inequality. By approximating first 7an[x[ ~ on [-2, 2] by some polynomials Tn2 of degree n 2 with error 1In (this is possible by Jackson's theorem), then taking the n2-th partial sum of the Taylor expansion of e -x, and then substituting here Tn(x) for x, we can get a polynomial Rn4 of degree at most n 4 such that uniformly in n and z E [-2, 2] ~

Using this and the classical Nikolskii inequality [IQ,~[I[-1,1] < Cnl/21IQnHL2[-2,2] with Qn5 = Rn'Pn we get for any Pn ][w"Pni[[-1,1]

<: C[[Rn.PnH[_I,I] <: Cn3HRn.Pn[[L2[_2,2] <: Cn3Hwnpn[IL2[_2,2] <: C.3HwnpnHL2

(actually the best exponent on the right is 1/2a, see [40], but we will not need this). This, Theorem B from the introduction and (3.9)-(3.10)easily imply with some c > 0 l=l>,~/= e-~=2nl'l=Pn2(x ) dx

< Cn3llw"P.ll2L=

e

dx+ ~o~ e-cnx dx) <

11lw"P.II22,

and this is (3.11).

3.3

The

|

Lp case

In [28] Lubinsky and Saff considered the weighted LP-extremal problem

En,p(w) := inf [[wPn[[Lp. P.EIL

20

Freud weights

Section 3

Note that when p = 2, then this is the same as the one in (3.1). They proved the following generalization of (3.3): if w(z) = exp(-Talxla), then (3.37)

2irnooEmv(w)~12n-l+l/Pen/an-(n+l/p)/a =

1,

where

~p = (r(l/2)r((v+ 1)/2)IF(p/2 + 1))lip In L2 Bernstein's formula takes the following form: Let 1 < co, and R2q a polynomial of degree 2q which is positive on ( - 1 , 1) with possibly simple zeros at 4-1. Then for n > q inf

ft

~/2-t/2

~l/p

P.ert. s_, "R-~q~ IPnlV)

(3.38) i.e.

= %2_,+l_l/Pexp(lf_llog(~t/2-1/2p/R~2 ~t/2 I/ ' E n,pw-" ~.,..,t/2-1/2p/~,l/2.~ /""2q J = ~ 2 - " + 1 - 1 / p G [,..,t/2-1/2p/~,t/2] [~ z""2q j "

Using this formula instead of (3.12) we can copy the proof of (3.3) and we can get (3.37) with minor modifications. For example, in the proof of the upper estimate (which correponds to the lower estimate on 7,~(w) discussed in Section 3.2) we have to use an obvious modification in the infinite-finite range inequality (3.11), and change (3.15) to h,(~) = e -'r~p"'*l~l~ IH,.,(x)l(1 - z2) (-1+1/p)/2, resp.

h.(,,,) = e - ~ , - n l

I IH._~(,,,)(1 - ~1~/~1(1 -

~,~1(-~+~/")/~,

(c.f. (3.15) and (3.34)) for which (3.35) and (3.36) can be achieved exactly as before. In a similar manner, only minor changes have to be done in the lower estimate (which correponds to the upper estimate on 7.(w) discussed in Section 3.1).

Part II

Approximation with general weights In the second part of the paper we will consider the approximation problem for general weights.

4 4.1

A general approximation t h e o r e m S t a t e m e n t of t h e m a i n r e s u l t s

Let ~ be a regular closed subset of the real line and w an admissible weight on E. We consider the problem of approximating a continuous function f by weighted polynomials w~P~. First we show that every such function must vanish outside the support 3~ of the extremal measure ft~ (see the introduction).

Theorem 4.1 Let us suppose that there is a sequence {Pn} of polynomials of corresponding degree n = 1, 2 . . . such that wn(x)Pn(x) uniformly converges to a function f on S,~ U {xo}, xo q~ S~. Then f(xo) = O. Let us note that the mere boundedness of w~P,, on 8~ does not necessarily imply that the sequence {wn(z0)Pn(x0)} converges to zero (for z0 ~ S,,). A counterexample is furnished by the weight w which is 1 on [-1, 1] and equals (z + z2v/~3-'L-'T 1)- -1 on (1,2] (consider the energy problem from the introduction on E = [-1, 2]), and the Chebyshev polynomials

1 In this case Pw is the arcsine measure ((Trlv/'~'2-~-z2)-ldz) on [-1, 1], and it is obvious that wnTn is bounded on 8~ = [-1, 1] but w'~(zo)Tn(zo) > 1/2 (and w " ( z o ) T , ( x o ) --+ 1/2) for all z0 E (1,2]. Next we turn to conditions guaranteeing approximation. Let O C E be an open subset of the real line. The space of continuous real functions that vanish outside O will be denoted by Co(O).

Definition. We say that t0 has the approximation property on the open set 0 if for every f E Co(O) there is a sequence of polynomials {Pn}n~176 such that tonPn converges uniformly to f on ~. Thus, what we have said above implies that we can hope for the approximation property on an open set 0 only if 0 _C ,q~, that is 0 should he part of the interior Int(Sw) of ~%. Our main result is that on the other hand, if #to has continuous and positive density function on the interior of 8w, then to does have the approximation property on Int(8~).

22

Approximation with general weights

Section

To formulate the main result of the paper we introduce the following definition: D e f i n i t i o n . Let S t~ denote the set of those points x0 where the equilibrium measure pw has continuous and positive density, that is dttw(t ) = vw(t) dt

in a neighborhood of x0, and the density function v~ is continuous and positive in a neighborhood of x0. This S ~~ is called the restricted support of #~. Thus, if p has positive and continuous density on Int(Sw), then S w = Int(Sw). On the other hand, if at xo we have v~o(xo) = 0, then this x0 does not belong to the restricted support. T h e o r e m 4.2 Let w be an admissible weight on ~ C It. Then w has the approximation property on the restricted support S w. In particular, if pw has continuous and positive density on Int(S~0), then every continuous function that vanishes outside Int(Sw) can be uniformly approximated on ~ by weighted polynomials of the form w n P , , where the degree of P , is at most n. As a corollary of Theorem 4.2 we get T h e o r e m 4.3 Suppose that E C I t consists of finitely many disjoint intervals lj, and w is an admissible weight of class C x+~ for some e > 0 such that Q = log 1/w is convex on every Ij. Then w has the approximation property on the interior of the support Sw. From the proofs of Theorems 4.2 and 4.3 the following result immediately follows. T h e o r e m 4.4 Suppose that w is an admissible weight of class C 1+~ for some e > O. Then w has the approximation property on the union of the interiors of the supports Sw~, )~ > 1. Now we show that Theorem 4.2 is sharp in a certain sense. To illustrate Theorem 4.2 let us consider the case when Sw = [-1, 1], and Pw has continuous density vw in ( - 1 , 1). We have seen that if this density is positive in ( - 1 , 1), then on Int(Sw) = ( - 1 , 1) the weight w has the approximation property, and in general this is the largest set where approximation is possible. Now what happens if vw vanishes at a single point, say at x = 0? We will construct an example in Section 7.1 where this single zero prohibits approximation in a very strong sense, namely approximation is possible only for functions that vanish at the origin. E x a m p l e 4.5 There e~ists a weight w such that the support of the corresponding extremal measure is [-1, 1], this measure has continuous density in ( - 1 , 1) which is positive everywhere except at O, and still no function that is nonzero at 0 can be approximated by weighted polynomials of the form wnpn.

Section $

A general approximation theorem

23

Hence, in this case the largest set for the approximation problem of the present section is the restricted support (-1, 0) U (0, 1). This shows that, in general, on no larger set than the restricted support can w have the approximation property; in other words, Theorem 4.2 cannot be improved. Theorem 4.2 does not tell us if approximation is possible on Int(Sw) if the density vw vanishes there. Example 4.5 shows that such internal zeros may prevent approximation, but this does not rule out the possibility of the approximation property on the whole Int(Sw) for a concrete function. Our next example together with Example 4.5 shows that, indeed, the situation is very delicate, approximation in the presence of internal zeros depends on the weight in a subtle way.

Example 4.6 There exists a weight w on [-1, 1] such that the support of the corresponding extremal measure Pw is [-1, 1], I~w has continuous density in (-1, 1) which vanishes at the origin, and still every continuous f that is zero at +1 can be uniformly approximated by weighted polynomials of the form wnPn.

4.2

Examples

and historical

notes

The type of approximation we are discussing has evolved from G. G. Lorentz' incomplete polynomials. Lorentz [23] studied polynomials on [0, 1] that vanish at zero with high order. That is, he considered polynomials of the form

Pn(x) = ~

(4.1)

akx k,

k=sn

and he verified that if shin -=*/9 and the Pn's are bounded on [0, 1], then Pn(x) tends to zero uniformly on compact subsets of [0, 92), and it was shown in [46] that [0,/92) is the largest set with this property. Although here there is no fixed weight, the resemblance to weighted polynomials wnP, with w(x) = x ~176 is obvious, and in fact, it is easy to transform results concerning incomplete polynomials into analogous ones concerning such weighted polynomials, and vice versa. In our terminology this result means that the support of the extremal measure for the weight w(x) = x ~ ~ = [-1, 1] is [/9~, 1]. The corresponding approximation problem, namely that every f E C[0, 1] that vanishes on [0, 192) is the uniform limit of polynomials of the form (4.1), was independently proved by v. Golitschek [8] and Saff and Varga [46]. In [44] E. B. Saff generalized the problem to exponential weights of the form exp(-cIxla), a > 1. With H. N. Mhaskar they proved in [36] that in this case the extremal support is

(4.2)

$~

=

[-7~/%-1/~, 7~/%-1/~]

(c.f. (1.1)), and they also determined the extremal measure (given by the Ullman distribution (2.4)) in this case. In [44] Sail" conjectured that every continuous

24

Approximation with general weights

Section J

function that vanishes outside (4.2) can be uniformly approximated by weighted polynomials wnpn. This was shown to be true in the special case a = 2 in [37] by Mhaskar and Saff, and, as we have mentioned in Section 1, by Lubinsky and Saff [29] for a l i a > 1. The missing range 0 < a < 1 was settled by D. S. Lubinsky and the author [30] proving that approximation is still possible if a = 1, and for a < 1 a necessary and sufficient condition that f be the uniform limit of weighted polynomials w~Pn is that f vanishes outside [-1, 1] and at the origin. It was also proven there (for the case a > 1) that even if we consider f only on the interval [-1, 1], it must vanish at the endpoints in order to be the uniform limit of weighted polynomials, i.e. approximation is not possible up to the endpoint for nonvanishing functions. More precisely, the following exact range for the approximation was established: Suppose that for n > 1 we are given closed intervals Jn symmetric about 0, and polynomials P~ of degree g n such that lim ]]w~Pn - 1]Is. = O. n---~ Oo

Then there exists a sequence {Pn }n~176with lim Pn = oo,

(4.3)

71--+OO

such that for infinitely many n

Jn C [-1 q- pn n-2/3, 1 - pnn-2/a]. Conversely, if {Pn}n%l is a sequence satisfying (4.3), then for every continuous f E C [ - 1 , 1] there exist polynomials Pn of degree at most n such that limo o [Iwp,Pn -

n--~

f[lI-l+p.n-2/s,l-p.n-~/q

= 0,

and sup I I W ~ P n ] I R < oo. n

In [37] the conjecture was made that even for general continuous weights w approximation by weighted polynomials wnPn is possible for an f if and only if f vanishes outside S~0. The necessity of the condition follows from Theorem 4.1. Its sufficiency is not true, a counterexample is furnished by Example 4.5 or by w(z) = exp(-[z[a), ~ < 1. In [3] the weaker conjecture was stated that at least for the case when Q -- log 1/w is convex, a necessary and sufficient condition for approximation is the same as before, that is that the function vanishes outside S~,. This conjecture of Borwein and Saff follows from Theorem 4.3 under the minimal smoothness assumption that Q is a C 1+*, e > 0 function on the support Sw (note that if Q is convex on an interval I, then it is automatically Lip 1 inside I). The sufficiency of the conjecture remains open for general convex Q's. We have already mentioned, that if w(z) = z ~176 then S~ = [02, 1] ([23]). The generalization to Jacobi weights was done in [45], where it was shown that if w(z) = (1 - z)a(1 + z) z, a , f l >_ 0, then the support of the extremal measure is -

-

,/S,

-

+

Section 5

Preliminaries to the proofs

25

where 01 := a / ( 1 + a +/3), 02 :=/3/(1 + a +/3) and A := { 1 - (01 + 02)2}{1- (01 - 02)2}. In this case the approximation problem was settled by X. He and X. Li [13]. As a "midway" ease between Jaeobi weights and Freud-type exponential weights let us mention the Laguerre weights w(x) = xae -x, a _> 0, E = [0, o~), for which (see [38]) Sw = [1 + a - ~/(1 + a~) - a 2, 1 + a + X/(1 + a2) _ a2]. In all these cases Q = log 1/w is convex, hence Theorem 4.3 can be applied and we can deduce the approximation property of the corresponding w on the interior of the support Sw. This is even true for the v. Golitschek-Saff-Varga theorem (note that in that theorem the function need not vanish at the right endpoint ofSw = [02, 1]), for it is enough to approximate f E C[0, 1] that vanish on some [0, 81] with 01 > 82, and for such functions the claim follows from the fact that the function F ( x ) = f(x01/02)) can be uniformly approximated on [0,02/01] by weighted polynomials of the form Xne/(1-O)Pn(x) (extend F to [0, 1] continuously so that it vanishes at 1, and apply Theorem 4.3 to F).

5

P r e l i m i n a r i e s to t h e proofs

To prove Theorem 4.2 we shall need a few preliminary results. First of all, let us note that properties (c) and (f) in Theorem A (see the introduction) imply the continuity of the equilibrium potential U u~ on its support S~, and hence everywhere by Maria's theorem [51, Theorem III. 2]. Thus, U g~ is continuous everywhere. We shall need to recognized the equilibrium measure in certain cases, and in such situations the following lemma comes in handy. L e m m a 5.1 Let w be an admissible weight on ~. If cr E A4(~) has compact support and finite logarithmic energy, and U~(z) + Q(z) coincides with a constant F everywhere on the support of a and is at least as large as F everywhere on Z, then a = Pw and F = Fw. The same conclusion holds if the assumptions are true with the exception of finitely many points (or even if they are true only quasi-everywhere).

Proof. For the proof we recall the principle of domination for logarithmic potentials (see [17, Theorem 1.27]): Let p and v be two positive finite Borel measures with compact support on C, and suppose that the total mass of v does not exceed that of ft. Assume further that P has finite logarithmic energy. If, for some constant c, the inequality (5.1)

U~(z) + c >_ UU(z)

2~

Approximation with general weights

Section 5

holds #-almost everywhere, then it holds for all z G C. T h e o r e m A (see the introduction) and the assumptions of the l e m m a imply t h a t UU~(z) < U~ - F + F,o for every z E supp(#w), hence the principle of domination gives that the same inequality is true everywhere. Letting z ---* c~ we can conclude that F < Fw. This argument can be repeated with #~ and a interchanged, hence we get t h a t F = F~, and then that the two potentials U u~ and U ~ concide everywhere. But then #w = a (see e.g. [17, Theorem 1.12']. The last statement in the l e m m a follows from the same considerations if we note that sets consisting of finitely m a n y points must have zero #~o- and a - m e a s u r e s because these measures have finite logarithmic energy.

|

Now we need the so called Fekete or Leja points (see [35]) associated with w. Let w be an admissible weight on the closed set E C_ R. For an integer n > 2 we set

zx,..-,z,~ EIB

l<'<j~_n

The s u p r e m u m defining 6~ is obviously attained for some set

5 , = { z l , . . . , z,} y These -~'n are called n-th Fekele sets associated with w, or shortly w-Fekete sets. For fixed n, the sets 9e, need not be unique; however in our consideration below we can use any choice of them. We shall need that the asymptotic distribution of Fekete points is the same as the equilibrium distribution #w: if u~,(A) :-- -1 n

Z

1-

I~'n n A] ,

xES,,nA

n

where A is any Borel subset of C (i.e. vT, puts mass 1/n to every point zj G .~',), then (5.2) lim v~, = #to in the weak* topology of measures (see [49, L e m m a 2.2]). W i t h this property at hand we can easily verify that 8~o is the smallest compact set S with the property that every weighted polynomial attains its norm on S (c.f. Theorem B in the introduction).

Lemma

5.2 Let w be an admissible weight on ~, and let S C ~ be a closed set. If, for every n - - 1 , 2 , . . . and every polynomial Pn with degPn _< n,

IIw"P.II then ,~to C_ 51.

= IIw"P.IIs,

Section 5

27

Preliminaries to the proofs

P r o o f . Consider the Fekete sets ~'n associated with w. We claim that for each n we can choose 5n C_ S. This will prove the lemma because then the normalized counting measures v~-, associated with ]-'n have support in S and converge in the weak* topology to the measure/~t0 (see (5.2)), hence Sw = supp(/~w) C_ S. Let 5n = { t l , . . . ,tn} be any w-Fekete set, and suppose that t l , . . . ,ts-1 E S but t s , . . . , tn ~ S. Consider the polynomial

Pn-l(z) := H ( z - t k ) . kgs By the choice of w-Fekete points

~a~lPn_l(zllw(z) "-1-- IPn_l(t.)lw(t.) n-1. By our assumption the weighted polynomial w'~-lPn_l attains its maximum modulus somewhere on S, hence there is a t* E S with

mea~lPn-l(z)lw(z)n-1 --IPn-l(q)lw(t,) *

* n--1 D

Thus, together with { t l , . . . , t , - 1 , t s , t s + l , . . . , t n } , the set { t l , . . . , t s - l , t * , t s + l , . . . ,tn} will also be a w-Fekete set, and the latter set has already s points in S. We can continue this process and eventually arrive at a w-Fekete set contained in S.

|

Now we are ready to prove a characterization of the points in the support that will be useful in our further considerations. L e m m a 5.3 Let w be an admissible weight on ~. Then z E ~ belongs to the

support ~qw of the extremal measure Pw if and only if for every neighborhood B o f z there exists a weighted polynomial wnpn, degPn < n, such that wnlPnl takes its maximum on ~ in B A ~, and nowhere else. P r o o f . Let z E Sto and let B be any neighborhood of z. By applying Lemma 5.2 to the set S = E \ B, we get a weighted polynomial Paw n with

Ilwnp.II~\B < Ilwnpnll~, which shows that wnlPnl takes its maximum on Z in B A ~, and nowhere else. Conversely, ifw n IP.I takes its maximumon ~ only in TAB, then by Theorem B (see the introduction) we must have B A $~ # 0, and of course if this is true for every neighborhood B of z, then z E $,0. I

As an immediate consequence we obtain L e m m a 5.4 IrA > 1, then

i.e. the sets 3w~ decrease as A increases.

28

Approximation with general weights

Section 5

P r o o f . Let z0 E 8~x, and let B be any neighborhood of z0. By L e m m a 5.3 there are a natural number n, a polynomial Pn of degree at most n, a point zl E B fq E and an ~/ > 0 such that [w"X(zl)Pn(zl)[ = 1 + r/, but outside B we have [wnXpn[ _< 1. Then w must be positive at zl, say w(zl) > m > O. Furthermore, let M be an upper bound for w. We may assume m < 1 < M. For a positive integer ! consider the polynomial P~. We clearly have

(W(X))tZn I+I IPn(x)[

> (1 + q)Zm,

while for z E Z \ B

(w(zl))

IPn(z )IZ <

M.

For large ! this means that the weighted polynomial w[lnX]-blp t, with deg Ptn < nl < [inl] + 1 takes its maximum modulus only in B. Since this is true for any neighborhood of z0, we can infer from Lemma 5.3 that z0 E Sto, as we have claimed.

I Now we need the concept of balayage measure. Consider in C an open set G with compact boundary OG, and let ju be a measure with supp(/~) C G. The problem of balayage (or "sweeping out") consists of finding a new measure supported on OG such that [1~1 = 11/~[1and (5.3)

UU(z) = U'~(z)

for quasi-every

z r G.

For bounded G such a measure always exists ([17, Chapter IV, ~2/2]), but for unbounded ones (with cap(0G) > 0) we have to replace (5.3) by (5.4)

Ut'(z) = U-if(z) + c for quasi-every

z r G.

with some constant c. Besides (5.3)-(5.4) we also know that (5.5)

U~(z) < U~(z)

respectively

(5.6)

V (z) < V.(z) + e

holds for all z E C. Furthermore, if G is regular with respect to the Dirichlet problem, then we have equality in (5.3) and (5.4). This is the case for example when G is the complement of finitely many closed intervals on the real line. Very often we have to take the balayage of/J out of G even if its support is not contained in G. In that case we take the balayage of the restriction of/J to G onto OG, and leave the part of ju lying outside G unchanged. Now the first part of the following lemma is an immediate consequence of these properties, Theorem A and Lemma 5.1. The second part follows from Lemma 5.3.

29

Preliminariee to the proofs

Section 5

L e m m a 5.5 Let w be admissible, and K C &o a regular compact subset of Sw. Then t~w[K = ttw,

where- indicates taking balayage onto K out of C \ K. In particular,

S~IK= K.

The lemma is true without the regularity assumption but we will not need this stronger statement. Recall that K is called regular if C \ K is regular with respect to the Dirichlet problem. Let us also mention that regularity is characterized by the Wiener condition (1.2). The next lemma can be esasily proved by the method of Lemma 5.1 if we use the properties (1.7)-(1.8) of the equilibrium measures. L e m m a 5.6 If )~ > 1 and Sto~ = Sw, furthermore this is a regular set, then

#~ = - ~

+

1-

ws~.

Again, here the regularity can be dropped, but we shall anyway need the lemma only in the case when the support in question is an interval. In what follows, let vw denote the density function of#w (wherever it exists). If the two supports S~ and Sw~, ;~ > 1 are not the same, then we only have inequalities for the corresponding extremal measures. L e m m a 5.7 IrA > 1, then

and

1

(1

:.

P r o o f . The proof is based on the following theorem of de la Vall6e Poussin from [5] (see also [4, 11, 71, and also [50]): Let p and v be two measures of compact support, and let fl be a domain in which both potentials U u and U v are finite and satisfy with some constant c the inequality

U"(z) < U~(z) + c,

(5.7)

z e n.

I r A is the subset off~ in which equality holds in (5.7), then UlA <_P[A' i.e. for every Borel subset B of A the inequality u(B) < p(B) holds. Consider the two potentials corresponding to the two measures/~w and/~w with some $ > 1. It follows from Theorem A (see the introduction) and (1.8) that with w = ws, (recall that this is the equilibrium measure of the set Sw)

1 -~(UU~(z)

-

Fw~) +

(1-

I)

(U~(z) - l o g ~ ) 1 cap(S

> U U , ( z ) _ Fw )

-

30

Approximation with general weightn

Section 5

for quasi-every z E S~, hence by the principle of domination (see the proof of Lemma 5.1) we have this inequality everywhere (recall that if a measure/, has finite logarithmic energy, then every set of zero capacity has zero/~-measure). Furthermore, Theorem A,(f), (1.8) and Lemma 5.4 imply that the equality sign holds for quasi-every z E S~x. Now each of the m e a s u r e s / ~ o , / ~ x and w have finite logarithmic energy, hence they vanish on sets of zero capacity, thus by applying the above theorem of de la Vall6e Poussin we can conclude the second inequality:

1

(1)

The first one can be shown with the same argument if we notice that with ~" = ws:~ we have 1 (UV- x (z) - Fwa) + (1 - 1 ) ( U ~ - ( z ) _ l o g

1 cap(Swx) ] -< Uu~ (z) - Fro

for every z E S~ox, hence by the principle of domination we have this inequality everywhere. Furthermore, by Theorem A,(f) and Lemma 5.4 the equality sign holds for quasi-every z 6 S~ox. Thus, the first inequality follows as before from the aforementioned theorem of de la Vall6e Poussin.

|

In order to apply the preceding lemma we shall need a convenient criterion for concluding that a point x0 from St0 belongs to some 8w~, A > 1, as well (recall Lemma 5.4 that these support are decreasing). L e m m a 5.8 Suppose xo is a point in the interior of Sw, vw is continuous in a neighborhood of xo, and vw(t) > co for It- xol < el for some eo > 0 and el > O. Then for A < 1/(1 - eoel) the point xo is in the interior of Swx, furthermore, vwx is also continuous in a neighborhood of xo. P r o o f . We begin the proof by the following observation. If for w we consider minimizing the weighted energy (1.5) on E and on some closed set Sw C E1 C_ E, then we arrive at the same extremal measure Sw. Seeing that Sto~ C_ 8~ (Lemma 5.4), this shows that in the proof we may assume without loss of generality that Z = S~o. Let v0 be the measure the density of which is e0 on I x 0 - el/2, x0 + el/2] and 0 otherwise, and consider the positive measure /41 ~

1 1

-

eo~ 1

(~

-

v0)

of total mass 1, and the weight function wl(x) = exp(U ~'(x))

31

Preliminaries to the proofs

Section 5

that it generates. By Lemma 5.1 the extremal measure corresponding to wl coincides with ul, and so x0 E Swl. Hence (see Lemma 5.3) if B is a neighborhood of x0, then there is a polynomial Pn such that w'~[Pn[ attains its maximum in B N ~ and nowhere else on Z. The potential of the measure

1 -

1 e0el

is symmetric about x0, attains its maximum at x0 and decreases to the right and increases to the left of x0. But then for the weight w2(x) -- w1(x)

exp(UV~176

the weighted polynomial wr{Pn can also attain its maximum only in B. Since this can be done for every neighborhood B of x0, it follows that x0 E ,gw~. However, the weight function =

exp(U"

-

and w A with A = 1/(1 - e0el) differ on Sw only in a multiplicative constant, which, together with the relation ~ = Sw, means that pwx = Pw2, and so Swx = Sw2. Thus, x0 E S~x as is claimed in the lemma (see also Lemma 5.4). Furthermore, the same proof can be carried out with Xl in place of x0 for every xl E [x0 - el/2, x0 + el/2] lying sufficiently close to x0, hence x0 is actually in the interior of Swx, and p)w ~ has a continuous density in a neighborhood of x0. It has left to show the continuity of the density function vwx at x0. Let I = [x0 - el/2, x0 + el/2], and let - denote taking balayage onto I out of C \ I. By Lemmas 5.5 and 5.6 we have

~[z = ~ ,

/ ~ [ =/J~ I

and 1 from which we get the formula

= Ap,~[i+ Ap~[( R \ I ) - ( A -

l)~i-p~l(R\

I1.

Now on the right each term beginning with the second one has continuous density in the interior of I, furthermore, by our assumption the first term also has continuous density at x0, and these prove that p ~ has continuous density function at x0.

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32

6

Approximation with general weights

Proof

of Theorems

4.1,

4.2 and

Section 6

4.3

After the preliminaries of the preceding section we are in the position to prove Theorems 4.1-4.3. P r o o f o f T h e o r e m 4.1. Let All be the set of the even natural numbers. By considering (wnpn) 2 we can see that for n E J~l there are polynomials Rn of degree at most n such that wnRn converges to f2 on Sw U {xo}, and here f2 is already nonnegative. Let the minimum and maximum of f2 on S~ he m and M, respectively. Let us suppose on the contrary that a := f2(x0) > 0, which will lead us to a contradiction as follows. We distinguish three cases.

Case I. f2 is not constant on S~ (i.e. 0 < m < M). We shall utilize the existence

of a polynomial U(x) without consta,U erm such that U(x) > 0 on [0, M] U U(m) # U(M) and on [0, M] U {~} the polynomial U takes its maximum at the point a. Such U's can he constructed as follows. Let K - max{2M, a}. If we choose the polynomial U*(x) as x(K - x)', then U* increases on the interval [0, K / ( s + 1)] and decreases on [K/(s + 1), K], hence by selecting s bigger than K / a and setting U(x) = U* ( K x / ( s + 1)a), the polynomial U will be nonnegative on [0, K], will increase on [0, a] and decrease on [a, K]. If U(m) ~ U(M), then we are ready. In the opposite case certainly m r 0, and we can set U(x) = U**(x)U*(Kx/(s + 1)a) with a polynomial U** that is nonnegative on [0, K], takes different values at m and M and takes its maximum on [0, K] at the point a. We claim that there is a subsequence Af2 C_ All and for each n E Af2 a polynomial Sn of degree at most n such that WnSn uniformly tends to g :-U ( f 2) on S w t J { x 0 } as n ~ cr n E Af2. In fact, let k be the degree of U. Since w2nRv.n --~ f2., n --* oo, we can see that for any j < k the weighted O$ converge to (f2)j 9 Thus, if U(x) = E j =kI ajxJ,. then polynomials .~ 2k!n*~2k!n/j the polynomials k

=

aj R2k !nlj (x) j=l

of degree at most 2k!n, will do the job (thus, we can choose Af2 = {2k!n I n =

1,2...}). By squaring again and considering g2 instead of g := U ( f 2) if necessary, we can also suppose that the polynomials Sn are nonnegative. Furthermore, the degree of Sn can be exactly n, for we can always add factors of the form (L - x)2/L 2 to Sn with sufficiently large L and these factors do not to change the properties of Sn on the compact set &0 U {x0}. In a similar manner, without loss of generality we may suppose that Sn does not vanish on the real line, for we can always replace any double real zero a of Sn by two conjugate complex zeros lying close to a. Note also that g is not constant on ,~w because we had V(m) r U(M) for the polynomial U.

33

P r o o f of T h e o r e m s 4.1, 4.2 and 4.3

Section 6

If we use Theorem A from the introduction we can see that 1

(6.1)

n ( U ~ ' ( x ) - Fw) - log S n ( x ) --* logg(x)

uniformly for all x E Sw t.J {x0} as n ~ c~, n E A/'2. If M1 denotes the maximum of g on S~o then it follows that there is a positive sequence {en} converging to 0 such that (6.2)

1

- n ( U U " ( x ) - Fto) + log s - f f ~ + l o g M l + en >_ O,

x e ,5~o.

Recall now the choice of U, according to which the function g attains its maximum M1 at the point x0 on Sw U {x0}. Thus, (6.3)

1

- n ( U "• (xo) - Fw) + log Sn(xo'--"~ + log M1 + en ~ 0

as n---* (x~, n E.N'2.

Let un be the counting measure on the set of zeros of Sn (counting multiplicity) and let ~ be its balayage out o f T \ S~0 onto S~0 with balayage constant d,, i.e. (see (5.4) in Section 5) m

UU"(x) = UU'*(x) + dn = log ~

1

+ dn

for quasi-every x E Sw. Then (6.2) goes into (6.4)

-n(U~'(x)-Fw)+UV"(z)-dn+logMl+en>O

for quasi-every x E St0, hence by the principle of domination (see the proof of Lemma 5.1) this inequality holds true for all x e C. On the other hand, from the nonincreasing character of taking balayage (see (5.6)) we can conclude from (6.3) that (6.5)

limsup

(-n(UU'(xo)

- r w ) + U~-Z(xo) - dn + logMx + en) _< 0,

n--*oo, n E2r

hence we must have here equality in the limit. But the function on the left is harmonic and nonnegative on C \ Sw, so I-Iarnack's theorem implies that lim

n--*oo, n~Af2

( - n ( U U ' ~ ( z ) - F,o) + U'a'Z(z) - dn + log M1) = 0

uniformly on compact subsets of C \ S~o. In particular, for z = ~ we obtain (6.6)

lim

n--*oo, nEAr2

(nF,~, - dn + log M1) = O,

where we used that the total mass of the measures npto and ~nn are both n. (6.1) yields with some monotone decreasing sequence {r/n} tending to zero that - n ( V U ' ( x ) - r w ) + VU"(x) - dn > - log(g(x) + r/n)

34

Approximation with general weights

Section 6

for quasi-every x E Sw, which combined with the preceding limit relation gives for n E Af2 -M1 (6.7) - nUf'~(x) + UV'(x) > log g(x) + qn Pn for quasi-every x E S~, with some sequence {Pn} tending to zero. Since the equilibrium measure w = ws~ of S~ has finite logarithmic energy, sets of zero capacity must have zero w-measure, hence the preceding inequality holds true w-almost everywhere. Thus, we can integrate this inequality with respect to w. Using the fact that the measures n~uw and ~ have finite logarithmic energy (recall that S~ did not have real zero, hence the balayage measure of its zero counting measure is of finite logarithmic energy) and that we have 1

V'~(x) = log cap(Sw) for quasi-every x E S~ (see (1.8)) we get from Fubini's theorem that f n U ~ d w = n f U~

= nlog cap(S~) = f U" ud'~ = f UV'~"dw, 1

and so during the integration of (6.7) against w the left hand side becomes zero. Thus, in the limit we get from the monoton convergence theorem that

f log M1

: o,

which can happen only if g(x) = All w-almost everywhere (recall that M1 was the maximum of g on S~,). But this is certainly not the case for g is not constant on Sw, so in some neighborhood O of an xl E St0 we have g < M1, and since cap(O N Sw) > 0 (recall that Sw was the support of the measure ~t0 of finite logarithmic energy), we must have w(O N Sto) > 0. The obtained contradiction proves the claim.

Case II. f2 is constant on Sw, but w is not. This case can be easily reduced to the previous one, for if w2nR2n tends to a constant on S,o, then w2n+2R2n will tend to w 2 which is not constant and deg R2n <_2n + 2. Case III. Both f2 and w are constant on S~ (in this case ~w = ws~). We can assume by proper normalization that w(x) = f2(x) = 1 on S~. By the maximum principle we have strict inequality in U~(z) < l o g - -

1

cap(S )

when z • Sw (see (1.7)), and since in the present case

W(Xo) <_exp (U~~

1

- log c a p ~ - ~ ) ,

Section 6

Proof of Theorems 4.1, 4.2 and 4.3

35

(see Theorem A,(d) in the introduction), we can conclude that w(xo) < 1. Hence by considering w2n+2R2n instead of w2nRu, if necessary, we can assume that f2(xo) =: a r 1. Choose now a polynomial U without constant term so that U(1) < 1 < 3 < U(a) is satisfied. Then, as before, there are some polynomials Sn, n E Af~ such that wnSn --+ U ( f 2) uniformly on $to U {x0}, i.e. wn(x)Sn(x) uniformly converges to U(1) < 1 on $w while it converges to U(a) > 3 at x0. This would mean that the sup norm of WnSn on S~0 is smaller than 2 for all large n E Af2, and at the same time these weighted polynomials take values bigger than 2 at x0 - - a contradiction to Theorem B in the introduction.

|

P r o o f o f T h e o r e m 4.2. Let f E Co(SW), We follow the proof of Theorem 1.1. In the present case Je denotes the set of points x E S ~ which are of distance > e form the complement R \ S ~~. Let J* be an arbitrary finite interval. Eventually we will choose J* so that in I2 \ J* we have (6.8) U"~ (z) > -Q(z) + Fw + 1 (c.f. Theorem A and the definition of the admissibility of w in the introduction), but for secure a free choice of J* later, it can be arbitrary at this point. First we show that it is enough to verify the following analogues of (2.1) and (2.2) for arbitrary J*:

(6.9)

w n ( x ) l Q . ( x ) l = exp(gL(X) + RL(X)),

x e J.,

where the remainder term RL(X) satisfies IRL(X)I _< CJL uniformly in x E Je with some CE >_ 1, and uniformly in x E ~ N J*

(6.1o)

wn(x)lQn(x)l

= e~

In fact, suppose this is true, and exactly as in Section 2 we apply it to w x instead of w with some A > 1. We can do this, because by Lemma 5.8 there is a A > 1 such that the set J~/2 is in the support Swx ofp~x and it has continuous density there, furthermore, exactly as U ' ~ the potential U ~ 9 is continuous everywhere; and these are the only properties that we shall use in deriving (6.9) and (6.10) below. Hence, by chosing A > 1 close to 1 we get that there are polynomials Q[n/x] of degree at most [n/A] such that with some gL and RL as above w"(x)lQE./~l(x)l

and (6.11)

= e x p ( g L ( x ) -- (n - A[n/A])Q(z) + RL(X)),

w"(x)lQ[n/,xI(z)l = e~

x E

Jc

x ~ ~ n J*

where now J* is an interval satisfying (6.8). Let 1/A < r < 1. We consider the polynomials Sn-[n/x] from the proof of Theorem 1.1, except that now we request that their degree be at most [ ( r -

36

Approximation with general weights

Section 6

1/A)n] (hence we write S[(r-ll~)nl for them), and we need their third property (2.3) only in the form (6.12)

_< 1,

x e J*

\

J,.

Since the disjoint sets J* \ J~ and J2~ consist of finitely many intervals, there is a 0 < c < 1 and for each m polynomials R~ of degree at most m such that (6.13)

IR~(x) - 11 < cm

(6.14)

IR~(x)l
and (6.15)

0_< Rm(x) < 1

for x ~ J2,, for x E d * \ J , for x e J ~ \ J 2 ~

(see e.g. [15, Theorem 3] where such polynomials were constructed for two disjoint intervals, from which the Rm's with the stated properties can be easily patched together). Finally, we set

Pn(x) = Q[nlX](x)S[(r_xlX),q(x)a[(l_~.),q(x), which has degree at most n. Exactly as in Section 2 we get that if r / > 0, then by choosing e > 0, A > 1 and L appropriately, for sufficiently large n the difference ]wn IPnl - f[ will be smaller than 3~1 on the set J* N ~. The only remark we have to make is that by (6.11), (6.12) and (6.14), the weighted polynomial wnPn is exponentially small on (~ N g*) \ J~. But by (6.8) and Theorem B (see the introduction) the same is true on ~ \ J*, and this proves that

Iwnlpn[- f[ < 3T! for every x E ~ provided n is sufficiently large. Thus, it has left to prove (6.9) and (6.10). The proof of (6.9) is identical to the proof of (2.1) given in Section 2, there is no need to change anything. As for (6.10), with the measure Pn from the proof of Theorem 1.1 we get by the continuity of the potential U ~'~ that I U " ' ( x - iLIn) - U " ' ( x ) l = o(1) as n ~ cx) uniformly in x E R, while the estimate log IQn(x)l + nU"'(x) < 31og 2On/L with D equal to the diameter of J* follows exactly as (2.9). These and

w(x) < exp(V#~(x) - Fro) (see Theorem A,(d) in the introduction) prove (6.10), and the proof is complete.

]

Section 6

P r o o f o f T h e o r e m 4.3. Theorem 4.2.

Proof of Theorems 4.1, 4.2 and 4.3

37

We show that Theorem 4.3 is a consequence of

Let ~ = U/j, where on the right we have a finite and disjoint union, and suppose that Q is convex on each of the I/'s. It was proved in [34, Theorem 2.2] that then S~ consists of finitely many intervals at most one lying in any of the Ij~S.

Let I be any subinterval of the interior ofS~. We recall Lemma 5.5 according to which

where - indicates taking balayage onto the right has continuous density inside by [49, Lemma 4.5] (c.f. also the proof pw]t is I, and w l i is a Cl+e-function

I out of C \ I. The second measure on I, and the same is true for the first one of Theorem 8.3) because the support of on the support. Thus, we can conclude

that/z~ has continuous density in the interior of S~. In order to be able to apply Theorem 4.2 we have to show that the density vw of the extremal measure Pw cannot vanish at any interior point of Sw. But this is an immediate consequence of Lemma 5.7 according to which we have

and this clearly rules out that the density of Pw vanishes at a point x0 unless x0 does not belong to the interior of Sw~. Thus, if x0 E St0 does belong to the interior of some Sw~, then at x0 the measure Pw has positive density. But every interior point x0 of Sto must belong to the interior of at least one Swx. In fact, since every 8 ~ consists of intervals at most one of which can lie in any Ij, it is enough to prove that in any neighborhood of any point x0 of Sto there is a point xl lying in some Sw~, because then this and the decreasing character of the supports St0x (Lemma 5.4) imply our claim concerning every point in Int(S~) lying in the interior of some Sto~. But if x0 E Sw, and B is any neighborhood of x0, then there are an n and a polynomial Pn of degree at most n such that w n [Pn[ attains its maximum in ~ at some point of B N ~ and nowhere outside of B. By continuity then the same is true of w An [Pn[ with some A > 1 sufficiently close to 1, and so Theorem B implies that B n Swx ~ 0. With this the proof is complete.

|

As for Theorem 4.4, compactness shows that it is enough to consider functions that vanish outside some Int(Sto~), and for such functions the preceding proof gives the appropriate approximation.

38

Approximation with general weights

7

Construction

of Examples

4.5 and

Section 7

4.6

In this section we construct the two theoretically important Examples 4.5 and 4.6.

7.1

Example

4.5

In Example 4.5 we have to construct a weight w such that the support of the corresponding extremal measure is [-1, 1], this measure has continuous density in ( - 1 , 1) which is positive everywhere except at 0, and still no function that is nonzero at 0 is the uniform limit of weighted polynomials. Hence, in this case the largest set for the approximation problem of Section 4.1 is the restricted support ( - 1 , 0) U (0, 1). The idea of the proof is to construct a density function v that is positive and continuous on ( - 1 , 0) U (0, 1) and tends to zero very fast at the origin. We shall construct an even w by inductively choosing density functions that will converge in some sense to the density of w. In fact, for each k _> 2 we shall construct a function vk with the following properties: Let Sk = [-1 + 2 -k, - 2 -~] U [2 -k, 1 - 2-k]. For every k >_ 2 1. vk is even, positive and continuous on Sk and zero elsewhere, 2. the integral of vk is 1, i.e. vk generates an absolutely continuous positive measure of unit mass with support on Sk,

3. vk(x) <_ 1/k if [z[ _< 2 l-k, furthermore vk(=l:2-k) _< 1/(k + 1), 4. Irk(x) - vk-l(x)l < 2-k if z E & - l , . there exists an nk > k such that if wk(z) = exp(U~k(z)), where U ~k denotes the potential associated with the measure generated by vk, then for every polynomial Pnk of degree at most nk the inequality

implies 1

w~(O)lP.~(O)l < ~. From here the construction of w with the desired properties will be simple, so first let us consider the construction of vk. For k = 1 define vk to satisfy the first three conditions, and now let us proceed with the construction of vk+l provided vk is already known. Let Ik --- [ - 1 + 2 - k - I , - I + 3 .

2 -k-2] U [ 1 - 3 , 2 - k - 2 , 1 - 2-k-~].

Section 7

39

Construction of Examples 4.5 and 4.6

Note that Ik C Sk+l is disjoint from Sk, hence if wsk is the equilibrium measure associated with Sk (see Section 4.1), then (7.1)

log

1 - maxU~Sk (x) =: Yk > 0 cap(Sk) xetk

(see (1.7)-(1.8) and apply the maximum principle according to which the potential of wsk is strictly smaller than log(1/cap(Sk)) outside Sk). Choose now a/3 =/3k > 1 arbitrarily. We claim L e m m a 7.1 There is a sequence On tending to 0 such that if Pn is an arbitrary polynomial of degree at most n, and

(7.2) and (7.3) then (7.4)

w~*(x)lPnr

~ 1,

x ESk,

w~'(x)lPn(x)lr n _< 1,

x EIk,

w~'(0)IPn(0)I _< 0n.

Recall that wk(x) = exp(U ~h(x)), P r o o f o f o f L e m m a 7.1. We begin the proof by reducing it to the case when Pn has a special form. First of all by considering (Pn(X) + Pn(-X))/2 instead of Pn we can assume that Pn is even. Let

and

P~(x) = la.I I I ( x 2 - I~1). i Since IPn(O)l = IP~(0)I, and for every x e R the inequality IPn(x)l >_ IP~(x)I holds because of the obvious inequality Ix 2 - a i l > Ix2 - lajll, without loss of generality we can assume Pn = P*, i.e. that the leading coefficient an is positive, and aj >_ 0 for all j, which also means that Pn has only real zeros. Let 0 _< O t l , . . . , ( ~ s < 2 -2k and a , + l , . . . _> 2 -2k, and now consider

j=l

x2 - aj

2 -2k

"

Again P,,(0) = Pn*(0), and for every I~1 ~ 2-k the inequality IPn(x)l > IPZ*(x)I holds because for such x the function Ix2 - a l / a decreases in c~ on the interval [0, 2-2k). Thus, (7.2)-(7.3) hold also with Pn replaced by P**, while the status

40

Approximation with general weights

Section 7

of the inequality (7.4) remains unchanged, hence without loss of genrality we can assume Pn = P,~*, i.e. that Pn does not have any zero on the interval Let M be arbitrary large. We distinguish two cases. Case 1. There are at least M zeros of Pn in Ik. If we set 1

Un(x) = log ipn(x) I then (7.2) takes the form (7.5)

nUVk(x),

Un(x) > 0

for every x e Sk = supp(vk). We can apply the principle of domination (see the proof of Lemma 5.1 in Section 5) to deduce (7.5) for all x E C. Now let us recall Harnack's principle, according to which if U is nonnegative and harmonic on a domain D, and Yl, Y2 E D, then there exists a K independent of U such that V(yl) < gV(y2). We set D = (C \ It) t.J ( - - 2 - k , 2 - k ) . Since all the zeros of Pn lie in R \ ( - 2 -k , 2-k), we can conclude the harmonicity of Un in D, and so for Yl = 0, Y2 = r there is a K such that

Un(oo) g gun(o)

(7.6)

independently of n. Now integrate (7.5) with respect to the equilibrium measure ws~ of Sk. Taking into account (1.7)-(1.8) we get from Fubini's theorem with the counting measure vn on the zeros of Pn 1 nlog cap(Sk)

=

1 n / U~Skvk = f nUV~dwsk < / log l-~ldws k

<

log - - + n log ca1 / "~ " - M~k, an pt~k)

1

-

1

where, at the very last step we used (7.1) and the fact that at least M of the zeros of Pn lie on Ik. This implies that (7.7)

Un(cx)) = log I

_>U y k ,

Gn

and so, in view of (7.6), we can conclude

Un(O) > M "rlk ~. _

Case 2. There are at most M - 1 zeros of Pn in Ik. Let J 1 , . . . , JM be disjoint open subintervals of Ik and t l , . 9 tM one-one point in each. Choose connected open sets Dj, j = 1,..., M containing the points 0 and t I such that Dj N R C_ Jj U ( - - 2 - k , 2 - k ) .

Section 7

41

Construction of Bxamples 4.5 and 4.6

By Harnack's inequality there are numbers K s such that if U is a nonnegative harmonic function on Dj, then

U(tj) < KjU(O). Now in this case there are at most M - 1 zeros on Ik, so these zeros have to miss at least one Dj, say Dj,, which means that Un is nonnegative and harmonic on D j , . Hence the preceding inequality can be applied with U = Un and j = jn, by which we get from the assumption (7.3) that

Un(O) > -~i1 Un(tj.) > ~ n1 l o g ~ > n -gi.

(min -

I ) logfl.

.....

This gives us for large n

Un(O) > M, which, together with (7.7), proves our lemma because M can be arbitrary large.

| Having this lemma at our disposal, we are proceeding with the construction of vk+:. Let us choose and fix an nk such that Onk < 1/k (see the previous lemma), and let Pk denote the set of polynomials Pn of degree at most nk that satisfy (7.8)

w~'(x)[Pn(z)[ _< 1,

x e Sk.

This set of polynomials of a fixed degree is compact in the supremum norm, hence we get from Lemma 7.1 that there are finitely many points Q , . . . , tN on Ik such that if for Pn E Pk the inequalities

(7.9)

w~'(t~)lPn(tj)lD n < 1,

are satisfied for every j, then

_<

(7.1o)

1

Let Hk be a subset of Ik containing the set {tj}N=: such that it is symmetric with respect to the origin, it consists of finitely many intervals, and cap(Hk) is some c~ that will we selected below together with the positive number p~. If w~ := w/~k is the equilibrium measure on Hk, then the potential of the measure pkwk is nonnegative on [--1, 1] (by the symmetry of Hk), it is of the order O(pk) on Sk, at each tj it has the value Pk log(1/ck), and at every other point of [-1, 1] its value is at most Pk log(1/ck). Thus, by appropriately choosing ck and p~ (to have, in particular Pk log(1/ck) = log~), we can achieve that the weight w~ = w~ exp(U pk~k) will satisfy the property that (7.71)

<_ 1,

9 e sk o zk,

42

Approximation with general weighta

Section 7

implies (7.12)

(wD"k (0)lPnk(0)l <

2

(see (7.8)-(7.10)), furthermore I log(w~)- log(wL)] _< log3. The point is that by adding the measure pLwk to vL we get a measure the potential of which is approximately the same as that of vk on SL (this is achieved by chosing pL sufficiently small), but at the points tj it is larger than the latter one by the amount log fl, and otherwise on [-1, 1] these two potentials differ by at most log3. Furthermore, the argument also shows that there is a p~ < 2 -L such that for every 0 < pL < p~ there is an appropriate cL (say PL log cL = log 3) with the property that for the corresponding w~ (7.11) implies (7.12). Now there is an eL > 0 such that if w is any function satisfying

[log(w(x))

-

log(w;~(x))l < eL

for every x E {0} L9SL O HL, which is a subset of {0) tJ SL+I, then

(7.13)

~""k(:OIPnk(x)l <

1,

x ~ &

o

h,

implies (7.14)

3 wnk(0)]Pnk(0)l _< ~.

Let hk be the density of wk. The function vk + pkhk is almost the desired vk+l, except that it has the following deficiencies: its integral is 1 + pk, that is slightly larger than 1, it has infinite singularities at the endpoints of the subintervals of Hk, and it is not everywhere positive on Sk+l. But all these are easy to rectify. In fact, it easily follows from the monoton convergence theorem and Dini's theorem about the uniform convergence of an increasing sequence of continuous functions provided the limit is continuous, that if we take h~ := min(hk, M), then for large M the potentials of this new function and that of wk differ as little as we wish. Then we can distribute the mass f(hk - h'k) that we saved in replacing hk by h~ to Sk+l \ (Sk U gk) so as to get a continuous continuation of vk that is smaller than 1/(k + 1) if Ixl < 2 -k and takes the value 1/(k + 3) at the points +2 -k-1. In other words, there exists a continuous function ~k+z with the following properties: it coincides with vk on Sk, coincides with h~ on Hk, continuous and positive on Sk+l, zero outside Sk+l (recall that Sk O Hk C Sk+l), is smaller than 1/(k + 1) if Izl < 2 -k, takes the value 1/(k + 3) at the points 4-2 - k - l , and it has integral 1 + Pk. Now 1 ~k+l :=

1 + pk ~k+l

with some small p~ > pk > 0 clearly satisfies all the properties that we required of vt for i = k + 1. Furthermore, this construction can be done in such a way that for the corresponding wk+l = exp(U ~k+l) the inequality (7.15)

~k

I log(wk+x(x)) - log(w~(x))[ < -~-,

x E [-1, I],

Section 7

Construction

of Examples

43

4.5 a n d 4 . 6

is satisfied with the ek chosen before, furthermore (7.16)

[ log(wk+l(x)) - log(wk(x))[ _< 2 log/3,

x E [-1, 1].

Until now we have not said anything of the constant /3 = /3k > 1. To ensure the uniform convergence of the sequence {wk} on [-1, 1] let us require that log/3k = min(ek_x/8, 2-k), which, together with (7.16) yields the uniform convergence of {wk}. We can clearly assume that ~k < ek-1/4 and ek < 2 -k are also true for every k. Then w(x)---- lim wk(x) k---* c<~

exists and continuous on [-1, 1] (actually on the whole real line), and it follows from (7.15), (7.16) and the inequality log/31 _< et-1/8 that (7.17)

[log(w(x)) - log(w~,(x))[ < [log(wk+a(x)) -- log(w~(~))[ +

I

- log(w,(

ek

))l < 7 +

l)k

1

< l>k

for every x E [-1, 1]. Our construction gives that the sequence vk uniformly converges on every compact subset of ( - 1 , 1), and if

v(x)-

lim vk(x),

k -....*oo

then v is positive on ( - 1 , 0)(J(0, 1), continuous on [-1, 1], v(0) = 0, furthermore

w(x)-- lim w k ( x ) = lim e x p ( U ' k ( x ) ) = exp(U'(x)) k---* oo

k---* oo

for at least every x E ( - 1 , 1). Thus, by Lemma 5.1 we get that the equilibrium measure associated with w will be v(x)dx, i.e. v is the density of tt~. Thus, it has left to prove that if a function f is uniformly approximable on [-1, 1] by weighted polynomials of the form wnP,~, then f must vanish at the origin. In fact, without loss of generality assume that If] _< 1/2. Let q < 1/2. If approximation is possible, then for large enough k there will be polynomials of degree at most nk such that I f - W"kPnk[ <__'7 on [-1, 1]. But then (7.13) is true in view of (7.17), hence (7.14) holds. Thus, we can conclude 3

If(O)l < Iwnk(O)Pn(O)l + If(O) - wnk(0)Pnk(0)l <_ ~ +~/, and since here , / > 0 is arbitrary and k can be any large number, it follows that f(0) -- 0 as we have claimed.

| In the previous construction we have considered the approximation problem on [-1, 1]. To get an example when w is defined on R all we have to do is to extend w to R \ [-1, 1] so that it be be admissible in the sense of (1.3) and be smaller than exp(U ~) there.

44

7.2

Approximation with general weights

Example

Section 7

4.6

In example 4.6 we have to construct a weight w on [-1, 1] such that the support of the corresponding extremal measure Pw is [- 1, 1], Pw has continuous density in ( - 1 , 1) which vanishes at the origin, and still every continuous function f that is zero at =kl can be uniformly approximated by weighted polynomials of the form wnpn. Here, as opposed to Example 4.5, we will construct a v which tends to 0 at the origin very slowly. Let {fi} be a countable system of continuous functions that vanish at =kl such that the linear combinations of {f]} is dense in C0(-1, 1). Then it is enough to approximate each f2. Without loss of generality we may assume that each fj is nonnegative smaller than one at every point of [-1, 1]. Let 1 1 _

vff-

x2

be the so called arcsine distribution, which is nothing else than the equilibrium distribution of the interval [-1, 1]. For each n let wn be a function that is continous and nonnegative on ( - 1 , 1), coincides with w on ( - 1 , - 1 / n ) t A ( 1 / n , 1), vanishes at 0, it is at most 2 on I - l / n , l/n], and has integral 1. Choose 1 < < 2, and let 7=

l -~

.

With some sequence {nk} to be chosen below we set oo

(7.18)

v(x) = 7 E l-'~wn'" 1=1

Then v will be a continuous function on ( - 1 , 1) that is positive everywhere except at the origin, where it vanishes. We will show that by appropriate choice of {nk} the weight (7.19) w(x) := exp(VV(x)) satisfies all the requirements. Since Lemma 5.1 easily implies that the density of Pw is exactly v, all we have to show is that every continuous function that vanishes at =k1 can be uniformly approximated by weighted polynomials wnpn. We shall determine the sequence {nk} recursively along with two other sequences {Nk} and {Mk} tending to infinity. Suppose that h i , . . . , nk- 1, N1, 9 9 Nk- 1 and M1, 9 9 M~_ 1 have already been chosen, and for m _> 2 let

,,:1

+k

Section 7

45

Construction of Examples 4.5 and 4.6

where ilk=

~

!- ~

and

7k=

l=k+2

(Z ]-' 1- ~

\l#k+l

,

]

and

\1=1

Then, if we set Wk,m = exp(U~k,'), it follows from Lemma 5.1 that #w~,,, has density Vk,m (note that Vk,m has integral 1). The number m stands for the next nt, i.e. for n~. We shall choose m only at the very end of our construction. Our aim will be to get estimates that do not depend on the actual choice of m. Let us follow the considerations of Sections 2 and 6. By the method of Section 2 the potential nUk,m of nvk,m can be approximated by potentials corresponding to the sum of n Dirac measures. We have to watch however the size of the remainder terms and have to carefully chose L because we want to get an estimate which is uniform in m. The details are as follows. Let e > 0, and consider the set Jc = [-1 + e, 1 - e] from Section 2, and the intervals Ij discussed there for the function Vk,m. The best constants c, C for which c/n <_ Iljl <_C/n for every Ij 13 J~2 r 0 can be easily seen to satisfy c _> e and C <_ 4~rk~-1, which follow from 1

2

1

k 1-" < v~,m _< ~ ~/T- x~

Note that these estimates do not depend on m. Thus, if L = k 3a/2-1, then in the present case we get for the sum in (2.8) the upper bound

(7.20)

C1

(Cln) 2

A~. (L/n)2 + (e(j - jo)/n) 2 $

oo

_< c, ~

~2a-2

k3~-~ + ~j~ + c, ~ x l ~ i

~

l~l~ -~ _< c~k~/~-~

liNJe2 =~

j=O

independently of m, for every sufficiently large n. Let Pn denote the measure that we obtain by translating vk,m(x)dx by iL/n (see Section 2). Recalling that vk,m and vk,oo differ by at most 2k - a , the argument of (2.5)-(2.6) yields that the potential of vLm and of p,~ differ by

uniformly in m, where IXk,m,,(x)[ <_ 2k -a (i.e. the o(1) is uniform in m, and the estimate on Xk,m,, is also independent of m). Note that because of L = k 3/2a-1, in absolute value the second error term on the right is at most nlXk,.-,,.(z)l

n

<

2kS/2a-lk-a n

_

2kS/2-1 _ _ n

46

Approximation with general weights

Section 7

which, after multiplication by n, is of the same order as the error term we obtained previously in (7.20) as the the analogue of the estimate for (2.8). Thus, altogether we get for large n the following analogue of (2.1): (7.21) w~',m(z)lQn(x)l =

exp(~k3"/2-%k,~(z) + Xi,m,n(X) + nk,~,n(z))

for x E Jr and large n, where

IXL.,n(~)I := nlXk,.n,n(x)l _< 2k "/2-' and

IRk,m,.(x)l _< C2k~/2-1, C

and these estimates are uniform in m. Hence, if we set e = k (~/2-1)/4, then it follows that

(7.22)

Ix~,.~,.(~)I < 2~ ~/2-~)/4

and (7.23)

IRk,m,,(x)l <_ C2k (~12-1)14.

This finishes the discussion of the approximation on Jr Outside J~ we used in Section 2 the estimate (2.2). Here the form of (2.2) we need is given by (6.10), i.e. we need it only in the form that

(7.24)

wLm(~)lQ.(~)l = eo(-)

uniformly in m, which is clear from the uniform equicontinuity of the functions wk,m (see the proof of (6.10) in Section 6). Now let us recall the definition of 7 and 7k from the beginning of the construction. Obviously, Ak :-- 7k/7 > 1. If we set k-1

)

\/=1

(note that now on the right we have/~k-1 to have integral 1) and

Wk, m = exp(U~k,,-), then it follows from the fact that the potential of w is constant on [-1, 1], that wk,rn and tt,,,* W k , m ]~xk differ only in a constant, hence together with (7.21)- (7.24) we get that for large n there are polynomials Q[*n/xk] of degree at most [n/)~k] such that (c.f. also Sections 2 and 6)

(7.25)

W* n (z)lQtn/~k](x)l * (~,.,) =

9 exp(~k 3"/2-1V ~k,| (~) + (n - ~k [-/~k]).i,~(~) +Xk,~,ln/~l ( X ) + nk,~,tn/~A~))

for x E J~, and

(7.26)

W* n ()lQtn/x~](z)l ~g * (~,,,,,) = e~

Section 7

47

C o n s t r u c t i o n of E x a m p l e s 4.5 and 4.6

and these are uniform in m. Let 1 _< j < k arbitrary, and consider the function fj from the system {fj} from the. beginning of the proof. Using (7.25) and (7.26) instead of (6.9) and (6.10), we obtain with the method of Section 6 that we can multiply this Q[*/xk] by a suitable polynomial of degree at most n - [n/Ak] so that for the so obtained polynomial Pkd,m,n of degree at most n we have I(w~,.,)"(x)lPk,~',.~,n(X)[

-

_<

f#(x) exp(x*k,m,[./Ak](X) + Rk,m,En/Xkl(X))[ Cayj,k+ o(i), z E [-i, 1]

uniformly in m, where ~/j,k is the maximum of fj on [-1, 1] \ J2e (recall that e was chosen to be equal to k('~/2-1)/4), and C3 is some fixed constant (in the preceding formula we assumed X* and R~,m,n extended outside Je so as (7.22) and (7.23) remain valid for all z E [-1, 1]). In fact, to be able to do this all we need is that the set of functions

{exp(sUVL=(z))[O

< s < 1, m = 1 , 2 , . . . }

is compact (see also the corresponding arguments in Sections 2 and 6), which is obvious from the definition of the functions win. If we also take into account (7.22) and (7.23) and that the fi's are at most i in absolute value, we can finally conclude that I(w~,m)nlP~a,.~,-l - l~l < c4(r

+ k(~/2-1)/4) + o(1)

with an absolute constant C4, where o(1) --* 0 uniformly in m as n --+ oo. Thus, there exists an N~ > Nk-1 such that if n >_ Nk, then independently of the choice of m we have

I(w;,,-,.,)n IPk,j,,-,,,,~l- f#l_< 2C4(~./,k + k (a]2-1)/4) for 1 <_ j < k, and this is how we choose Nk. Now let us return to k,m

=

7

I - a w n , q- k - a W m "4"

1-aw

9

l=k+l

\1----1

As we have already mentioned, here m stands for nk, so the only difference between v*k,m and v (see (7.18)) is that in the latter one the terms l-~w with i _> k are replaced by I-~wm. Consider now only those n for which Nk-1 < n < Nk. If we replace the terms I - ~ , ! > k by some l-a~n, in v*k,trt, then if these nt's are sufficiently large, we get from the preceding estimate applied to k - 1 rather than to k (in which case v~_l,.k_~(z ) = V~,oo(Z)) that

(7.27)

fr

___3C'4(T/Lk-1 + (k - 1) ((~/2-1)/4)

will be true for every Nk-1 < n < Nk (note that Nk has already been fixed) and 1 < j < k. Thus, there exists an Mk > Mk-1 such that i f m = nk, nk+l, nk+2,...

48

Approximation with general weights

Section 7

are all larger than Mk, then for w defined by (7.19) the estimate (7.27) is true. This gives the choice Mk. Now all we have to do is to select nk = m > Mk and nk > nk-1. We emphasize again that this choice does not influence the choice of Ark and Mk, so our parameters can be selected in the order 9.., nk-i, Nk, Mk, nk, Nk+l, Mk+l, n k + i , . . 99 We claim that this choice is appropriate. Let j > 1 be arbitrary, and n > j some large number. Then there is a k such that Nk-1 < n < Nk. (7.27) implies that

IwnlPk-l,i,nk-,,-I-

fil --< 3C4(~/1,}-1 + (k - 1)(a/2-1)/4).

Here k tends to infinity as n does so, and then the right hand side tends to zero, hence the uniform approximability of f7 by the weighted polynomials w 2n [Pk- 1,j,nh_l,n [2 follows. Since the linear combinations of the f7 's is dense in C0(-1, 1), it follows that uniform approximation (on [-1, 1]) of every continuous f with f ( + l ) = 0 is possible by weighted polynomials of the form w2np2n. But this implies the uniform approximability by weighted polynomials of odd degree exacly as we mentioned it in Section 2 (divide through by w, approximate and multiply back).

1

Part III

Varying weights In several problems weighted polynomials of the form Wn Pn appear where { W,~} is a sequence of weights (see e.g. Section 12), i.e. the weights are not powers of a fixed weight function. In such a case we set wn = W2/'* and consider weighted polynomials w~Pn with varying weights wn. The method of the preceding sections allows us to deduce convergence results in this setting. The applications of these results will be given in Chapter IV.

8

Uniform approximation by weighted polynomials with varying weights

Let us begin with the analogue of Theorem 4.2. T h e o r e m 8.1 Suppose that {wn} is a sequence of weights such that the extremal support ~%~ is [0, 1] for all n, and let 0 be an open subset of (0, 1) for which the set [0, 1]\O is of zero capacity. If the equilibrium measures lho~ are absolutely continuous with respect to Lebesgue measure on O: /.tw,,(x) = vn(x)dx, and the densities vn are uniformly equicontinuous and uniformly bounded from below by a positive constant on every compact subset of O, then every continuous function that vanishes outside 0 can be uniformly approximated on [0, 1] by weighted polynomials wnpn, deg Pn < n.

Actually, the sequence {w,~Pn} can be constructed in such a way that the convergence w~Pn ~ f holds uniformly on some larger set [-0, 1 + 0], 0 > 0 (provided of course the weights are defined there). This easily follows from the proof. The conclusion is false for every O for which [-1, 1] \ O is not of positive capacity, but we shall not show this. For conditions in terms directly on wn themselves which guarantee the assumptions in the theorem see Theorem 8.3 after the proof. P r o o f o f T h e o r e m 8.1. We set ~ : J* : [0,1], J~ : { x [ ( x - e , x + e ) C_ 0}, and copy the proof of Theorem 4.2. This can he done word for word with one exception, and this is the estimate (6.10). In fact, the heart of the proof of Theorem 4.2 is Lemma 5.8 which is valid in the following form with virtually the same proof: If v,~ are uniformly equicontinuous on an interval [x0 - el, x0 + el] and Vn(t) >_ eo there, then for A _< 1/(1 - eoq/2) the interval [x0 -- q / 2 , x0 + e1/2] belongs to the interior of Sw~ , and the densities vw~ are also uniformly equicontinuous there. W h a t goes wrong with (6.10)? In the present case we could claim (6.10) only under the assumption that the potentials U ~ (z) are uniformly equicontinuous

50

Varying weights

Section 8

on [0, 1] as functions of the complex variable z (c.f. (2.7), which has been replaced in the proof of Theorem 4.2 by a similar relation where the left hand side is o(1)), and this may not be true. In any case we have (6.10) in the form

(8.1)

w2(x)lQ.(x)l

<

Coec~

Note however, that the densities are uniformly bounded on compact subsets of O, which easily implies that (6.10) is true inside O, i.e. (8.2)

w,~(x)lQ,(x)l = e ~

uniformly on compact subsets of O. In Section 6 the estimate (6.10) is used in conjunction with the estimates (6.13)-(6.15), i.e. with

(8.3)

IR~(x) - 11 < cm,

for ~ e J2~,

(8.4) IR~(-)I < c~, for 9 e S* \ J~, and 0 < Rm(x) _~ 1, for x E J ~ \ J 2 ~ , (8.~) where c was some positive number less than 1. Now if this c was actually smaller than exp(-C0) from (8.1), then the proof in Section 6 would be valid in the present case, as well. The proof also shows that any J, with some fixed but small ~/> 2e can stand in (8.3) and (8.5) instead of J2~, furthermore in (8.3) we do not actually need geometric convergence, i.e. (8.3) can be replaced by (8.6)

[Rm(x) - 11 =- o(1)

for x e J , ,

as m ---, oo. It is also clear from the proof that (8.5) can be replaced by the uniform boundedness of Rm(x): (8.7)

[Rm(x)l _< C

for all m and x E J*.

Thus, it is enough to show that in the present case for arbitrary ~1 > 0 and c > 0 we can choose an e > 0, such that (8.4), (8.6) and (8.7) hold for some polynomials Rm of degree at most m whenever m is sufficiently large. The assumption that [0, 1] \ O has zero capacity implies that the capacity of J* \ Je tends to zero, hence our claim follows from the next lemma by setting S = J, and K = J* \ J,/2 if we apply it to the sets L = J* \ J~ with e < ~//2, e --* 0. Thus, the verification of the lemma will complete the proof of Theorem 8.1. L e m m a 8.2 Let S and K be two disjoint compact subsets of [0, 1]. Then there is a constant 6 > 0 such that for all compact subsets L of K and suj~iciently large n there are polynomials Pn of degree at most n such that

(8.8) (8.9)

IP.(x)l _< 2, IP,~(x)

-

x e [0,1],

11 <

,

~, e s

and

(8.10)

[Pn(x)[_ (cap(L)) ~",

x E L.

Uniform approximation by weighted polynomials with varying weights

Section 8

Proof.

51

Let = II(zj--1

be the polynomial of degree m that has all its zeros zj in L and minimizes the norm IIT IIL among all such polynomials (these are the so called restricted Chebyshev polynomials associated with the set L). The classical proofs for the Chebyshev polynomials given e.g. in [51, Theorem III. 26] can be easily modified so that we obtain (8.11) lim IIT,.,,]IiL/'n= cap(L). $T~ - - * O O

Since all the zeros of Tm belong to [0, 1], we also have (8.12)

ITm(x)l < 1, 9

x 9 [0,11.

Let now Sp and Kp be the set of points on the plane the distance of which to S and K, respectively is at most p, and choose p so small that the sets Sp and Kp be disjoint. Consider the function fro(z) which is defined to be 1 on Kp and 1~Tin(z) on Sp. This fm is analytic on Sp U Kp and we have the bound

II (z)l

<

dist(S, K)

=: Gin

for it, hence by a classical approximation theorem of Bernstein ([52, Theorem 5, Sec. 4.5, p. 75]) there is a r < 1 and there are polynomials Rk of degree at most k such that IRk(z) - f,~(z)l _< C?r k,

z E Sp/2 U gp/2.

We set here k = rm, where r is so large that r r < 1/4C1 holds. Thus, (8.13)

z 9 Sp/2 U Kpl~.

[R~(z) - f,,,(z)[ <_ 4 - ~ ,

We also get from the Bernstein-Walsh lemma ([52, p. 77]) that with some constant C2 > 1 (8.14)

IRk(x)l < C~C'~ =: 07,

x E [0, 1]

(note that here C3 _> 1). We have already used in (6.13)-(6.15) the consequence of [15, Theorem 3] that if there are two disjoint systems of subintervals of [0, 1], then there are polynomials that take values in between 0 and 1 on [0, 1] and geometrically converges to zero respectively to 1 on the two systems of intervals. Thus, we can choose atr and for all sufficiently large l polynomials Qz of degree at most I such that

IQz(x)l _< ,d,

z 9 [0, 1] \

Spl2,

and - 11 < ,J,

9 9 ,.,r

52

Varying weights

Section 8

We set here 1 = s m with an s such that gs < 1/2C3 holds, by which we get

(8.15)

IQdx)l <

(1) ~

,

x 9 [0,1] \ S;i 2

and

(8.16)

IQz(x) - 11 <

,

x e S.

Finally, we set Pn(x) = T m (x)Rk(x)Ql(z) which has degree at most (r + s + 1)r,. On S we have f,-,~(x)T,~(x) = 1, hence [Rk(x)Tm(x) - 11 = I(Rk(x) - fm(x))Tm(x)[ < 4 -m ' 1 <_ 2 -'~ by (8.13) and (8.12). If we take into account (8.16) then we can conclude that IP,(x) - 11 < 3 . 2 -m on S, which proves (8.9). In the same fashion, on L the product [Rk(x)Qt(x)l is at most 2 -'~ by (8.14) and (8.15), and we have ITm(x)l < (cap(L)) m/= for large enough m, hence HPnHL <_ (cap(L))m/2 proving (8.10). If x r Sp/2, then (8.12), (8.15) and (8.14) imply that ]Pn(x)I <_ 1 < 2. Finally, if x E Sp/2 then the same conclusion follows from (8.16), (8.12) and (8.13), because the latter two imply

ITm(~)R~(~) - ii = fTm(~)[IR~(~) - f ~ ( ~ ) l < 4 - ~ . These prove (8.8), and the proof is complete.

|

It is clear how we should modify the proof in order to achieve convergence on some [-0, 1 + 0]: all we need to do is to add the sets [-0, 0] and [1, 1 + 0] to L -= J* \ Je. As e, 0 --* 0 the capacity of the new L will tend to zero, and this is all we needed above.

|

Now let us discuss some conditions that guarantee directly in terms on the weights Wn that Theorem 8.1 can be applied. We shall always assume that the weights are normalized so that the support Sw. of the corresponding equilibrium measure is [0, 1]. T h e o r e m 8.3 Suppose that {w,}, w, = e x p ( - Q , ) is a sequence of weights such that the extremal support $w. is [0, 1] for all n, on every closed subinterval [a, b] C (0, 1) the functions { Q . } are uniformly of class C 1+~ f o r some ~ > 0 that may depend on [a, b], and the functions tQ',(t) are nondecreasing on (0, 1) and there are points 0 < c < d < 1 and an ~ > 0 such that dQ~n(d) >_ cQ~(c)+y for all n. Then every continuous function that vanishes outside (0, 1) can be uniformly approximated on [0, 1] by weighted polynomials wnnPn, degPn < n.

Section 8

Uniform approximation by weighted polynomials with varying weights

53

Being uniformly in C x+~ means that the derivatives satisfy uniformly a Lipshitz condition IQ'(x) - Q'~(y)I < CIx - yl',

x ~ [a, b], y ~ (0, 1)

with constants C and e = ea,b > 0 independent of z and y. Note that our assumptions require only C 1+~ smoothness on Qn inside (0, 1). We can conclude again that wnn Pn "-* f holds uniformly on some larger set [-0, 1 + 0], 19 > 0 (provided of course the weights are defined there). C o r o l l a r y 8.4 Suppose that {w,}, Wn = exp(-Qn) is a sequence of even weights such that the eztremal support Sw. is [-1, 1] for all n, and on [0, 1] the functions satisfy the conditions of the preceding theorem. Then every continuous function that vanishes outside (-1, 1) and also at the point 0 is the uniform limit on [-1, 1] of weighted polynomials w~Pn, deg Pn < n. What happens around 0 (i.e. what is the situation if the function to be approximated does not vanish at the origin) is quite complicated (see Theorems 12.2 and 12.3 for more details): i f w , ( x ) for all n is the Freud weight e x p ( - 7 ,1[(~ Ixl a ), c~ > 0, then clearly all the conditions of the Corollary are satisfied, but an(y) f that vanishes outside (-1, 1) but not at the origin is approximable by weighted polynomials w~,Pn if and only if a >_ 1 (see [30], and also the discussion in Section 12). We begin the proof of Theorem 8.3 by a lemma.

Lemma 8.5 Let w(z) -- exp(-Q(x)) be such that ,5to = [0, 1] and that tQ'(t) nondecreases on (0, 1). Then the density of the equilibrium measure dltw(t) = v(t)dt is given by 1 D 1 ~ fl sQ'(s)- tQ'(t) ds + (8.17) v(t) = -~V - 7 - Jo s -~ ~ / s ( 1 - s) ~ ' where ~2 ~011]~_SsQ,( s)ds, D = 1____1

and here D >_O. P r o o f . Let f(x) = Q(x2)/2, x c [-1,1]. It was shown in [28, Lemma 5.1] that the integral equation /_111og ~x-~g(t)dt = - f(x) + C where C is some constant has a solution g(Q of the form g(t)

2 X]/~'~__~2 [ 1 SI'(s) -- tf'(t) ds + D1 r-if'- J0 (1-s2)l/2(s ~ - t 2) 1 V ~ - - t~ ~ '

54

Section 8

Varying weights

where

Di

_ 1

1 fl 8f'(8) ,if2 J_ 1 ~ '

7r

furthermore g is even and has total integral 1 over [-1, 1]. If we set h(t) = g(v/t)/v/t, t E [0, 1] then h will have total integral 1 over [0, 1], and by the symmetry of g its potential satisfies

i

1

I

1

1

fo l~ l~- ulh(U)du = 2 fo l~ lx 1---t21g(t2)dt= 2 / - l l~ ix/~- t' g(t)dt = - 2 I ( V ~ + C = -Q(x) + C for every x E [0, 1]. On applying Lemma 5.1 we can conclude that dttw(t) = h(t)dt i.e. h(t) = v(t) provided we can show that h is nonnegative (to be more precise the equality d#w (t) = h(t)dt automatically follows from the properties above, but we shall need to prove the nonnegativity of D anyway, and this easily implies the nonnegativity of h). The nonnegativity of h will follow from the last statement of the lemma that we are going to show in a short while. In fact, we can see by integration by parts that the two constants D and Dt are the same, so the relation D > 0 is the same as Dt _> 0. Furthermore, the first term in the expression of g is nonnegative in view of the fact that sQ'(s) nondecreasing. Hence, the nonnegativity of D implies that of h, and this, as we have seen before, implies v(t) = h(t). Now if we carry out the substitution f(s) = Q(s2)/2 and u = s 2 in the formula for g, we obtain the form of v stated in the lemma. Thus, it has left to show that D > 0, i.e.

-1~01 7r l~__ss Q ' (s)ds < 1.

(8.18)

First we show that for every t E (0, 1)

(8.19)

si/2(i-1

~1 ~oilog ~

s)312ds = i.

Weshall do this by examiningthe integral 1 [,i 1,=~.u

1- t log ~

where * indicates that we skip a small during integration. If

X,,t(u) =

1 si/2(1- s ) a / 2 d s = 1 ,

(t - e, t + e) C_ (0, 1) neighborhood of t

(u -- t) -1 0

if It - u] > e

otherwise,

then

9 log ~1 - t

= f l X,,t(u)du

Section 8

Uniform approximation by weighted polynomials with varying weights

for all s which belongs to the range of integration in parts yields

I~,

55

hence integration by

As e ~ 0 this tends to the principal value integral

1dp v ~s I ' s -sTt ~r Unsing now that

s/(s - t) = 1 + t/(s - t)

(8.20)

PV

and that

f01s - 1- t

= 0,

for all t E (0, 1) (see [39, p. 251, (88.9)]) we obtain (8.19). Since in (8.19) the integrand is negative to the left of 2t - 1 and positive on (2t - 1, 1), we obtain for all a < 1 and t E (0, 1) the inequality 1 21r

log

~

s1/2(1 -

s)a/ids < 1. -

Here the integrand is bounded from below by the function Ca T

min

O, log

,

Ca = (1 -

a) -3/2,

integrable with respect to the product measure ds dtzw(t), hence we can integrate this inequality with respect to dlzw(t), and we can change the order of integration. Taking into account that the potential of p~ equals - Q + const on (0, 1) (see Theorem h in Section 1), we obtain that

1 f0 a sl/2(1 Q(1)-Q(s)ds_ _ s)3/2 < 1. Our assumptions imply that Q is monotone in a left neighborhood of 1 (tQ'(t) is either positive or negative in such a neighborhood), so by the monotone convergence theorem we can conclude from the preceding inequality the same f o r a = 1:

1 f l Q(1)-Q(s)

J0 sl/2(1 _ s)a/2ds_<

1,

and from here (8.18) follows by integration by parts (verify this separately for the cases when tQ~(t) is negative or positive, respectively in a left neighborhood of 1).

]

After that we can turn to the

56

Varying weights

Section 8

P r o o f o f T h e o r e m 8.3. All we have to do is to show that the conditions of Theorem 8.1 are satisfied with O = (0, 1). Under the conditions of Theorem 8.3 the functions sQ~n(s) uniformly belong to Lipc for some e on every compact subinterval of [a, b] C (0, 1), and it is well known from the theory of the singular integrals with Cauchy kernels (see e.g. the Plemelj-Privalov theorem in [39, p. 46]) that then the same is true of the integrals in (8.17) (with Q replaced by Q,, of course). Hence, the uniform equicontinuity of the densities vn on compact subsets of (0, 1) has been established. It has remained to show that they are uniformly bounded away from zero on every compact subset of (0, 1). Since the second terms in (8.17) are nonnegative by the last statement of the lemma, it is enough to show that even the first terms in (8.17) (again with Q replaced by Qn) remain above a positive constant on (0, 1). We set g(t) = tQ~n(t), and it is enough to show that if g is nondecreasing on (0, 1) and g(d) > g(c) + rl, then for all t e (0, 1)

~01 g(s) - g(t) 1 s -t ~ d s

> O,

where 0 > 0 depends only on c, d and 7. In fact, it is enough to prove this for continuously differentiable g, in which case the claim follows from the fact that

1 g(s) - g(0

1

f01 g(s) -

(0 ds,

and here

1 g(s) - ~ ( t ) g s

f0

8

=

~oI~ 1 ~t g'(u)duds

~f~g'(u)Iogt -~ud~ + ~1E(u)Iogu-tdu l -

=

_> ~

f

t

g'(u)du = ~(g(d) - g(~)) ~ ~,7

with a = min (log 11- ~ _c, log 1 } 9

| The Corollary immediately follows from Theorems 8.1 and 8.3 (applied to the interval [-1, 1] rather than to [0, 1] and to the set O = ( - 1 , 0) U (0, 1)) if we use the substitution applied in the proof of Lemma 8.5.

Modification of the method

Section 9

9

57

M o d i f i c a t i o n of t h e m e t h o d

In this section, we modify the method that we used above. This modification allows us to get better aproximation around the endpoints of the interval. These are needed if we want to handle infinite singularities inside the support of the generating measure, or if the approximation has a second constraint that frequenly appears in applications (see Theorem 10.1 and 10.2 below). The modification is roughly as follows: In the first two parts of this work we approximated a given potential by first translating the generating measure and then appropriately discretizing it. The idea here is similar, but instead of translation, that can be viewed as projection onto a segment, we shall project it onto a curve, which is closer to the support exactly where the generating measure is larger. We start with a technical lemma. L e m m a 9.1 Suppose that {Un} is a sequence ofnonnegative functions on (0, 1) satisfying the following conditions: {un} is uniformly equicontinuous on every compact subinterval of (0, 1), 01 Un -=-

1,

and for some constants A > O,/3 > - 1 and ~o

(9.1)

un(t) < A(t(1 - t)) ~,

(9.2)

Un(t) > A(t(1 - t ) ) p~

t E (0, 1),

1

t E (0, 1).

Then there is an Lo > 1 such that for every L > Lo there are polynomials Qn of degree at most n such that for large n, say n >_ nz

(9.3)

o _< loglQn(x)l+nU ~

"(x)

BL21ogn BL 2 log 1/(x(1 - x))

if x e [0, n -1] U if n - l < x < l - n

[1-

n -1, 1] -1,

and with some continuous functions gn that are uniformly bounded and uniformly equicontinuous on compact subsets of (0, 1)

(9.4)

Ilog [Qn(x)l + nUU"(x) - Lgn(x)l <_ B L - 1

if L -6 < x < L -6.

Here B is an absolute constant depending only on A,fl and flo. The same conclusion holds if we assume the inequality in (9.2) only on the interval [n - r , 1 - nr].

The lemma is true in more general situations (allowing several intervals or zero or infinite singularities in the weight), we shall comment on that after the proof.

58

V a r y i n g weights

Section 9

P r o o f . We shall only concentrate on the behavior on the interval [0, 1/2], the other half being symmetric. We shall indicate at the very end of the proof the necessary modifications we have to make to cover the whole interval [0, 1]. Without loss of generality we can assume that ~ < 0 and ~0 > 0. Fix a constant L. We select a continuously differentiable function Vn that has the same size as un on [L -9, 1 - L-9], as follows: for x 9 [L - t ~ 1 - L -1~ we set 1

(9.5)

[~+dL -J~-dL un(t) dr,

vn(x) = ~

where the constant dL satisfies 1 (9.6) ~ < Un(tl)/Un(t2) < 2

for tl,t2 9 [L -11, 1 - L-11], It1 - t~ I < dr.

for all n (such a dL exists because the functions Un are uniformly equcontinuous and uniformly bounded below by a positive constant on compact subsets of (0, 1)). Let 0 < r0 < 1/(~0 + 1) if (9.2) is true for all t 9 (0, 1), and 0 < r0 < min{r, 1/(/~0 + 1)} if we assume (9.2) only for t 9 In -y, n r] (in the former case we may set r = o~). For an n let us choose consecutive intervals Ij, j = 0, 1 , . . . , n - 1 starting at 0 such that on Ij the function un has integral equal to l/n:

I j

'Un ~

1

~. n

As in Section 2 let (j be the weight point of Un on Ij :

Let j = 0, 1 , . . . , N be those indices for which Ij C_ [0, L-~]. Set now

llil (9.7)

mj =

flj un(t)/v,(t) dt

ifj = 0,1,...,N if j > N.

We claim that the polynomials n--1

Qn(x) = H ( x - ~ j

+iLmj)

j=0 satisfy all the requirements of the lemma. Before we embark on the proof, we make a few preliminary observations. In doing so let us agree that in what follows c, C denote positive absolute constants depending exclusively on A, ~ and ~0. However, we allow C and c to change from line to line. As before, the symbol R ,., S will indicate that c < R / S < C with some such c, C.

Modification of the method

Section 9

59

It follows from assumption (9.1) that (9.8)

~j >_c(J--~l) 1/(1+t3),

and for all j (9.9)

,Ij,>c(j+l)-MO+t~)n-1/(l+'),

mj -~ IIj I,

0 _< j < n.

(9.2) gives

(9.10)

m.axll#l<

Cmax{n-', n -1/(z~

<

C n -r~

3

regadless if (9.2) holds for all t or only for t E [n-r, 1 - n-r]. The choice of v0 implies via (9.10) that ifz E [n-r~ 1/2] and I is an interval of length _> z/8 that intersects (x/2, 2x), then ~dn > - - . n

This means that there is at least two-two lj lying stricly in the intervals (x/2, x) and (x, 2x). For the length of every such an Ij we obtain from (9.1) and (9.2)

(9.11)

A(4X)-a/n -< Ibl < A(x/4)-a~

1~ c

(x/2, 2x).

Finally, we mention the formula (9.12)

logtQn(x)l+U""(x)=~n ~j

log

1=0

x - (j -+ iLmj un(t)dt.

I

z

t

t

T h e lower e s t i m a t e

First we prove that log IQ,(x)l + u~"(x) > o. In view of (9.12) it is enough to show that

n f , l~

+ iLm~ lu"(t)dt >

for all x and j. Let x EIjo. If j = j0, then the quantity under the logarithm is at least as large as

L rni >_Ll-~il>_l,

Ig-:--/-t

for sufficiently large L, where we used (9.9). If, however, j r j0, then the quantity under the logarithm is at least as large as

60

Varying weights

5'ection 9

and by the concavity of the logarithm function log Ix - t I for n[

t EIj

we get

< logix

logix-tiu,(t)dt

J1

These inequalities prove the lower estimate. The upper estimate We distinguish two cases according to the location of x. Case I. x E [0, n-r~ ranges:

Let x E Ijo. We divide the sum in (9.12) into three jo-l-1

Z + Z + Z =Zl+Z2+Z3"

j<jo-1 j=j0-1 j>jo+l We estimate these three sums separately. In ~ 2 we have (at most) three terms. The second one is at most as large as n f/Jo log

x-~J;+-iLm~~

_ ftjo log I]Ij~ ~iC~[IJ~

< llog((l+C2L2)lljol)+n

I

log-ff~_tlu,(t)dt. 1 Jo

We break the last integral into two parts, for integration over Ix - t[ < n -(1+0) and the rest. In view of (9.1) we can get the following upper bound for this integral: n [n-(l+~) J0

(logl) at, dt + log nO+a)

<_Cntl+alt=_(l+.)

logn (1+#) + (1 + fl) logn < Clog n.

This shows that the second term is at most Clog Ln, and similar estimates hold for the first and third terms. Thus, altogether we have Z 2 < C log Ln

< CL log n.

Now let us consider ~ a . For j > j0 we set

& = I/Jo+ll + " " + Ibl. Then for t EIj we get from (9.9)

l~

log I& + iCLIZjlt Aj-1 log~+~log

I+C2L

2 . .

Section 9

61

Modification of the method

Thus, E3 <

E

I~

I~1 ~

E

j>j0+l

-

A-'7"."-- E 3 1

j>jo+l

§ E32"

Here the first sum is 1 An-1 - - ' - _< C log n. E 3 1 = mgAjo+.---~ < log 7~0+1 9

(9.13)

For the second one we get (9.14) E 3 2 -< CL2

y.

[Ij [

<_CL 2

"-' E.~o5o+1

- dt < CL 2 log n. Jo+t[ t

-

These prove E a < CL2 log n. The estimate of ~ 1 is completely analogous if we set for a j < j0 - 1 Aj = IIj[ + . . . + II~o-xl, and use for t E I j log

x-~j+iLrnjl IAi+iCLII~II x--t _< log Aj+I

Thus, we obtain in this case the upper bound CL 2 logn in (9.3). Case II. x e [n - ' o , 1/2]. Let again x EIjo. Let M0 be the smallest index j for which 1i g (0, x/2) and ml the largest one for which Ij g (2x, 1]. We split the sum in (9.12) into three ranges: M1 - 1

E § E § j>>_Mt E --El+E

j<_Mo j = M o + l

§

The estimation of the first and third terms is completely analogous to the estimate of )-~a in Case I above. In fact, consider e.g. ~3. Proceeding as in Case I we get IX~l 2

E a < E log A, +CL2 E - - = E a l + E a 2 " j>M1 Aj-I ./_>M~ A~ Now the analogue of (9.13) is E81

=log

11

_
62

Section 9

Varying weights

while that of (9.14) is

CL2j>_M1

<

IIjl

< CL 2

IZ, I

1 dt < CL 2 log 2. AM 1-I

t

-

X

These prove

Z

< CL 2 log 1

3 --

X'

and the estimate of ~ 1 is analogous. Now let us consider ~ 2 - We write with 5 = 2(fl - fl0)

where the range of the summations in ~21 is j = j0 5: 1, in ~22 it is 1 < [j - j 0 [ <_ x - t , while in ~ 2 3 it is x -6 _< [ i - J 0 [ , and recall that in every case we have Ij C_ (x/2, 2x). In )-~22 we can utilize once more the technique of (9.13) and (9.14). In fact, if M2 is the largest integer that is smaller than x -6, then the part ~ of ~2~ which corresponds to the indices J0 + 1 < j < j0 + M2 can be estimated with the method of (9.13) and (9.14) as

Z;3

<

--

< -

log A i ~

+ CL 2

Aj~

/ AjO+M~ 1 JII~o+~l

CL 2 /a~~

- dt t

l dt = CL21ogA, o+M~

JlZJo+~l

Ajo+l

Here we can use (9.11) according to which

Ajo+M 2 < M~A(xl4)-#~

< A(xl4)-aox-61n

and Ajo+l > A ( 4 X ) - a / n , These imply via the preceding formula

Z~2 < CL 2 log 1 3--

X

The other part of ~2~ (corresponding to the indices jo - M2 g j < jo - 1) can be similarly handled and we arrive at

Z

< CL 2 log 1. 23--

X

In ~'~21 we have three terms. As before, we get for the middle one

o

x - - t

--

e

--

Section 9

63

Modification of t h e m e t h o d

+ c2L )) + "/,,0 log

dt

Since th~ integral of u. on the interval 5 is 1/n, there must be a point tl 6 5 such that un(tl) < 1/nllil. But for any t 9 5 , 5 C (z/2,2z), the ratio u,,(t)/un(tl) is less than CA2z/3-~o (see (9.1)-(9.2)), hence for any t 9 5 we have nun(t) <_ CA2x~-~~ Thus, if we break the last integral into two parts, for integration over Ix - t ] < ]5Ix ~ and the rest, then we can deduce the following upper bound for this integral: 5 log 1 + CA2x~-~~ L

log I/S~ dt <

5 log 1 + CA2x~_~ox~ log 1. X

X

This shows that the second term in ~21 is at most CL 2 log 1/x (recall the choice ofh: 5 = 2(/30 -/3) >/30 -/3), and similar estimates hold for the first and third terms. Thus, altogether we have

E

< CL 21ogl.

21--

X

Thus it has left to deal with the sum ~23- For the j's in ~23 we shall make use the inequality: if y _> - 1 / 4 or if 10[ _> [~[, then

Ilog]l+y+iO[-~l=lllog(l+2rt+O2+02)- ~ _<2(72+02).

(9.16)

When t e 5 then It - ~SI <_151 <- Lms, and so by the preceding inequality log

-

+iLrn s

--~

=log 1+

t-

+iLm s

--~

t-~s

< x - t +2

[t-~s[2+ 2 2

(~_t)2

The first term on the right is

t-~s+

(t-~s)2

t-~j

= x - ~j + O ( l I J [ 2 / d i s t ( x '

5)2).

Thus, if we multiply the last but one inequality with nun(t) and integrate on 5, then by taking into account the fact that the integral of the term

~-:-~ un(t) vanishes because of the choice of ~i, we finally get that

E23 <- C L 2

E

I/-io[>M2, /jC(~/2,2x)

1512 dist(~, 5)2"

64

Section 9

Varying weights

Taking into account that by (9.11) here

ISsl < A(xl4)-Z~ while dist(x, Ij) >_Aj-1 > (j - jo - 1)A(4X)-~/n, we can continue the last estimate as

Z23

<- CL2A2x2(~-[3~

1

E

(j _ jo)2 < CL2A2z2(~-~~

]j-Jol>M~

< CL2A2x2(~-~o)x ~ = CL ~ by the choice of 6 = 2(~0 - ~). The inequalities that we have proved so far verify the upper bound in (9.3) for x > n-r~ With this the proof of the upper estimate in (9.3) is complete. The asymptotic

estimate

Let x >_ L -6. We are going to verify (9.4). Let j = 0 , . . . , N1 - 1 be those indices for which

Ij C [0, L-s], and let I be the union of the rest of the Ij 's: I=

-1 Ujn =NI lj.

Let u~ be the restriction of un to I, and similarly, let n-1

Q*(x) = H (x - ~j + iLmj), j =N1 i.e. in Q* we only keep those terms from Qn that correspond to indices for which Ij is lying in I. First we remark that <_ (log [Q.(x)[ + nU""(x)) - (log [Q:(x) + nUn:(x)) N1 - 1

--

E

[

1=o JIj

log z - ~ j +iLmj

I

z---t

un(t) dr.

Here the terms under the integral sign are bounded by l o g ] l + t - ~ j +iLrnj l < t - ~ j + C ( 1 + L2)IIjl 2 < t - 5 ~ z t [ - x-t dist(z, Ij) 2 I

+ c (1 + LU)lI./I 2 dist(x, Ij) 2 '

Section 9

Modification of the method

65

where we used (9.16). The integral against un(t) dt of the first term on the right vanishes again because of the choice of ~j, hence, (9.17)

0 __5(log IQ.(x)] +

iI~1 z dist(x, 1i)2

Nt-1

< cn2 E

i=o

nU~'(z)) - (log ]Q:(z)] + nU":(z)) 21X, !

< eL 2

\

i=o

<-- C L 2

I/

,

<

uniformly in z E [L-6, 1/2]. Thus, it has left to consider log IQ:(x)[ +

nUU:(x).

Fix an zo E [L -a, 1/2] and consider the functions

u.(~)=/logz-t+

~.~.(t)~.(t)dt

and

iL l ~.(t) at. nv.(~0) Seeing that now z0 is fixed and the weights un are uniformly equicontinous, Un,.o(x) = ~ log [x _ t +

the argument applied in Section 2 gives that

uniformly in x E [L -6, 1/2], and it is also easy to see that this limit relation uniformly holds also in x0 E [L -6, I/2]. In particular, we have

(9.18)

n

(Un,xo(xo) + UU~,(xo)) -- I rTLun(xo),. --(l + o(1)) v.(~0)

uniformly in x0 E [L -a, 1/2]. Now let us consider

iL

vn(t) -1 - vn(xo) -1

~(U~(~o)- U~,~o(~O))= n/ 'ogl1+ "C~o: ~ - ~ ~ _ ~ Since here

v"(t)- 1 - v.(~o)-I

independently of x0 E[L -a, 1/2] and n, for large enough n we get

nlogli + iL

vn(t)-l-vn(xo)-I

n ~o--~7~v.~-1

[

<- 2KL.

l~.(t)at.

66

Varying weights

Section 9

This and the limit

iL l i m . l o g 1+ n

vn(t) -1 - vn(x0) -1 I 0775n -11 = ~

imply via the dominated convergence theorem that lim n(Un(zo)

-

Un,.o(XO))

= 0

n --* o0

uniformly in xo 9 [L-s, 1/2]. Thus, altogether we get from the last limit and (9.18) that

n (Un(x) + U~'*(z)) = r L ~ ( l + o ( 1 ) )

(9.19)

uniformly in x 9 [L-6, 1/2]. Note that the functions

(9.20)

gn( )

vn(x)

on the right are lying in between 7r/2 and 27r for n > nz (see (9.6) and the definition of vn in (9.5)), and they are uniformly equicontinuous on every closed subinterval of (0, 1) (in particular, on [L-6, 1/2]) by our assumption concerning the functions un. From our considerations it will be clear that these are the functions gn in the lemma. Next we estimate the sum

(9.21)

log IQ ,(x)I-

for x E [L-s, 1/2]. Let x 9 Ijo. Then j0 is a large index (larger than const-n) in view of the fact that by (9.2) we have vn(t) >_ L -s~~ if 9 [L-7, L-s], hence there are at least cLn, CL > 0 intervals Ij lying in 9 [L-7, L-S]. We can write this difference in the form

'l~

j~nftJ

-

+ \j=N1

Z i=jo-L+l

+

Z

/ n

j=jo+L]

l~

-- ~J + iLmj

--_-~i~i--~mj

log

Un(t) dt I

un(t)

Z

First we estimate )-~2. The definition of mj (see (9.7)) shows that for the j's appearing in the sum ~'~2 there is a ~j E I j such that Vn((j) -1 = nmj, and

Section 9

67

Modification of the method

this easily implies that for t 6 Ij (9.22)

rnjl

Ivn(t)-~n -1 -

=

!lvn(t)-l-v(o)-ll n

<

cl!jl Iv'n(,%)l < CL2Z"

-

n

vn(~j)

2

-

n

Thus

E2I. <- 2(L + 1)1_log IIi 12 + L2rny(1LZm~ + O(n-1)) < ~'C where we have also used that by (9.9) mj ,.~ lit I. The sum I)-~3[ can be written in the form n[

log 1

t-~3' + iL(v"(t) - l n - 1 -

mj)

dt

We would like to bring the estimate (9.16) into play, so first we have to show the lower bound - 1 / 4 for the real part of the ratio standing after 1:

~%t - ~i + iL(vn(t) - l n - 1 - mi ) x - ~j + iLmj

(t-~j)(z-~jz)+L2(v.(t)-ln-X-rni)mj

IZil

>

=

---~J)2+L2rn]

-

~j-x

c=

> 1

n--5'

provided L is sufficiently large (depending on A and/3), and n is sufficiently large depending on L. At the last but one step we used (9.22) and at the last step we used that ~j 6 Ij with index j > j0 + L, hence for large L we have -IIi[/(~j - z) > - 1 / 8 by the uniform equicontinuity of the functions u, on compact subsets of (0, 1). Thus, we can apply (9.16) to the integrands in the sums in )~3 and we obtain

~_

~'~ j =j

~t-~i +iL(vn(O-ln -x -m~),,n(oe t j

Z - ~j + iLmj

--1

ilil ~ +

< 2 ~

CLL2(mj/n)2

( x - ~ ) 2 + L2m~ '

j=jo+L

where we have used again (9.22). By the definition of the numbers ~j and mj each integral

f z ~tt -~J + i L ( v n ( t l - l n -1 - m j l u.(t) d ~ i z - ~j + iLrnj =

(~ (

,

- ~j + i L m i

1

z-~j + i L m /

)/,

(t - ~ i ) ' ~ . ( O dt -

)/(vn(t)-'n-'-rn,)Un(t)dt ati

68

Varyingweights

Section 9

vanishes, hence we can continue the above inequality as

~-'~31 <- 2 y ~

11112+CLL21IjIgn-2

1I~12 Ij) ~

< 3j>jo+LEdist(z,

dist(z, [j)2

j>_jo+L

if n is sufficiently large. Now we write the last sum as ~31 + Z:32 where in Y~31 we sum for all indices j _> j0 + L for which I t C_ (x, x + dL) with the constant dL from (9.6). The choice of dL yields for Ij, It C_ (x, x + dL) 1

_< Ibl/ISzl _< 2,

hence

OO

E

1

jo+L (j --

c

j~

<- T"

For the second sum we get from (9.10)

~ s ~ < m~x~ II~l ~ l I j l < C"-~~ if

d~

~ < L -1

-

-

n is sufficiently large. The estimate of ]~11 is similar:

ilil 2 + CLL21I~I2n_ ~

jo-L

E1

-< 2 E

dist(z, Ij)2

j=N,

<

C

E j <_jo- L

_< 3 E

j=N,

1 maxj Iljl j0)------------~ + E

(j

d2L

--

i/il 2

jo-L

dist(z, ij)2

I/Jl < c L'x. -

Our estimates from (9.21) show that for n > nL (9.23)

C I log IQ~,(x)I - nUn(x)l < T

for x 6 [L -6, 1/2] with a constant C independent of L provided L is sufficiently large, say L _> L0. Finally, from (9.17), (9.19) and (9.23) we obtain (9.4) in our lemma on the interval [L -6, 1/2] with the functions (9.20). It is obvious what modifications we have to make to cover the interval [1/2, 1], as well. Practically the only necessary change is to choose now mj = Ibl

if Ij C_ [ O , L - 9 ] U [ 1 - L-9,1],

otherwise let

mj = fs u.(t)/vn(t)dt. i

Section 9

Modification of the m e t h o d

With this change the proof goes over with trivial modifications.

69

|

Closer inspection of the above proof shows that the estimates in Lemma 9.1 hold in larger ranges: 0 < log IQn(x)l + nU"'(x)

(9.24)

<-

BL logn if dist(x, {0, 1}) <_ n - r BL 2 log (1 + 1/dist(x, {0, 1})) for all other z,

and with some continuous functions gn that are uniformly bounded and uniformly equicontinuous on every compact subset of (0, 1) and vanish outside [0, 1] (9.25) Ilog IQn(x)l + nUU"(x) - Lgn(z)l _< s n -1 if dist(x, {0, 1}) > L -6. This remark allows us to extend i e m m a 9.1 to several intervals or to have zero or infinite singularities of the weights Un inside their support. In fact, let us assume that (ak, bk) are finitely many disjoint intervals with possibly common endpoints. We set Z0 = tg(ak,bk) and X = {ak, bk}. Suppose that {Un} is a sequence of nonnegative functions on G0 satisfying the following conditions: {Un } is uniformly equicontinuous on every compact subinterval of G0,

/~3 un = l, 0

and otherwise on every (ak, bk) the functions Un satisfy inequalities that are analogous to (9.1)--(9.2) with some ~j, fl0j, A and r > 0. Then there is an L0 > 1 such that for every L > L0 there are polynomials Qn of degree at most n such that for large n, say n > nL (9.26)

-C

_< l o g l Q n ( z ) l + n U - - ( z )

<-

BL21ogn BL 2 log (1 + 1/dist(x, X))

if dist(x,X) < n -1 for all other x,

and with some continuous functions gn that are uniformly bounded and uniformly equicontinuous on every compact subset of G0 and vanish outside the closure of G0, Ilog IQn(X)[ + nUU"(x) -- Lgn(x)[ < BL -1

if dist(x, X) > L -8.

Here B is an absolute constant depending only on the parameters in the assumptions on the un's. The proof can be based on Lemma 9.1. In fact, if for an n the function un has integral of the form lk,n/n with some integer ik,n on each (ak, bk), then we can construct the polynomials in Lemma 9.1 for the weight function un](ak,~ bk) with the modification that now the degree has to be lk,n (to match the integral of nun [l(ak, bk)). Then by the above remarks the product

70

Varying weights

S e c t i o n 10

of these polynomials will be appropriate (actually with C = 0 on the left of (9.26)). In general, however, the integral ak,n := f unl(ak,bk) is not of the form lk,n/n. In that case let us fix for each k a nonnegative continuous function sk with compact support in (ak, bk) and with integrM 1, and consider the weights u:,(t)

:=

Un(t) - ~

~ , n -- ~

k>l

\k>l

These satisfy the assumptions that their integral on any (ak, bk) is of the form lk,n/n, so for these there are polynomials Qn with the required properties. Finally, all we have to mention is that the functions n(u~.(~)

-

uu:(~))

are uniformly bounded and equicontinuos on the real line. Hence the above

Qn's are suitable for the un's, as well (this is where we need possibly zonzero C on the left of (9.26)). We could have even allowed the set Z0 vary with n, but we do not pursue this direction any further.

10

Approximation

in geometric

means

In this section we prove two theorems that are in some sense refinements of the theorems in Part 2 in the special case considered here. T h e o r e m 10.1 Suppose that {Vn} is a sequence of nonnegative functions on (0, 1) satisfying the following conditions: {Vn} is uniformly equiconlinuous on every compact subinterval of (0, 1), ol vn =

I,

and for some constants A, ~ > -I and ~o (10.1)

Vn(t) <_A(t(1 - t)) ~,

(10.2)

Un(t) > A(t(1 - t ) ) #~

t e (0,1), te (0,1),

and set

wn(x) := exp(U'"(x)).

Approximation in geometric means

Section 10

71

If 1 > 7 >__0 and u(x) is any positive continuous function on [0, 1], then there are polynomials Hn of degree at most n such that for

(10.3)

hn(x)

w

=

"(

)lHn(x)l(x(1

-

we have hn(x) > 1,

(10.4)

x e (0, 1),

lim hn(x) = 1 . - - - * OO

uniformly on compact subsets of (0, 1), and

(10.5)

lirn

L 1 log hn(x) ~ - ~ - - 7 ) dx = O.

It will be of utmost importance that the degree of I-In can be somewhat smaller than n, namely we can have deg(Hn) = n - i , where i , ~ oo. This will follow from the proof (with in = n * for some ~ > 0). Here, exactly as in Lemma 9.1, we actually need the inequality in (10.2) only for t E [n - r , 1 - n-r]. Proof.

For some p > 0 let An = l + n -p,

and consider the weights W.

:-- W n

and the corresponding weighted energy problem of Section 4.1 with extremal measure/z* and ,S* = supp(/~). We shall again concentrate on the behavior around the left endpoint x = 0. Fix a Co and set an = Con -p/(a~ Since for x E [anl2, 112] we have on the interval [x/2, 3x/2] the inequality

vn (y) >

1

o

and since A

1 vaox /~0

"-'0 >- A.8/~o+ln p'

we get from Lemma 5.8 that for large enough Co the interval [a,/2, 1/2] is part of 8~. Let Jn = [an~2, 3/4]. Lemma 5.7 can be applied to deduce (10.6)

dl~(t) < AnV,(t) dt < 2At ~ dt

for all t E (0, 1/2]. In particular, each/J* is absolutely continuous. By Lemma 5.7 we also get that dtt*(t) > AnVn d t - (An - 1)d/~l~(t),

t ~

[a.12, 3/4],

72

Varying weights

Section 10

where the last measure 1 d p j , (t) = -

1

f(t

-

~nl2)(314

-

t)

dt

on the right is the equilibrium measure of the interval Jn. Thus, on [an, 1/2] we get 1 (10.7) (p~(t))' _> vn(t) - n-P(t - an~2) -I/2 >_ ~vn(t) for large enough n and t _> Con-./(t+~o) which follows from (10.2) and the choice of the numbers an. Lemma 5.7 together with the assumptions on vn also show that the functions (10.8)

Utn/~.J(t ) := (,~(t))'

are uniformly equicontinuous on compact subintervals of (0, 1/2]. Thus, in summary we can say that {un} satisfy on the interval (0, 1/2] the assumptions of Lemma 9.1 with r = p/2(1+/30) where we consider (9.2) only for t 9 [n -r, 1 - n -r] (here the factor 1/2 is put into the expression o f t = p / 2 ( l + ~ 0 ) to get rid of the constant Co). The same analysis can be done on the interval [1/2, 1), and we can conclude that the densities {un} from (10.8) satisfy all the assumptions of Lemma 9.1 (with r = p/2(1 + ~0)) for large enough n. On applying Lemma 9.1 to u[n/x,] and to In~An] rather than to n we get that for every L > L0 there is a sequence of polynomials Q[n/X,] of degree at most [n/)~n] such that for sufficiently large n (n >_ nL) we have

(10.9) for every z ~ (0, (10.10)

wn(x)Xdnl~.]lQtn/x.](x)l >_ 1 1), wn (z) x'[n/x'] lQ[n/x,] (x) l = e Lg'@)+~

for L -6 < x < 1 - L -6,

(10.11)

( 1 "~O(L2) wn(z)XdnlXdlQ[nlX"](x)[ = ~,x(1 [- z ) ]

forn -r
-6orn -r_l-x_
(10.12)

-6,and

wn(x)xdn/X"llQtn/x.l(x)l = nO(L2)

for 0 < x < n - r or 0 _< 1 - z < n - r , where the 0 is uniform in L and n >_ n L and the functions gn >_ 0 are uniformly bounded and uniformly equicontinuous on compact subsets of (0, 1). The weights w~ "In/x"] and w~ differ only in a multiplicative factor w~", 0 _< 6n := n - An[n/An] _< 1 which are also unifomly bounded on (0, 1) (c.f.

73

Approximation in geometric means

Section I 0

the assumptions of the theorem concerning vn) and uniformly equicontinuous on compact subsets of (0, 1). It is now easy to find for every q > 0 polynomials R~-[n/x4 of degree at most n - In/An] such that for sufficiently large n the following estimates hold with some absolute constant CL independent of q and n: lim .--~

R~_[n/A4(z)eLa~(')w~'(z)=(z(lu(x)z)) ~

uniformly on [L -s + q, I - L -s - q],

1 < P~_[n/A4(z)eLg~(X)w~(z)< CL on the intervals [L -6, L -s + q] and [1 - L -6 - q, 1 - L-6], and 1 < R~_t./~4(x ) < 2

on [0, L -6] and on [1 - L -6, 1]. In fact, the functions

{ e-S-'g"(=)w~''(x)(x(1 r.(~) -

-

x))'~lu(x)

i f x e [ L - 6 + q, 1 - L - 6 - q]

1

if x E [0, L -6] U [1 - L -6, 1]

linear

on [L -e, L -6 + q] and on [1 - L - 6 - q, 1 - L - 6 ]

form a compact subset of C[O, 1] (they are uniformly bounded and uniformly equicontinuous in n), hence for every e > 0 there is an m such that for every n there are polynomials R~,n of degree at most m such that era,. := Ilrn

-

R~,,,nllv[0,1i < e.

This means that if we choose Rn-[n/x:] as * X R.-tn/~4(~) := Rn-tn/~.],.( ) + ~n-tn/~4,. +

e -CI/L,

with some appropriate but fixed C1 (that depends only on the constants in the preceding inequalities), then (by n - [n/A,~] > nP/2) this choice will satisfy the above inequalities. Thus, althogether we get for the polynomials

gn(x) = QtnI~,,](i)R.._tnI~.](x) of degree at most n for every large n, say n > nL the estimates 1 < w."(x)lHn(x)l(z(1

- x))-~u(i) _< e c ' L - '

for L - 6 + q < x < 1 - L - 6 - q , 1 < w."(z)lHn(z)l(z(1 - x ) ) - ~ u ( r )

< C . CL

74

Varying weights

for L -6 < x < L - 6 + q and L -6 _< 1 - x

S e c t i o n 10

_< L -6 + q , 1

l <_ wn(x)lHn(x)[ <_ ('x"~l(1-__x) ) cL2

f o r n - r < x < L -6 a n d n - r <. 1 - x < L - 6 , a n d 1 < w,"(x)lnn(x)l < n CLa for 0 < x < n - r or 0 < 1 - x < n - r , where the constant C is independent of q and L and n > nL,q. Since these inequalities actually give estimates on hn (see (10.3)), and they show that hn > 1, all we have to verify that the geometric means in (10.5) are as small as we like. The preceding estimates give for this mean and for large n the inequality 0

<

[ 1 loghn(x) dx

-

Jo

<_ (2

X/x(1-x)

(

n-rl2L21ogn + L 2

/0

log 1/z _ _

~- q log

CL -]- L - 1).

Choosing now first L large, then q small, we can make the right hand side as small as we wish. With this the proof is complete.

|

Next we prove the following companion to Theorem 10.1. T h e o r e m 10.2 Suppose that {Vn} is a sequence of nonnegative functions on (0, 1) satisfying the conditions of the preceding theorem, and

Wn(X) := exp(UV"(x)). If 1/2 > 7 > 0 and u(x) is any positive continuous function on (0, 1), then there are polynomials Hn-x of degree at most n - 1 such that Hn-1 does not vanish on (0, 1), and for the function

(10.13) we have (10.14)

(10.15)

h,.,(x) = w~(x)IHn_I(x)X/'x-"~

_< 1,

- x)l(x(1

9

- x))-'ru(x)

(0, 1),

lim hn(z) = 1

n---* O0

uniformly on compact subsets of (0, 1), and

(10.16)

lirn f01 ~log - - Thn(x) l dX = O.

Note, that in the present case we need the factor X / ~ - x ) in hn when 7 > 0 in order to achieve (10.14). Again, we can have deg(Hn-1) = n - in where in --* oo (say in = n 6 with some & > 0).

Approximation in geometric means

S e c t i o n 10

75

P r o o f . Proceed as before in the proof of Theorem 10.1 until (10.9)--(10.12). Choose with a large D, to be specified below, the number N = DL 2, and consider Q[*/x,](x) := Q [ , / x ~ ] ( x ) ( x + 1 ) N ( 1 - x + 1) N 9 If D is large (depending only on the constants in (10.10)--(10.12)), then we can easily get from (10.10)--(10.12) that

w.(x)~t"/~"]lQ~,v~.](x)l <_ 1 for all x E [0, 1],

w~tn/~.](~)IQ~n/~=](.) I = f o r L -6 < x < 1 - L

~,:(=)+o(L-')

-6,

~.(,)~["/~=] for 0 < x < n -~ or 0 _< I -

IQ['./~=](*)I

> (,(1 - ,))~nL=

x _< n -~, where the 0 is uniform in L and

n

>_ n L ,

and the functions

g~(x) = Lg,(x) + N log((x + 1/n)(1 - x + l/n)) are uniformly bounded and uniformly equicontinuous on [L -6, 1 - L-s]. With the argument applied in the preceding proof we can find polynomials R , , - l _ [ , , / ~ . ] - 2N of degree at most , ~ - 1 - [n/)~n]- 2N such that for sufficiently large n the relations *

~

)imoo R~_[,V~,,](x)e".(~)wd'(x)Vx(1

x)

-

-

(x(1 - - x ) ) u(x)

7

uniformly on [L -6 4- q, 1 - L -6 - q], * 6 1 > R._t./~](~)~g-(~)w.~(~)

on [L -s, L -~ 4- q] and [1 - L -6 - q, 1 - L-~

_>

1 CL

and

1 _> R._[./~.](~) _>

1

on [0, L -6] and on [1 - L -6, 1], hold with some constant CL independent of n and q. We set

S,~-l(x) = Q[,~l:~,,](x)Rn_l_[n/:~,d_2N(x)e -c'/L, which has degree at most n. The estimates

1 >_ w~,(x)lHn_l(x)X/~ - x)l(x(1 - x))-'ru(x) >_ e -c/L

76

Varying weights

Section

10

for L-6 + q < x < 1 - L-6 - q, 1 > w , ~ ( z ) l g n - l ( z ) V / ~ ( q ' - x)l(z(1 - z ) ) - * u ( x ) > (x(1 - ~)) 2 D L 2 1 --

--

for L -6 < x < L

-6+qandL

-6_
eL

-6+q,

1 > wn(x)lHn_l(X)[(x(1 - x ) ) - T u ( x ) >_ (x(1 - x ) ) 2DL2 for 0 < x < L -6 and 0 _< 1 - x _< L -6 hold with some constant CL independent of q and n >_ nL, and some absolute constant C. These tell us first of all that hn < 1 (see (10.13)), and then that O_>f0

~logh,~(x) / x ( 1 - x) dx

-C

L2

~ l oag l / xx

, + qlogCL + L -1

.

Choosing first L large, then q small we can make the right hand side as small as we wish. The proof is complete.

|

In order to be able to apply Theorems 10.1 and 10.2 we need convenient criteria in terms of the weight w itself (note that these theorems refer to the density function). In the rest of this section we discuss what smoothness conditions on w ensure that the assumptions of Theorems 10.1 and 10.2 are satisfied. We shall do this with conditions similar to those in Theorem 8.3. T h e o r e m 10.3 Suppose that {Wn}, Wn = e x p ( - Q , ) is a sequence of weights such that the extremal support S, on is [0, 1] for all n, the functions tQ" (t) are uniformly of class C ~ on [0, 1] for some e > O. Suppose further that the functions tQ'n(t ) are nondecreasing on [0, 1] and there are points 0 < c < d < 1, and an

T1 > 0 such that dQ'( d) > cQ'(c)+rl for alln. Then the conditions of Theorems 10.1 and 10.2 are satisfied and their conclusions hold. For example, the conditions of this theorem are true if all t Q ' ( t ) coincide with a single C e function, say if Qn(t) = t a for an c~ > 0. P r o o f . We use the representation (8.17), and set gn(t) = tQ'n(t ). The uniform C ~ property of the gn's easily imply that the integrals

In(t) =

fo

gn(s)-gn(t) s

t

1

~ d s

are uniformly bounded by a constant multiply of (x(1 - x)) -1/2+'/2, hence condition (10.1) is true. We have already used in the proof of Theorem 8.3 that the uniform C e property of the gn's implies that of the integrals In (t) on compact subset of (0, 1) (see e.g. the Plemelj-Privalov theorem in [39, p. 46]), and this is more than

Approximation in geometric means

Section 10

77

the uniform equicontinuity of the densities un of the corresponding extremal measures/tw. Therefore, it has left to verify condition (10.2). But we have seen in the proof of Theorem 8.3 that the integrals In (t) are uniformly bounded from below by a positive constant, and this implies (10.2) via the formula (8.17).

|

We shall also use the following corollary (c.f. Theorems 10.1 and 10.2). C o r o l l a r y 10.4 Suppose that {Wn}, Wn = e x p ( - Q n ) weights such that the extremal support S,o, is [-1, 1] for functions satisfy the conditions of the preceding theorem. is any positive continuous function on [-1, 1], then there degree at most n such that for

is a sequence of even all n, and on [0, 1] the If 1 >__7 > 0 and u(x) are polynomials H,~ of

hn(x) = w~(x)lHn(x)l(1 - x2)-~u(x) we have (10.17)

hn(z) > 1,

x E ( - 1 , 1),

lim hn(x) = 1 n ---~ (:X3

uniformly on compact subsets of ( - 1 , 1), and

lira=~o I loghn(x) ~ dx =

O.

In a similar fashion, i f 7 <_ 1/2 then the conclusion also holds for some Hn-1 and h.(x)

= wa(x)ln._l(,)x/1

-

21(1 - , = ) - ' u ( , )

with (10.17) replaced by hn(z) < 1,

z e ( - 1 , 1).

Exactly as in the case of Theorems 10.1 and 10.2 the degree of Hn can be somewhat smaller than n, namely we can have deg(Hn) = n - in where in ~ oo. Proof. This corollary is an immediate consequence of Theorems 10.1 and 10.2 and the discussion made at the end of Section 9. In fact, by using the transformation z --* x 2 applied in the proof of Lemma 8.5 we can see that the assumptions of Lemma 9.1 hold true on ( - 1 , 0) and on (0, 1), and at the end of Section 9 we mentioned how to use these facts to conclude the statement of Lemma 9.1 for the union of these intervals. Now the proof of Theorems 10.1 and 10.2 were based on Lemma 9.1, hence the proof can be copied to yield the corollary.

|

Part IV

Applications In this chapter we shall briefly discuss some applications of our results. They are here for illustration. Some of these have been achieved in less generality by different authors using different techniques, we shall give proper reference at those places.

11

Fast decreasing polynomials

In this section, we shall discuss an application of the method developed in the preceding chapters. We call polynomials Pn, degP,, < n, fast decreasing on [-1, 1], if they attain the value 1 at x = 0 and decrease fast away from the origin: (11.1)

Pn(0) = 1,

IP,(x){ < e - ~

x E [-1, 11.

We shall discuss the problem with what 9 and n this is possible. The significance of such fast decreasing polynomials lies in the fact that they approximate the "Dirac delta function" as best as possible among polynomials of a given degree, hence for example these are the best polynomial kernels for convolution operators to reproduce the identity. By integration we can get from the above polynomials good polynomial approximants Sn of the signum function in the sense (11.2) [signz - Sn(z)[ _< e ~ z e [-1, 1], which in turn can be used to construct well localized "partitions of unity" (c.f. the construction on p. 156 in [14]) consisting of polynomials of a given degree n,

The problem can be formulated in two different ways: one can ask what possible decrease (i.e. what O) is possible for a given degree, or, alternatively, for a given 9 what is the smallest degree n for which there are polynomials with properties (11.1). Let no denote this degree. For symmetric O's which are increasing on [0, 1] this problem was completely solved up to a constant in [15]: Let 9 be an even function, right continuous and increasing on [0, 1]. Then 6 N o _< n| _< 12No, where No = 0 i f 0 ( 1 ) < 0, and No = 2

sup

O-a(O)_<x
~) X~

+

b = min(O-l(1), 1/2), otherwise.

[112 O(x) dx + x2

Jb

r

sup 112<_x<1- log(1 - x)

+ 1,

80

Applications

S e c t i o n 11

This estimate is given for all r in particular, r can depend on n. As an immediate corollary we get that if ~ is even and increasing on [0, 1], then there are polynomials Pn of degree at most n satisfying (11.3)

Pn(O) = 1,

IPn(x)[ _< Ce -~nv(~),

x E [-1, 1], n = 0, 1,...

for some constants C, c > 0 if and only if

f01

~(u) u2

du

< oo.

In [49] a potential theoretical method was developed for obtaining sharp results for the largest possible c in (11.3). It was shown there that if ~ is even, increasing on [0, 1], and ~(v/~) is concave on [0, 1], then there are polynomials satisfying (11.3) only if c2 t 2 ~_ t 2

Ir J0

,it < i -

holds, and if we have here strict inequality then such polynomials do exist. This can be applied to any ~(t) = cltl ~ _< 2 and we obtain that there are polynomials Pn with (11.4)

P,,(0) = 1,

IPn(x)l < Ce -n~l~r,

x e [-1, 1],

only if a > 1 and c < v/-~F ( ~ ) / F ( ~ ) , and conversely, if the strict inequality holds, then the existence of polynomials with property (11.4) was proven in [49]. The existence of the polynomials in question when the equality holds has remained open (it was resolved in [31] for a - 2). Now the discretization method of Sections 2 and 3 enables us to settle this problem. T h e o r e m 11.1 Let e < 2. Then there are polynomials Pn with property (11.4) if and only if a > 1 and c< -

V;r (~) r

Let us mention that the latter result is no longer true for ~ > 2. It is an open problem to determine the largest constant c that allows (11.4) when a > 2. o f T h e o r e m 11.1. The necessity of the condition was proved in [49, Theorem 3.3], so it is enough to prove that if 1 < a < 2 and c = V/~F (-~)/F ( - ~ ) , then there are polynomials with the property (11.4). Let us consider the energy problem on ~ -- [-1, 1] with weight w(x) -exp(c[z[ ~) (note the positive sign in the exponent which makes this weight essentially different from the Freud weights). First we show that the corresponding extremal measure is given by the density function Proof

(11.5)

(1 -

u2)3/2du+ d~

)

,

Section 11

81

Fast decreasing polynomials

where the constant d~ is

01 1-

d~ =

u 2-a

(1 - u2)3/2 du.

To get this form let us consider the function l

(O~ -- 1) O~ / 1

f ( x ) = ~Tr lv/T-Z-~_z2

U~-I

,

7r Jlxl u2v/'ff'f'~-x~du

built up from the Chebishev and Ullman distributions (see (3.4)), and recall that the Ullman distribution corresponds to the energy problem with respect to the weight exp(-7alxl a) (note the negative sign, which is not the case with the energy problem we are discussing now). This function has total integral 1 over [-1, 1], and by (3.6) its logarithm for x e [-1, 1] is of the form const + (a 1)7a Ixl a, where

r(~)r( 89 2r ( ~ - ~~ )

But using that r(t + 1) = tF(t) and r(1/2) = v ~ we easily obtain that a

1

(~ _ 1)~o = (~_ l~r (7) r (~) ' 2r (~_~1)

~r(~)

:

r (~5-~)

: c,

and so the potential is of the form const + clxl ~'. Thus, if we can show that the function f is nonnegative, then we can invoke Lemma 5.1 to conclude that f is nothing else than the density of the equilibrium measure in question (actually, the same conclusion can be derived from the principle of domination without referring to the nonnegativity of f, but we shall need the following consideration anyway). Clearly, it is enough to consider positive values of x. If we write the integral in f in the form 1/x

Xv~-I

.,1

Ua-1

~2v/'ffrZ-'f-1du

and integrate by parts then it follows that f satisfies the differential equation f t ( x ) -- (~

x

r(1-x2)z/2 +

~ - If

x

(x)

with initial condition f(0) = 0. We can solve this linear equation and get with some constant d that r

( 1 - u~)a/2du + d

)

.

The value of d follows from the condition that f has integral 1 over [-1, 1]. This means that we must have d + f01X(~_IC~ [ x U 2-(~ --i lrJo ( 1 - u 2 ) 3 / 2 d u d x = 2'

82

Applications

S e c t i o n 11

which easily yields the value do for d. Since do is nonnegative, the nonnegativity of f follows from the preceding expression for f ( x ) , and the same expression verifies (11.5). When a = 2 then do = 0 and (11.5) takes the form 2

=

x2

-

7r

(1

:~2)1/2'

-

while for 1 < ~ < 2 the constant do > 0 and in this case the density v has order ,~ [x[ a-1 as z approaches 0. Thus, if 6 = a - 1 if 1 < a < 2 and 6 = 2 if = 2 then we can conclude that the density v of the extremal measure satisfies v(t) ,~ It[ 6 as t ~ 0 and v(t) ,,~ (1 - t z ) -1/2 as t ~ + l , and otherwise v is continuous and positive. This is all we lined of v. Let now p(t) = v(t)dt be the extremal measure. By Theorem A from the introduction we have U ' ( x ) = c[x[ ~ -~ F~o for every z E [-1, 1]. If we can construct polynomials n--1 j=O

such that (11.6)

-

log

IR

log

IR

(

)I

-

nU (x)

C

for all x E [-1, 1], and (11.7)

-

(0)I -

nU (O)

<

C,

then Pn(z) = Rn(x)/R~(0) will satisfy (11.4). This is where the discretization technique of Section 2 enters the picture. Let n be an even number (when n is odd, use n - 1 in place of n below). Let us divide [-1, 1] by the points - 1 = to < tl < ... < tn = 1 into n intervals /j, j = 0, 1 , . . . , n - 1 with p ( I j ) = 1/n, and let ~j be the weight point of the restriction of p to Ij. Set n-1

R (t) = 1-i(tj=0

We claim that these satisfy (11.6) and (11.7). We write (11.8) - log

IRn(~)l- n U " ( x )

= ~

n

log .i

v(t)dt =:

L./(x). j=0

The proof of (11.7) is very simple: since ~0/2-1 < tn/2 = 0 < ~ / 2 , and the function log ] 0 - t ] is concave on every Ij, we get that every term Lj(x) in (11.8) is at most 0. This proves (11.7). It is left to prove (11.6). Let x E Ijo. The individual terms in (11.8) are clearly bounded from below when j = j0,j0 =t: 1. For other j's the integrands

Fast decreasing polynomials

S e c t i o n 11

83

are bounded in absolute value by an absolute constant independent of n, x and

j ~ jo,jo + 1, hence the integrals themselves are also uniformly bounded, for the integral of v on each Ij equals 1/n. As we have done in Section 2, we write for x E Ijo, and j ~ jo,jo • 1 the integrand in Lj (x) as

1 ~j-t ~j-t ( ~j-t ?~ log + _ , ~ = =_,~ +0 L ~ J' which holds because

~# -t >_ -q >-l,

t e I#,

with an absolute constant 0 < q < i. Thus, we have

(11.9)

Lj(x)

=

n

J,

O .l=_~i I

J

= o ((~,j 11~12 _ ,%)2, because the integrals

x-~j vanish by the choice of the points ~j. We have to distinguish two cases according as x is closer to one of the endpoints or it is closer to 0.

Case L x is close to an endpoint. [ - 1 , - 1 / 2 ] . We have to estimate

Let us suppose for example that x E jo-2

&(=) := ~ IL#(=)I j=O

and

n-1

s=(=) := ~ ILi(=)l. is+2 We shall only do the first one, the second one being similar (in view of (11.9) the part of $2 corresponding to the indices for which ~j > - 1 / 4 is less than (11.10)

c~,lI~l

2 < C~,lI~l

J

< C.)

J

For j _< j0 - 2

1 - 1 - ~ j , , , ( 2J ~ l-) n2

'

40

-4

~

-

,

Application8

84

Section 11

hence

jo--2 Sl(X) ~ C E

(.~)2

io/2

1o-2

+~

=

=: K1 + K2.

j=o

Here jo/2 gl_
= 0(1)

and

io-~x_., K2

C

Z,,

i=iol2

J~ ((j _ jo)jo)2

- 0( 1) ,

and these verify that

Sl(x) =0(1).

Case IL x is close to 0. Now suppose that x E [-1/2, 1/21, say x e [0, 1/21. Let ~ : ~./2+1-1, 13 -" In12+1-1, L; = Lnl2+1-1. Then ~'1+1 = - ~ ; , I*-i+1 = - I ~ , and for j > 0 (j)

~.7 "

1/(6+D

1

,

IX;I "~ nll(6+l)j61(6+1),

where df is the number chosen above, i.e. 6 = (~ - 1 if 1 < a < 2 and df = 2 if a=2. If r E I~o, j0 > 0 then we have to estimate

jo--2

sl(~) := ~ IL;(~)I, j=l

s2(~) := ~

IL;(~)I

1=-n/4 and n/4

E

i=io+2 because the contribution of the rest (like that of v,-nl4~ z--,-n/2J is easily seen to be bounded (use the argument of (11.10)).

Section 1~

Approximation by W(a,,x)Pn(x)

85

We shall estimate Sl(x) + S2(x), the sum S3(x) can be similarly handled. Now (11.9) yields 1

(nl/($+l)j6/($+l))

jo-2

((n)

-(n)

Jo12 j-261(~+1)

jo-2

J--'

<_ c

2

jo/2

)

jo-2

,=,

j-261(~+1)

E

+ ~C - - = o(1) j=l j~/(6+1) j=jo/2 ((jo - j)jo~/(t+D) 2

and 1 2 (nl/(6+l) (ij i+l)a/($+l))

0

s2(x) _
-

(

~ J

1


0

) 0

+ ~

j=-jo

,_-_,o

1)_26/(6+1)

(lYl+ ~2/(t+1)

= o(1)

JO

| 12

Approximation by W(anx)Pn(x)

Let W(x) = exp(-Q(x)) be a weight function on the real line. In this section we shall consider the problem of approximating functions by weighted polynomials of the form W(anz)Pn(x) with some appropriately chosen normalization constants an. This type of approximation has been the key to many recent results concerning the orthogonal polynomials with respect to W(z), and the monograph [28] contains the basic results concerning it. If W(z) = exp(-c[x[ a) is a Freud weight and an = n 1/'~, then W(anx)Pn(x) = W(x)nPn(Z), hence in this case our problem is just the one considered in the rest of the present work. In general, we shall set w,(z) = W(anz)Un, so that W(a,z)Pn(z) = Wn(z)nPn(z), and the approximation problem in question is the one considered in Section 8 with varying weights {wn}. On W(z) = exp(-Q(z)) we shall always assume that Q is even, the derivative of Q(z) exists in (0, ~ ) and zQ'(z) /2 oo as z ---} ~ . As normalizing constants we shall use the so called Mhaskar-Rahmanov-Saff numbers an, that are defined for sufficiently large n as the solution of the equation

2 / 1 antQ'(ant)

"=7

~

at.

It is known ([34]) that the supremum norm of weighted polynomials of the form W(x)Rn(x), degRn < n lives on I-an,an]. So by contraction the norm of

86

Applications

S e c t i o n 1~

weighted polynomials W(anx)Pn(x) lives on [-1, 1], hence if a function f is the uniform limit of such polynomials then it must vanish outside ( - 1 , 1) (see also the end of the proof of Theorem 12.1). Now the problem we face is under what condition is it true that every continuous function f that vanishes outside ( - 1 , 1) is the uniform limit of weighted polynomials W(anx)Pn(x) (on R, or what amounts the same on some interval [-1 - 0, 1 + 0], 0 > 0). If the answer to this problem is yes, then we say that the approximation problem for W of type II (as opposed to the problem we have considered previously) is solvable. We shall see that special role is played by the point zero, so we start with a result in which the approximation is guaranteed with a restriction at the origin. T h e o r e m 12.1 Let xQ'(x) /2 cr as x --* oo, and suppose that there are C > 1 and e > 0 such that CQ'(Cx) >_2Q'(x) and Q'((1 + t)x) < Q'(x)(1 + Ct') are satisfied for x > xo and 0 < t < 1. Then every continuous f that vanishes outside ( - 1 , 1) and at the origin is the uniform limit of weighted polynomials of the form W(anx)Pn(x), deg(Pn) < n. This settles the approximation problem under a rather weak (Cl+~-type) smoothness assumption on Q provided we assume f(0) = 0 for the function f that we want to approximate (compare this with the results of [28] - - especially [28, Theorem 12.2] - - where similar results are proved under more restrictive conditions). What happens if we want to approximate a function f that does not vanish at the origin is rather interesting (note that it is enough to approximate some function of this type, then every other one can be approximated in view of the preceding result). For completeness we shall briefly discuss the situation along the arguments of [30]. The proof of Theorem [30, Theorem 1] shows that this type of approximation is closely connected with the problem of S. N. Bernstein if for every continuous g with the property o

as

9

there are polynomials Sn such that IIW(g -

Sn)llR

0

as

n

It is known (see e.g. [1, Theorems 3,5]) that in our case (i.e. when xQt(x) increases to infinity) the necessary and sufficient condition for a positive answer to Bernstein's problem is

F

oo

log W(t) i+

t2

dt=c~.

Approximation by W ( a . x ) P , , ( x )

Section 1~

87

T h e o r e m 12.2 If in the case

log w(t)

F

co 1 + t~

dr
there are polynomials Pn of degree at most n such that IIW(anx)P,(x) - f ( x ) l l R --' 0

as

n --* cr

for some f that does not vanish at the origin, then W -1 must be an entire function. On the other hand, we have T h e o r e m 12.3 Let

W(t) Flog dt = co, co l + t 2 and suppose that Q(x) is twice continuously differentiable for large x, and the function T(x) = (xQ'(x))' Q,(x)

'

lies in between two positive constants: (12.1)

0 < A < T(x) < B

for x > xo and is of slow variation in the sense that . T( ;~x) lm ~ = 1 x-.co T ( r ) for all )~ > O. Then the approximation problem of type H is solvable for W . These results can be applied for example to the Freud weights W ( x ) = e x p ( - e l x [ " ) , c , . > O, in which case

an =

1

1/",

:= r ( 7 ) r ( 7 ) / ( 2 r ( 7

a

1

+ _))

is the number from Section 1. We can conclude that for all a > 0 every f that vanishes outside ( - 1 , 1) and at the origin is the uniform limit of weighted polynomials W(anx)Pn(x). When f(0) • 0, then this is the case if and only if a>l. P r o o f o f T h e o r e m 12.1.

We set

Qn(x) = Q(anx)/n,

and

Wn(X) = e x p ( - Q n ( x ) ) .

Our conditions immediately imply that Q~(x) ,,, Q~(y) if x ~ y, x, y ~ ~ , and that for any fixed x0 > 0 we have anxoQ~(anxo) ,~ n, which can be translated

88

Applications

S e c t i o n 12

as Q~(xo) "~ 1 (here A ... B denotes that the ratio A/B stays away from zero and infinity in the range considered). Taking into account also that xQ~(x) is nondecreasing on (0, 1), we can easily conclude from the assumptions of the theorem that the Qn's uniformly belong to C 1+* on every compact subinterval of (0, 1), furthermore there are constants 0 < c < d < 1 and t / > 0 such that dQ~(d) > cQ~(c) + 71. Hence, Corollary 8.4 can be applied and we obtain the statement of the theorem, at least for concluding uniform convergence on [-1, 1]. However, we know that in Corollary 8.4 the convergence is actually true on a larger set [ - 1 - 0, 1 +0], 0 > 0, and outside this interval weighted polynomials W(anx)Pn(x) that are bounded on [-1, 1] automatically tend to zero under our conditions. This can be seen as follows. For any equilibrium measure/~ =/~w with w(x) = e x p ( - Q ( x ) ) , xQ'(x)/z, 8~ = supp(/~,o) we have that

j~_l x (U"(x) + Q(x))'

=

-

xx--_tdl~(t)+ xQ'(x) 1

increases on [1, oo), furthermore this expression is nonnegative around z = 1 (see Theorem A,(d) and (f) in Section 1). Since for all t E [-1, 1] and x > 1 + 0 / 2 x

>

1 + 0/4

x--t-- l+O/4--t

+

0 16'

we can conclude (see also Theorem A,(f)) that for z > 1 + 0 the sum U~'(z) + Q(z) - Fw is at least as large as 02/64. Now we can make use of Theorem B from the introduction according to which for such z IW"(x)Pn(x)I _< M exp (n(-Q(z) - UV(z) + Fw)) <_exp(-n02 /64). Apply now this to w(x) = W(anx) 1In.

|

P r o o f o f T h e o r e m 12.2. Suppose that f(0) # 0, say f(0) = 1, and there are polynomials Pn with W(anx)Pn(x) - f(x) uniformly tending to zero on R. Set .In(x) = f(x/an) and R~(x) = Pn(x/an). Then we get that IIA - W R ~ I I R -~ 0

as n --* cx~, and hence IIw(W-lfn

-- R ~ ) I M -~ 0,

IIW2(W-lf,

- R~)IIR -~ 0.

which implies We also have

2 -1 I l m t - , , ~ l

liT w

-~ o

as 7 -+ oo,

89

Approximationby W ( a n x ) P n ( x )

S e c t i o n 1~

furthermore the functions fn uniformly tend to f(0) = 1 on compact subsets of t t (note that an -'* oo as n ~ oo). These imply I I W 2 ( W - 1 - R n ) I I R --' O,

as

n

~

c~.

But then the polynomials {R~} are uniformly bounded on every compact subset of the complex plane (see e.g. [1, Theorems 5,7]), hence we can select a subsequence from them that converges to an entire function on the whole plane. But R~(x) -* W(x) -1 for every real x, so W -1 has to be the entire function in question. II

P r o o f o f T h e o r e m 12.3. First of all let us mention that in view of Theorem 12.1 it is enough to show the following: for every e > 0 there is a continuous function X (that may also depend on e) such that X(0) = 1, and for all sufficiently large n there are polynomials Pn of degree at most n such that IW(x)Pn(x) X(x/a,,)l < e for all x E R. In fact, suppose this is true and we want to approximate an f which vanishes outside i - I , 1). Then, in view of Theorem 12.1, for a given e > 0 we can approximate f ( x ) - f(0)X(X) by a W(anz)Rn(x) uniformly on R with error smaller than e. The sum Pn(an X)+ Rn (x) multiplied by W(anX) will then be closer to f than 2e. We shall need a lemma, which is a variant of a result of D. S. Lubinsky [24]. L e m m a 12.4 Under the conditions of Theorem 12.3 there is an even entire function H with nonnegative McLaurent coefficients such that W(x)H(x) ~ 1 a8 x ----+ o o .

P r o o f o f L e m m a 12.4. The lemma can be proven with the method of [24, Theorems 5,6]. We shall only indicate how the construction goes and what changes are necessary in those proof. As before, let W(x) = e x p ( - Q ( x ) ) , and for x > 0 let the number qx be the solution of the equation qxQ'(qx) = 2x. Then (12.2)

qn ~ an.

With the function T ( x ) from the theorem we set

H(x)

~

\'q"~n]

v f ~ T ( Q ( n ) ) eq('')"

n----O

T h i s H will satisfy the claim in the lemma. The proof is an adaptation of that of [24, Theorem 5], we are not going into the details.

|

Now let us return to the proof of Theorem 12.3. Let Rm be the m-th partial sum of the McLaurent expansion of the entire function from the preceding

90

Applications

S e c t i o n 1~

lemma. Since the coefficients of H are nonnegative, we have for any x the inequality 0 <__ Rm(x) _< H(x), furthermore, Rm(x) ---* H ( x ) uniformly on compact subsets of R. As for the Mhaskar-Rahmanov-Saffnumbers an, we can conclude from the assumption (12.1) that there is a positive constant C such that an <_ a2n <_ Can holds for all large n. This and (12.2) easily imply (see also the computation for qx in [24]) that given any e > 0 there are numbers r and t such that

IH(x),~ )

<<_e

for I x ] < r a m

and

for Ixl_ tara.

IR (x)l < H(x) -

Now W ( x ) H ( x ) - 1 tends to zero at infinity, hence by the solution to Bernstein's problem to every e > 0 there exists a polynomial S such that for all xER [ W ( x ) ( W - l ( x ) - H ( x ) - S(x)) I < e. This gives that

[1 - W ( x ) ( H ( x ) - S(x)) I < e, and so in view of of the preceding inequalities we obtain that (12.3)

[1 - W(x)(Rm(x) - S(x)) I < (1 + M)e

(12.4)

W(x)IRm(x ) - S(x)I <_ Me

and (12.5)

for Ixl _< ram,

for Ixl >_ tara,

W(z)lRm(x ) - S(x)l < M ( M + 2)

otherwise, where M is an upper bound for W H on R. The assumptions on Q imply (see also the computation for q~ in [24]) that there is also an L such that 2ta[n/L] < ran is also true. Now if we set I

k=l

then we ge from (12.3)--(12.5) that

IW(x)P.(x) - x(Ixi/a.)l < (1 + M)e + M(M + 2) where the function X(x) is defined on [0, ~ ) as follows: x(L -k) = 1 - 1/k for k = 1 , 2 , . . . , i - 1, X(0) = 1, X(X) = 0 for x > 1 and X is linear otherwise. Hence the polynomials Pn and the function X satisfy the requirement from the beginning of the present proof if we choose l > M ( M + 2)/e.

|

Extremal problems with varying weights

S e c t i o n 13

13

91

Extremal problems with varying weights

Let {Wn} be a sequence of weights on the interval [-1, 1], u a fixed nonnegative function and for a 1 _< p < ~ consider the extremum problem (13.1)

Em,p(wnu) :=

inf

IIPrnw~ulILP

P,.EIIm

where again IIm denotes the set of polynomials {x'~+ -- -} with leading coefficient 1. We have already discussed this in Section 3.3 in the particular case when each w, is the same. From Theorems 10.1 and 10.2 and Bernstein's formula (3.38) we can easily get the following strong asymptotics for Em,p. T h e o r e m 13.1 Let 1 < p < oo, {wn} a sequence of weight functions on the interval [-1, 1] such that the corresponding extremal measures pw, have support [-1, 1], they are absolutely continuous there and if we write d~,~.(t) = v,(t) dr, then the functions vn satisfy the conditions (13.2)

vn(t) < A(1 - t2) p,

(13.3)

vn(t) > A(1 - t2) #~ ,

t 9 (0, I), t 9 (0, 1)

for some constants A, ~ > - 1 and fig. Let furthermore u be a positive continuous function. Then for every k = O, + 1 , . . . (13.4)

E

n ) n+k,p(wnu

(1 + o(1))~p2-"-k+I-X/PG[wn]nG[u]

as n ---* oo. Actually, the relation (13.~) is uniform in k > - K for every fixed K. The result is also true for even weights {wn} satisfying on [0, 1] the conditions of Theorem 10.3. Recall that

G[V]= exp (l f_11l~

dx)

are the geometric means from (3.13) and

= (r(t/2)r((p + l)/21/rcp/2 + 1)) 1/'. If we integrate the equality in Theorem A,(f) with respect to the equilibrium measure of the interval [-1, 1] and use Funibi's theorem as well as (1.7)-(1.8), then we get that G[wn] = exp(log 2 - Fw,), hence (13.4) can be written in the form E n+Lp(wnu) n = (1 + o(1))ap2-k+l-1/pG[u]e -"F~. The result holds for somewhat more general u's, but the exact conditions on u are not clear. Below we shall prove a more general result for p = 2. This case corresponds to orthogonal polynomials and appears in several situations.

92

Applications

S e c t i o n 13

P r o o f . The proof is very similar to the argument of Section 3. First let us assume that k > 0. The assumptions imply that Theorems 10.1 and 10.2 can be applied on the interval ( - 1 , 1) rather than on (0, 1). We set 7 = (1 - 1/1))/2. Then by Theorem 10.1 there is a sequence { Hn } of polynomials of corresponding degree n = 1, 2 , . . . such that if

hn(x) = w'~,(x)u(z)IHn(x)I(I - x2) (-1+1/p)/~, then hn(z) > 1 and lim G[hn] = 1. Thus, by Bernstein's formula (3.38) lim inf En+k,p(W~U)/ (%2-n-k+x-llPG[wn]nG[u]) n --~. O 0

>_

=

/ (G[w.]"C[u]),

and if we consider that here the fraction on the right hand side is 1/G[h,], it follows that

limmfEn+k,p(wnu

[wn]nG[u

>_ 1.

n ---* O 0

The proof of the upper estimate is completely symmetric if we use Theorem 10.2 and the polynomials R2n(x) = IHn-I(x)I:(1 - x 2) there which are positive in ( - 1 , 1) with simple zeros at • (c.f. (3.38)). Since the degree of Hn in the proof can be smaller than n, say n - in with in -'* cr it also follows that the preceding argument actually holds for all k because eventually we shall have k > - i n . The uniformity of the convergence in k > - K also follows from the proof. Finally, the last statement follows from Corollary 10.4 since the proof used only the existence of the polynomials guaranteed by this corollary (in Corollary 10.4 the degree of the polynomials Hn can again be n - in where in ---* ~ ) .

|

We have already mentioned that the quantity En,2(W) gives the reciprocal of the leading coefficient of the n-th orthonormal polynomial with respect to the weight function W 2. Thus, the case p = 2 is of special interest and is connected with many other problems in mathematics. Our next aim is to extend Theorem 13.1 in this case to more general weight functions u. This extension is connected to multipoint Pad6 approximation that we shall briefly discuss in Section 16. T h e o r e m 13.2 Let p = 2. Then with the assumptions of Theorem 13.1 the

asymptotic relation (13.~) holds for every measurable u that is positive almost everywhere on [-1, 1].

Section 13

93

Extremal problems with varying weights

We also add that in this case a2 = V ~ ,

E n+k,~(WnU n )=

(1 +

so (13.4) has the form

o(1))v/~2-n-kG[wn]nG[u].

Let us also mention that if u does not satisfy the the so called Szeg6 condition /_~ log u(t)

(13.5)

dt > -r

1

then formula (13.4) gives only an upper estimate. Thus, in finer asymptotic problems we shall assume (13.5) exactly as is done in the classical case. P r o o f . We can assume that u satisfies the Szeg6 condition (13.5). In fact, then the general case when this is not so follows by adding to u a positive 6 and then letting ~ tend to zero. The case when u is continuous and positive follows from Theorem 13.1. The general case will be deduced with the aid of a theorem of G. Lopez [19]. Lopez proved the following: let 2n-in

v;. = H (x- z.,j) j----1

be polynomials of degree 2n - jn with jn ---* c~ which are positive on ( - 1 , 1), and let p be a measure which has positive (Radon-Nikodym) derivative almost everywehere in ( - 1 , 1). We set dpn -- (V2*n)-ldp,.and assume that Pn is a finite measure on ( - 1 , 1) for every n. Then we can form the orthonormal polynomials P*,k with respect to Pn:

f'_" P~,kP~,mdpn =

6k,rn.

1

Assume further, that if ~(z) = z + ~ ' 2 - 1 is the conformal map that carries the complement of [-1, 1] into the unit circle, then for the zeros of the V~n'sthe condition 2n-j,,

li_moo E

(1 -I~(zn,r

=

o~

5=1

is satisfied. Under these conditions we have ([19, Theorem 9]) for n --* cr

(13.6)

,.

[

1

9

2

,,~,(Pn,n+k) (t)

nl~rn~L1 Ytt)

1

fl

V~n(t~ dp(t) = ~ J--1 f ( t ) ~

1

dt

for every bounded f and every fixed k = 0, + l , . . . . In other words, the measures 9 2 (Pn,n+k) dpn converge weakly to the arcsine distribution in the stronger sense indicated.

Applications

94

Section 13

Let 7n,k be the leading coefficient of P*,k, i.e. 9

= 7n,kX

k

+'".

As we have already mentioned, it is immediate from the orthogonality of the polynomials Pn,k that the extremal polynomial in the minimum problem (13.1) (with p = 2) for the measure (V~n)-ldp is Pn,JTn,k, i.e. inf

/

,

2

dp =

[Pn [2

It also follows from the considerations of [19] and [21] that (see also [19, (43)]) (13.7) 7*,n+k = (1 + o(1))V~2n+kG[V~n]n/2G~u'] -1/2. In fact, the corresponding result for the unit circle was proved in [21] under the assumptions that the zeros of certain polynomials (the analogues of V2n) are real, but the proof works just as well for the general case if one uses the results of [19]. From here the transfer of the result to the real line to obtain (13.7) is just the standard technique (c.f. [33] or [19] and use also [19, Theorem 3]). After these preliminaries we turn to the proof of Theorem 13.2. By Theorem 10.2 there is a sequence {Ha-l} of polynomials such that if

ha(z) = w~(z)lHn_l(x)~/1 - z21(1 - x2) -I/4, then ha(z) < 1 and lim G[hn] = 1.

n--~ oo

Actually, as we have remarked after the lemma, the degree of Ha-1 can be n-in with in --* OO. Furthermore, the construction of Lemma 9.1 shows that at least half of the zeros of the /am are of distance Lncn -1 from the interval [-1, 1], where Ln --* oo as n -+ oo. Thus, if we set V~n(z ) = IHn_1(z)12(1 - z 2)

and dp(t) = (1 - t2)l/2u2(t) dr, then all the properties of V~n mentioned above are satisfied (note that if dist(z, [-1, 1]) > cLn/n, then > Cl min(1, Ln/n)),

1 -[r

hence for the corresponding orthogonal polynomials P~,k we have (13.7). But hn _< 1 implies

E

n

n+k,2(Wn ) _< En+k,2

-..-( --U~01/4

and so it follows from the formula u~ I/4 ,~ _-

(13.8)

En+k,2 (V~n)i/2 ]

1

7~*,.+k'

)

,

Section 14

Asymptotic properties of orthogonal polynomials with varying weights

95

the asymptotic relation (13.7) and

G[wnlnG[u]/G[~'/4u/(V~n) 1/~] = G[hn] = 1 + o(1), ! that limsup E,,+k,2(w~ u)/(~22-"-k+X/2G[w,l"G[u]) < 1 The proof of the lower estimate is similar if we use Theorem 10.1 instead Theorem 10.2. This completes the proof.

l 14

Asymptotic properties of orthogonal polynomials with varying weights

Let again {wn} be a sequence of weights on [-1, 1] as before, and u a measurable function satisfying the Szeg6 condition (13.5). We denote by

p.,k(x) = ~.,~(x)xk+...,

~.,k

> 0,

the k-th orthonormal polynomial with respect to "tU - n2n."0~2., _1 --

--

Pn,kl)n,m

w2nu 2 n

:

~k,m.

1

Note the square in the weight. For the monic polynomials 1 qn,k :--

7n,k

Pn,k

we have (14.1)

J

q n , k q n , k W n~nU 2 =

E~,2(w"u),

where Ek,2 is the extremal quantity discussed in Section 13. This means that the monic orthogonal polynomials are the optimal ones in the extremal L 2 problem of Section 13 for the case p = 2, and we also have (14.2)

7n,k = 1/Ek,2(wnu).

In this section we discuss asymptotics on Pn,k. For fixed weights (wn - 1) all these are classical, and some of the results below were proved by G. Lopez [19, 21] for varying case when the "tUn ...2n, S are reciprocals of polynomials. For simpler notation let Wn be the weight wnn u. With some positive T/> 0 we choose a stricly positive continuous function u* that coincides with u outside a set E v of measure smaller than 7/. By Theorem

96

Section 14

Applications

10.2 (see also the remark after it) there is a sequence { H , - I } of polynomials of degree n - in, in ~ oo, such that that they do not vanish on ( - 1 , 1) and if

h;,(x) = w,~(z)u*(z)lH,,_l(x)v~then h*(x) _< 1 and (14.3)

lim

n - - + OO

We also set (14.4)

:1(1 - :)-~/4,

G[h*] = 1.

X(x) = min{u*(x)/u(x), 1)

which again belongs to the Szeg6 class. By chosing ~ sufficiently small (and u* appropriately) the geometric mean of u* can be as close as we wish to that of u, hence the geometric means of xWn and Wn will also be close. Now with 9

(14.5)

~1/4

w,.: = iH._ll~/2,

~(x)

---- 1 -

x 2

we have

X(x)Wn(x)IW~(x) < h*(x) < 1,

x E [-1, i]

and the geometric means of the three weights Wn, xWn and W* can differ by as small amount as we wish if we choose y and u* appropriately and n is sufficiently large. These properties and Theorem 13.2 on the asymptotic behavior of the leading coefficient of orthogonal polynomials with respect to varying weights easily imply the following: for every e we have polynomials Hn of degree n - in with in --+ oo such that for some set Ee of measure at most e we have the relations (14.6) Wn(x)/W*(x) - 1 + o(1), x • E~, and for every fixed k -- 0, :t:1, 999and large n (14.7)

(14.s)

En+k,2(xWn) ( 1 1 ) 2 En+k~(Wn) + En+k,2(W~) > 1 - e , 2 Et,+~,2(xW,,) <_ (1 + OEt,+~,~(W;) ~ *

and (14.9)

E~+k,2(XW.)

< -

(q,,,.+k)*2(XW.)2 < f ( q.,.+k) 9 2(W.:) 9 2 * E~+k,2(W~) <_( 1 + e)E~+k,2(xWn),

q*,m denotes the m-th monic orthogonal polynomial corresponding to (w;) ~.

where

We have already mentioned that by [19, Theorem 9] the functions

(pT,,.+k w;) ~

Section I$

Asymptotic properties of orthogonal polynomials with varying weights

tend to (d~

.--

97

1 1 r x/1 - t 2

in the sense that for every bounded and measurable f (14.10)

l i m f f(Pn,n+kWn) * , 2 = f fw.

n "* O0

We start with a simple observation. We apply the parallelogram law

l f (Pn,.+.--Pn,n+.) (XWn) + f (7(Pn,n+k+ "

(xw.) ~

if( Pn'n+k)2(XWn)2 § ~if(,Pn,.+k) (XWn

= -2

and observe that the first term on the right is at most 1/2 by X < 1, the second one is at most (1 + e)2/2 by (14.9) and (14.8) (see also (14.2)), while the second term on the left is at least as large as (1 - e) 2 by (14.7) and (14.2). Therefore, we can conclude that (14.11)

(Pn,n+k

* 2 (XWn) 2 <_ 12~, - Pn,n+k)

in particular,

(14.12)

] JE

(Pn,n+k - - P n", n + k ),2--,2 IWn

<~ 12e,

where E~ := [-1, 1] \ E, denotes the complement of E, in [-1, 1] (recall that

x(~) = 1 for all 9 9 E,'). Let now T be a measurable subset of [-1, 1] not intersecting E,. We can easily get from (14.12), (14.6), (14.10) and Schwarz inequality that

l i m ~ f fT(P,,,+kWn)2 >_fTw--12e-- 2 V ~ , which gives for e ~ 0 that for every measurable subset T of [-1, 1]

On applying this to the complement of T in [-1, 1], it follows that here the liminf can be replaced by limit (recall, that the corresponding integral over [-1, 1] is 1). Since every bounded function can be uniformly approximated by linear combinations of characteristic functions of sets, we finally arrive at T h e o r e m 14.1 Let {w,} be a sequence of weight functions on [-1, 1] such that the corresponding eztremal measures #w, have support [-1, 1], they are abso-

lutely continuous there and if we write dl~w,~(t) = Vn(t) dt, then the functions Vn

98

Section 14

Applications

satisfy the conditions [13.2} and (13.3). Let furthermore u satisfy Szeg6's condition (13.5). If pn,m are the orthonormal polynomials with respect to .2n. 2 then for every fized k = O, +1,... and for every bounded and measurable f

The result is also true for even weights {wn) satisfying on [0, 1] the conditions of Theorem 10.3. For fixed weight, i.e. when Wn --- 1 this was proved in [33, Theorem 11.1], while, as we have seen and used above, for varying weights when wn2~ is the reciprocal of a polynomial with zeros not too close to [-1, 1], by Lopez [19, Theorem 9]. It would be equally easy to extend other results from [33] and [19] to the present case at least under SzegS's condition on u. Probably the results also hold if we only assume that u(t) > 0 almost everywhere on [-1, 1]. Instead of pursuing this direction further we seek stronger asymptotics on the orthogonal polynomials. As one can expect, we shall get strong asymptotics away from [-1, 1], and on [-1, 1] in L 2 norm. To this end we shall need the following corollary to Theorem 14.1. With the notations applied before the theorem we have (14.13)

limsup n - - - * OO

,n+kWn --pn,,+kvv,)

< V/~

J

as n ---, oo. In fact, with the set E~ used before we have

f(p.,n+kWn-p.,n+~vvn) _2 (fE (p.,n+.W.)2+ /~ (pn,.+.W;) ) *

"

*'~

<

*

*

2

w;)) ~, +2f g (pn,n+.Wn-V.,n+kWn) * 2 +2 f (V.,n+~(Wn , -

and by the preceding theorem, (14.10), (14.12) and (14.6) we get the bound

7

~o+24~+o(1) e

for the right hand side, and this is easily seen to be smaller than ~ for large n (and small e). To formulate our next theorem we need a definition (see [48, (12.2.3)], where an additional factor 1/2 appears on the right which we put into our functions). If V is a nonnegative function on [-1, 1], then let Fv(x)=-~

1

fl

1

logV(O-logV(~) ~

x

(1-x~]1/2 \1-~2)

d~,

where the integral is understood in principal value sense. It is easy to see that Fv(cos0) coincides for 0 < ~ < r with the trigonometric conjugate of log V(cos 8) (see [48, (12.2.3)]).

Asymptotic properties of orthogonal polynomials with varying weights

Section 15

99

T h e o r e m 14.2 Let wn satisfy the assumptions of Theorem 1~.1, and suppose

that u satisfies Szeg5"s condition (13.5). Then for fixed k = O, =l=l,... the difference

2

1

1

tends to zero in L2[-1, 1]. P r o o f . We continue to use the functions and notations from before Theorem 14.1. Recall that there we have chosen to an e > 0 a function u* and polynomials Ha-1 with properties (14.3)-(14.9). It follows from [48, Theorem 2.6, (2.6.2)] for p(x) = H~_I(Z ) that

p~,.+k(cosO)W~,(cosO)(sinO) 11~

=

{eic-+k)0exp(irln._,l(cos0))}

for all large n (so that n - in < n + k). The last factor on the right is c o s ( ( , + k)0 - rlx._,,(cos 0)), so in view of (14.13) it is enough to prove that the difference of cos ((n + k)O- rln._,l(COS 0)) and

in L ~ [ 0 , 7r] is as small as we wish for small e. Since the conjugate function of log I sin 0[ is 0 - (~r/2) for 0 E [0, r] and (~r/2) + 0 for 0 E I - r , 0], the last but one expression is actually

and so it is enough to show that the two functions r w : (cos 0) and

r .(cos0) are close on a set of almost full measure on [0, r] provided n is sufficiently large. But the difference of these two functions is rh~u/u. (cos 0), and so recalling that Fv(cos 0) coincides with the trigonometric conjugate of log V(cosS), it follows

100

Applications

Section 14

from the weak (1,1) property of the operator of trigonometric conjugation that the measure of the set

is at most Ct/-1 fo"

[log h~, (cos 0) 1dO =

C~?-1 (~o 1 II~

dx + fo I II~ ~/1 - x 2

dx).

Since for fixed ~/the first term on the right hand side tends to zero as n ---* cr (see Theorem 10.1) and the second term is as small as we like by appropriately choosing u*, the proof is complete.

|

Finally, we prove a strong asymptotic formula for Pn,n+k away from [-1, 1]. To this end we introduce the so called Szeg5 function

Dv(z):exp ( z2x/TC~ll /_l1 l~z - x

dx

provided log V ( x ) / ~ is integrable, i.e. provided V satisfies the Szeg6 condition (13.5). This form of the Szeg6 function appears in [22], and can be deduced from the more familiar one corresponding to the unit disk by the standard conformal mapping between C \ [-1, 1] and the unit disk. The Szeg6 function for the unit disk is (see e.g. [48, Ch. 10]) /gg(~)=exp

( i~ f

) . e eit+~logK(t)dt "-~

[,~[ < 1,

This DK is the outer function associated with K 1/2 normalized by D(0) > 0; in particular, /gK is not zero in the unit disk, /~K E g 2, and K(0) = [DK~ei~ 2 almost everywhere, where DK (e i~ denotes the nontangential boundary limit of /)K at the point eia. If we set g(t) = Y(cost), then Dv(z) = DK(~), = z - v / ' ~ - 1, where, as usual, we choose that branch of the square root that is positive for positive z. Hence, Dv is not zero in C \ [ - 1 , 1] and V(x) = [Dv(x)l 2 for almost every x E [-1, 1], where again the last quantity is a boundary limit. The function F that we used above gives the argument of DK on the lower part of the cut C \ [-1, 1] (see [48, (12.1.7)] and note that x = cos0, 0 < 8 < on the lower part of the cut corresponds to the point e i~ under the mapping = z - V r ~ - 1): (14.15) D~(z)/lDv(x)[ 2 = e'rv(').

Section 1j

Asymptotic properties of orthogonal polynomials with varying weights

101

T h e o r e m 14.3 With the assumptions of Theorem 1,~.1 we have Pn,n+k(Z) = (1 + O ( 1 ) ) ~ 2 (Z + ~ - -

1) n+k ( D r , ( z ) ) -1

uniformly on compact subsets of-C\ [-1, 1], where

Yn(t)

= w.

n(t)u2(t)vff-- t2.

We can put Theorem 14.3 into a somewhat different form (c.f. [43]). For simplicity let us assume that u is identically one (this can be attained in the most interesting cases by looking at WnU1In instead of wn), in which case we will not need the Szeg6 function in our asymptotic formula. T h e o r e m 14.4 With the assumptions of Theorem 1~.1 we have that the polynomials pn,n+k(z) are asymptotically equal to 1 V~.~(z+kl~z2-1)'+I/9exp(nFio.-nilogzl--_tdPw.)(z2-1) -I/4

uniformly on compact subsets o f - C \ [-1, 1], where pro, and Fw, are the equilibrium measure and the equilibrium constants from Theorem A, Section 1. If u is not identically one, then the fixed multiplier

(D.2(z)) -1 also appears on the right. This form gives via standard arguments the following asymptotics for the zeros of the orthogonal polynomials: with the assumptions of Theorem 14.1 let Vn be the normalized counting measure on the zeros of the orthogonal polynomials Pn,n+k. Then lim Iv,(/) - pw.(I)l = 0 n---~so

uniformly in the intervals I C_ [-1, 1]. Of course, to conclude this we do not need the full force of the strong asymptotic formula in the preceding theorem, n-th root asymptotics would suffice (see e.g. [47, Chapter 3]). That the two theorems Theorem 14.3 and 14.4 are equivalent can be easily seen from the fact, that the Szeg6 function associated with x/1 - x 2 is

+ ezr=-r-1/

'

and, by Theorem A in the introduction the function

102

Section 14

Applications

coincides with the outer function D~. (on the domain C \ [-1, 1]) associated with w . . Thus, it is enough to prove one of them, but before we do that let us utilize once more the preceding formula. By what we have said about the relation of Fv and DK in (14.15) it follows from the just metioned identity that

where vt0. (t) is the density of the equilibrium measure pw.. Since the argument of (z - t ) -1 on the lower part of the cut of ~ \ [-1, 1] is r for t > ~z and zero otherwise, it readily follows that F~,(x) = ~r

v ~ , ( t ) d t - arccos x,

and so we get the following variant of Theorem 14.2. T h e o r e m 14.5 With the assumptions of Theorem 1~.1 f o r k = O, :1:1,... the difference p.,.+k(~)w~(~)~(~) tends

cos

t: + i / a r c c o s

x + n~r

(t) dt + r ~ (x) -

to zero in LZ[-1, 1].

P r o o f o f T h e o r e m 14.3. We use again (14.11). If we recall the properties of the Szeg5 function this can be written with ~ = z - z2v/~"S~-1, i.e. z = 89 + 1/~) in the form ~

2

I=1 ]P"'"+k(z)~"+kbK=(~) --P~'n+k(z)~n+~DK"(~)

dl~l _< Se,

where K . ( t ) = X 2 ( c o s t ) V . ( c o s t ) = X 2 (cost)wn2 n (cost)u 2 (cost)[ sinth

n > ne, and, as before, P*,k denote the orthonormal polynomials with respect to W,~ (see (14.5)). But here both functions under the integral sign belong to the Hardy space H 2, therefore we obtain from Cauchy's formula, that uniformly on compact subset of the unit disk {~[ [~[ < 1} the difference (14.16)

pn,n+k(Z)~n+kDK.(~)

--

* n+k D~g . ( ~ ) pn,n+k(Z)~

tends to zero if n > nE and e ~ 0. By the choice of the functions H . and hn we have for

g~(t) = 1/IHLl(cost)l"

Section 15

103

Freud weights revisited

the identity

ff)/b , if) = b(x .)(cos)ff)2, and since the L~r norm of log[x(cost)hn(cost)] is as close to zero as we like by choosing e > 0 sufficiently small and n large (see Theorem 10.1) and of course selecting the function u* in (14.4) appropriately, the ratio on the left of the preceding formula will be as close to 1 as we like. Hence it is enough to examine the behavior of

P* n+k(z)~n+kbK: (~). It is known that

Pn,n+k(z ) , = ~1

(~n+~Dg.(1/~)_ 1 +

~_n_k/~K.(~)_l)

(see [48, Theorem 2.6, (2.6.2)] where this formula appears for Iz[ = 1 - - actually in the form of the first displayed equation in the proof of Theorem 14.2 - - which clearly implies the same formula for all z because both sides are polynomials in z), hence it is left to show that here the second term is the dominant one. But that is easy: the ratio of the first and second terms on the right has absolute value 1 on the unit circle. Since the Szeg5 function of a nonegative trigonometric polynomial of degree ! is a polynomial of degree ! ([48, Theorem 1.2.2]), it follows that the ratio in question has a zero at the origin at least of order 2(n + k) - 2(n - in), where n - in is the degree of Hn-1. Thus, by Schwarz's lemma the ratio of the first and second terms tends to zero uniformly on compact subsets of the unit disk as n --* co because in --* co. This proves *

Vn,n+k(z)~

n+k

"

1

DK*(~)--* x/~"

From here we obtain the theorem if we use the aforementioned fact that the difference in (14.16) is as small as we like provided we choose e small and n sufficiently large (and u* in (14.4) appropriately), furthermore that with Un (t) = Vn (COSt) = w~n (cos t)u2(cos t)l sin t l the ratio DK,,(~)/Dv.(~) (which is nothing else than b•162 2) can also be as close to 1 as we like (see the definition (14.4) o f x and recall that its geometric mean can be as small as we want).

|

15

Freud

weights

revisited

Let wa(x) - e x p ( - T a IXla), C~> 0 be the Freud weights we considered in Section 3, and consider the orthonormal polynomials with respect to w2:

=

+....

104

Applications

Section 15

We set for all n x E [-1, 11.

wn (x) = exp(-Tc, I~1~), Note that w, is defined on [-1, 1]. Let Pn = 1 + n -7/12 and w~(x) = w~(pnx) : e -~"~l~l~ ,

x ~ [-1, 1].

It follows from the infite-finite range inequality (3.11) that (15.1)

7,(wa)n('~+l/2)/'~p(n+l/2)/c' = (1 + o ( 1 ) ) / E n , 2 ( ( w ~ ) n ) ,

where En,2 is the extremal quantity from Section 13, (13.1) (see also the connection (14.2) in between this quantity and the leading coefficients of orthogonal polynomials). Here the support of the equilibrium measures associated with w~ is and not [-1, 1] (see Section 3). But recall the discussion at the end of Section 9, where we proved that the estimates of Lemma 9.1 actually hold true on a larger range (see (9.26)), hence Corollary 10.4 is also true for symmetric weights {w*} which have support Sw~, = [-~n, ~n] with some ~n satisfying 1 - n - r < ~n < 1 for some ~" > 0, and which otherwise satisfy the assumptions of of Theorem 10.3 on [0, ~n] rather than on [0, 1]. In particularly, this is true for the weights w* we are considering now. But the proof of Theorem 13.1 was based on approximation like in Corollary 10.4, hence Theorem 13.1 is also true for such weights, in particular, for our w n s. On applying Theorem 13.1 we immediately get 7n(Wa)n(n+Zl2)lap(n+ll2)laa22-n+l/2G[w~] n = 1 + o(1). Here a2 = X / ~

and, as we have already seen in Section 3.1, (3.19), 1

[1

Ixl

dx

1

which yields n~ln G,twnJ Pn

\

-

e-n/,~ e x p [I ( n / a ) ( ( 1 - pn) + logpn)) = (1 + o(1))e-"/~.

Thus, we finally arrive at (15.2)

lim % ( w ~ ) r l / 2 2 - n e - " / ~ n

(n+V2)/~ = 1,

n - - - * OO

which is the extension of (3.3) to all a > 0. Let now p~,~(x) = nV2apn(wa; n l / a x ) ,

x e [-1, 11,

Section 15

105

Freud weights revisited

and let P,,,k be the orthonormal polynomials associated with wnn, like in the preceding section. We apply the parallelogram law

1j:

1

(Pn,n+k -- P-*n , n + k )~2.Wn2n q-

1/1

/1(1, 1/_l 1

~2w2n l~Pn'n+k} n -~'~

:--2

* Pn,n+k "{-Pn,n+k

,),

Wn2n

i . ~2 2n l~Pn'n+k) Wn ,

and observe that the first term on the right is 1/2, the second one is at most

1/2 since

j_l

* ~2 2n tP.,.+k) wn <

1

F

p . ( w ~ ; ~ ) 2 w ~ ( z ) d ~ = 1,

oo

while the second term on the left is 1 + o(1) because the ratio of the leading coefficients ofp~,n+k and Pn,n+k tend to 1 (see (15.2) and Theorem 13.1, and also recall (14.2)). Thus, it follows that (15.3)

(Pn,n+k - - P n*, n + k ) ,2 Wn2n = O. 1

Using this relation instead of (14.13) everything that we have proven in the preceding section on w n can be carried over to the Freud polynomials. For example it follows from Theorem 14.5 that for fixed k = 0, :1:I,...the difference

nZ/2apn+k (wa ; n l / a x ) exp(-nva [z I=)

-

W ~ ffl 1- z 2 cos ( ( 1k) + ~ arccos z + n~r f l va(t) dt - 4 )

tends to zero in L2[-1, 1], where va is the Ullman distribution (3.4). We also mention that in [26] it was proved that pointwise asymptotics of this form are also valid (at least for a > 3, see also [43] for another proof which covers every a > 1), i.e. the above difference tends to zero uniformly on compact subsets of (-1, 1) not just in L 2 norm. On the other hand, it follows from Theorem 14.4 and Fw. -

log ~ - t

dttw" = log(z + ~

- 1) +

~01

Ztct-1

z2.v,t~-~W_ t 2 dt

(c.f. the computation in Section 3) that

nl/2apn+k (wa ; n l / ~ z )

is asymptotically equal to +

exp

zt

-:

(2 -

106

Applications

S e c t i o n 16

m

uniformly on compact subsets of C \ [-1, 1]. In fact, using (15.3), the proof of Theorem 14.3 can be copied to yield the preceding asymptotic relation from that of the one in Theorem 14.4. We could easily extend these asymptotic formulae for orthogonal polynomials with respect to a weight W 2 where W satisfies the conditions of Theorem 12.1. In fact, Theorem 10.3 can be applied in such case and otherwise the proof is just the same as before. Compare this with the results of [28] where similar results were proven under more restrictive conditions.

16

Multipoint Padd approximation

In this section we briefly discuss the problem of multipoint Padd approximation which is intimately connected to orthogonal polynomials with respect to varying weights. In fact, this area was the main motivation for A. A. Gonchar and G. Lopez [10] (see also [21]) for considering orthogonal polynomials with respect to varying weights and our discussion would not be complete if we did not touch this aspect of the theory. Historically orthogonal polynomials originated from continued fractions, and one of the classical results in the analytic theory of continued fractions is Markov's theorem to be discussed briefly below. A function of the form (16.1)

f(z) : c + / d..~(._x)x_ z

is called a Markov function if p is a positive measure with compact support S(p) C_ R i.e. Markov functions are Cauchy transforms of positive measures p with compact support in R. For functions of type (16.1) A. Markov [Ma] proved that the continued fraction development

bl

(16.2)

bg

z -- al + z-a2+... B

of f at infinity converges locally uniformly in C \ I(S(p)), where I(S(p)) is the smallest interval containing S(p). In what follows we shall assume that the support of p lies in [-1, 1]. It is well known that the n-th convergent is the [ n - 1/n] Padd approximant to the function (1.1). Hence the convergents of (1.3) are rational interpolants with all interpolation points being identically infinity. Gonchar and Lopez considered rational interpolants with more general systems of interpolation points. For every n E R we select a set

A . = { x , , o , . . . , x.,2.} m

of 2n + 1 interpolation points from C \

I(S(~)),

which are symmetric onto the

S e c t i o n 16

107

Multipoint Padd approximation

real axis. The points need not to be distinct. We set 2~

(16.3)

II

w,(z) :=

j=o x,.jr The degree d, of wn is equal to the number of finite points in An. Denote by 7~,, the set of all rational functions with complex coefficients with numerator and denominator degree at most n. By rn = r n ( f , A , , .) E Tr we denote the rational function that interpolates the function f of type (16.1) in the 2n -{- 1 points of the set An = {x,,,0, 9 9 xn,2n}. If some of these points are identical, then the interpolation is understood in Hermite's sense. It is easy to see that this is equivalent to the assertion that the left-hand side of

f(z) - r.(f,A.;z) is bounded at every finite point of An and at infinity it has the indicated behavior. We note that interpolation at infinity has not been excluded. It can be shown (see [10] or [47, Lemma 6.1.2]) that there exists a unique rational interpolant

rn(z)=r.(f, An;z)= q.(z) ETin p.(z) of the above type, and Pn satisfies the weighted orthogonality relation

/p,(,),kdv(x)

=

0

for

k

= 0,...,n-

1,

i.e. they are orthogonal polynomials of (exact) degree n with respect to the varying weights wn(z)-ldp(x). Furthermore, the remainder term of the interpolant has the representation

(16.4)

(f - rn(f, An; . ) ) ( z )

=

~.(z) p2n(Z)f ~p~.(~)d,(.) : Z)

for all z E C \ [-1, 1]. By homogeneity we can clearly assume that the Pn is the n'th orthonormal polynomial with respect to w,~(x)- 1d/~(x). Suppose that S(/~) C [-1, 1]. Now this is a typical situation when the assumptions in the results of Section 10 hold true, at least if the points of An are not too close to [-1, 1]. In fact, the function 1/Iwn(z)l can be written as 1

_ eV~(,),

Iw,,(z)l where Vn is the measure that has mass 1 at every point of An. Thus, if we set

~.(~) = ~.(~)i/(2.+i),

108

Applications

S e c t i o n 16

then the equilibrium measure/~w, corresponding to wn is nothing else than the balayage of , , / ( 2 n + 1) out of C \ [-1, 1] onto [-1, 1] plus 1-

dn/(2n + 1)

times the arcsine measure (equilibrium measure of [-1, 1]). It is easy to verify that if the points of An stay away from [-1, 1], then the collections of all such measures has the property, that the corresponding densities are equicontinuous in compact subsets of [-1, 1] and they satisfy the conditions (9.1) and (9.2) with fl = - 1 / 2 . Hence, the results of Section 14 can be applied provided the density u 2 of # is in the Szeg6 class (see (13.5), and from the asymptotics there we can easily get strong asymptotics for the error away from [-1, 1]. in view of (16.4) provided/~ is absolutely continuous and its density u satisfies the Szeg6 condition (13.5). In fact, we know from Theorem 14.3 that p,(z) asymptotically equals 1 z+ ~ v~-~( - 1)n+l/2(z 2 _ 1)-1/4 (D~2(z)) -1 times the Szeg6 function (with respect to the domain C \ [-1, 1]) ofwn. Let d,~ be the degree of w,. Then

wn(z) = (z -k V~Z2 - 1) d" ( ( z - ~/z 2 - 1)d"w,(z)) = h,(z)Hn(z), and here the second factor H,~(z) on the right belongs to the Hardy space H 2 ( C " \ [-1, 1]), and the square of the Szeg6 function associated with wn is just the outer function associated with Hn. Recalling that the ratio of Hn and that of its outer function is the Blaschke product (with respect to C \ [-1, 1]) associated with the zeros of Hn (note that there is no singular part in Ha), it follows that what remains in the ratio in front of the integral in (16.4) is

2r(z + V~z2 - 1)-l-2'~+d"V~Z2 - 1Du2(z) times the Blaschke product associated with the zeros of wn. The integral itself converges to if 1 1 1

7J by Theorem 14.1, so we have full description on how the remainder behaves away from [-1, 1]:

(f - rn(f, An; .))(z) = (1 + o(1))TrD~2(z)(z + ~ -

1) -1-2n+d" H

r

- O(z,,,j) ,

where ~ is the canonical conformal map of the complement ~ \ [-1, 1] of [-1, 1] onto the exterior of the unit disk, and the product is taken for the zeros of w•, i.e. for the finite points in the system An.

Sect~or~ 17

Concluding remarks

109

The results of G. Lopez ([19]-[21]) give a different asymptotics that supersede the above one in the sense that the points in An can approach the interval [-1, 1] so long as the sum ~-](l(O(xj,n)]- 1) tends to infinity. It is worth mentioning the connection of the above asymptotic relation with the problem of minimizing the norms of Blaschke products on compact sets. In fact, suppose that V is a compact subset o f T \ [-1, 1], and we want to construct good rational approximants to the Markov function f on V. By picking some sets An of 2n + 1 points the above discussed multipoint Pad6 approximants are certain one of the candidates. The problem is how to choose optimally the interpolation points in An so that the approximation be as good as possible. In view of the asymptotic relation given for the error, our problem is to minimize the uniform norm of the Blaschke product in the error on V in the presence of the weight function ID,~(z)l. A result of Parfenov [41] is relevant here, which asserts that if V is an ellipse with foci at 4-1 (or the exterior of it), then the minimum behaves like (r + r2v~Y':-~1) -2'*-1 times the geometric mean of the weight, where r is the sum of the half axes of V. It is plausible that the points of An can be chosen so that this asymptotic is attained. It is an interesting problem to investigate the same question for other, less symmetric V's. Another interesting problem is how far the so obtained bound for the above rational approximation to f on V is from the best one.

17

Concluding remarks

As we have seen, the method of Section 2 gives good approximation for logarithmic potentials by logarithms of reciprocals of polynomials provided the generating measure has continuous density. The method was sufficiently strong to settle the approximation problem for weighted polynomials w"P, for a general class of weigths w and to considerably relax the conditions of [28] concerning approximation by weighted polynomials of the type W(a,~.)Pn(.). It is possible that finitely many logarithmic type singularities in the density (these arise for example at the origin if one considers w(x) = exp(-Ixl) ) can be handled by appropriately adjusting the correction polynomials (like S,~_[,~p,] in Section 2) to have appropriate order of interpolation at these 'bad' points. The situation is much worse if the infinite singularity is not of logarithmic type. For example, if w(x) = exp(-clxl ~) with 0 < a < 1, then the density of the extremal measure has a singularity of the form ~ t a-1 at the origin, and indeed, we know that in this case approximation is not possible. Internal zeros in the density function constitute another problem. We have seen in Example 4.5 that even a single zero may rule out the possibility of approximation in the sense of Theorem 4.2. On the other hand, Example 4.6 shows that in some cases approximation is possible even in the presence of an internal zero, and it seems to be a very delicate problem to clear the role of internal zeros on the approximation problem for given individual weights. The

110

Application8

Section 17

problem with internal zeros is that if the density function has a zero at z0 in the interior of Sw, then in general z0 will not belong to any Swx, A > 1 (c.f. the argument at the end of Section 6), and usually we need to apply the approximation technique to some w x instead of w in order to be able to handle the effect of the singularities in the density v that may appear around the endpoints. A typical example of this kind of difficulty is encountered if we consider the weight w(x) = ex2 (note the positive coefficient in the exponent) considered on := [-1, 1]. It can be shown that Sw = [-1, 1] the density v of #w is given by v(t) =

2t 2

which has a zero at the origin, and has a (1 - z2) -1/2 type singularity at +1 (see Section 11). If A > 1, then S ~ will miss a neighborhood of 0. We do not know e.g. if for every function f 9 C[-1, 1] with f(:kl) -- 0 there is a sequence of polynomials pn of degree at most n such that

uniformly on [-1, 1]. Let us also mention that recently some efforts have been done to find a 'soft' approach to the approximation problem considered in this paper. In some restricted cases such an approach is possible, for example in [3] and [10] simple sign change counting was used to prove such theorems (this works for example if w(x) = e-~2). The paper [18] should also be mentioned that contains a construction for related "one point" polynomials.

Although I have not discussed the contents of this paper in details with D. S. Lubinsky and E. B. Saff, I would like to express my appreciation to them, because the present work was motivated by some of their results.

References [1] N. L Akhiezer: On the weighted approximation of continuous functions by polynomials on the entire real axis. AMS Transl., Set. 2, 22(1962), 95-137. [2] N. I. Achiezer: Theory of Approximation, (transl. by C. J. Hyman), Ungar, New York 1956. [3] P. Borwein and E. B. Saff: On the denseness of weighted incomplete approximations, Proceedings of the First US-Soviet Conference on Approx. Theory, Tampa 1990, Springer-Verlag, (to appear). [4] M. Brelot: Sur l'allure des fonctions harmoniques et sousharmoniques h la fronti~re, Math. Nachr., 4 (1950-51), 17-36. [5] Ch. J. de la Vall6e-Poussin: Potentiel et probl~me g6n~ralis~ de Dirichlet, Math. Gazette, London, 22(1938), 17-36. [6] G. Freud: On the coefficients in the recursion formulae of orthogonal polynomials, Proc. Roy. Irish Acad. Sect. A, 76(1976), 1-6. [7] B. Fuglede: Some properties of the Riesz charge associated with a ~subharmonic function, (manuscipt) [8] M. v. Golitschek: Approximation by incomplete polynomials, J. Approx. Theory, 28(1980), 155-160. [9] M. v. Golitschek, G. G. Lorentz and Y. Makovoz: Asymptotics of weighted polynomials, Proceedings of the First US-Soviet Conference on Approx. Theory, Tampa 1990, Springer-Verlag, (to appear). [10] A. A. Gonchar and G. Lopez: On Markov's theorem for multipoint Pad~ approximants, Mat. Sb., 105(147)(1978), English transl.: Math. USSR Sb., 34(1978), 449-459. [11] A. F. Grishin: Sets of regular growth of entire functions (Russian), Teor. Funktsii, Funktsional. Anal i Prilozhen. (Kharkov), 40(1983), 36-47. [12] L. L. Helms: Introduction to Potentid Theory, Wiley-Interscience, New York 1969. [13] X. He and X. Li: Uniform convergence of polynomials associated with varying weights, Rocky Mountain J., 21(1991), 281-300. [14] K. G. Ivanov: E. B. Saff and V. Totik: Approximation by polynomials with locally geometric rates, Proc. Amer. Math. Soc., 106(1989), 153-161. [15] K. G. Ivanov and V. Totik: Fast decreasing polynomials, Constructive

Approx., 6(1990), 1-20.

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Index approximation property 26 approximation property 21 balayage measure 28 Bernstein's formula 13, 20 Bernstein's problem 86 Bernstein-Walsh lemma 51 Blaschke product 108 Chebyshev polynomials 51 conformal map 93 energy integral 3 equilibrium or extremal measure 5 of a set 4 extremal or equilibrium measure 6 extremum problem 91 /2-extremal problem 19 fast decreasing polynomials 79 Fekete or Leja points 26 Freud weights 1, 7 geometric mean 14 Hardy space 102 incomplete polynomials 23 infinite-finite range inequality 13 Jacobi weights 24 Laguerre weights 25 Lipshitz condition 53 logarithmic capacity 3 logarithmic energy 3 logarithmic potential 2 Maria's theorem 25 Pad~ approximation 106 Nikolskii-type inequality 19 orthogonal polynomials 11, 91

asymptotics for 99 leading coefficients of 11 recurrence coefficients of 11 outer function 100 parallelogram law 97 principle of domination 25 quasi-everywhere 3 rational interpolant 107 remainder term of 107 regular set 2 restricted support 22 strong asymptotics 10, 12 Szeg6 condition 93 Szeg6 function 100 trigonometric conjugate 98 Ullman distribution 8 weight function 3 admissible 3 weighted polynomials 1 Wiener's condition 2