Israel Journal of Mathematics, Volume 4, Issue 3 (1966)

ON A DIFFERENCE EQUATION ARISING IN A LEARNING-THEORY MODEL BY BERNARD EPSTEIN* ABSTRACT

An analysisis presented of the equation f ( x + a) -- f ( x ) = e - x {f(x) -f.(x--b)}. Here a and b denote arbitrary positive constants, and a solution is sought which satisfiesthe followingconditions: f(-- ~ ) =0, f( + cx3)= 1, 0 =3q<(x) =<1• Existence and uniqueness,of solution, are established,, and then an analytical form of the solution is obtained by use of bxlateral Laplace transform. 1. This paper is devoted to a study of the difference equation (1)

f ( x + a) - f ( x ) = e - X { f ( x ) - f ( x - b)},

which arose in connection with a certain model for a/earning process. (See [1], [2].) Here a and b denote positive constants and f ( x ) denotes, for each value of x, a probability of absorption, so that the inequalities 0 < f ( x ) < 1 must be satisfied. The appropriate boundary conditions are given by f ( - o o ) = 0, ( + oo) = 1. Since f does not denote a cumulative distribution function, monotonicity is not required. However, it will be shown that there exists a unique bounded function which satisfies (1) and the prescribed conditions at + 0% and that this solution is monotone; thus, the inequalities 0 < f ( x ) < 1 will certainly be satisfied. It is immediately seen that (1) possesses at most one continuous solution satisfying the boundary conditions. For, if g(x) were the difference of two distinct such solutions, it would satisfy (1) and vanish at _ 0% so that it would attain a positive maximum (or negative minimum) on a non-empty compact set S. Setting x in (1) (with f replaced by g) equal to the maximum of S, one immediately obtains the desired contradiction. A simple modification of this argument suffices to show that uniqueness also holds under the weaker hypothesis of boundedness. Suppose now that f ( x ) is a function of bounded variation which satisfies (1) and the boundary conditions. From the continuity of the factor e -x it follows that the right-hand limit, f ( x + 0), also satisfies (1), and the same is then true of the right-hand discontinuity, Received, March 8, 1966. * Research supported by the National Science Foundation, Grant GP-2558. 145

146 (2)

B. EPSTEIN

[September

s(x) = f ( x + O) - f(x).

If f(x) possessed any right-hand discontinuities, s(x) would assume non-zero values on a non-empty set {x~, x2, ""}, finite or denumerable, and the inequality ~ Is(

(3)

)l < oo

would hold. Assuming that s(x) takes positive values, we choose for x the point at which it attains its maximum (or the largest of these, if there are several); we thus obtain s(x + a) - s(x) < O, s(x) - s(x - b) > O, contradicting (1) (with f ( x ) replaced by s(x)). If s(x) is supposed to assume negative and zero values only, a similar argument is employed. Therefore, s(x) must vanish identically; similarly, f ( x ) possesses no left-hand discontinuities, and so it must be continuous. We shall now obtain, by two different methods, a monotone and a continuous solution, respectively, of (1), both satisfying the boundary conditions. It will then follow from the above remarks that the two solutions coincide. 2. In this section we shall construct by iteration a monotone solution of (1). It will be convenient to re-write (1) in the form

(4)

f ( x ) = w(x)f(x + a) + (1 - w(x))f(x - b),

where w(x)= eX/(1 + eX). It is natural to attempt to construct a solution by iteration, beginning with a more or less arbitrary initial function fo(x): (5)

fn+l(x) = w(x)f,(x + a) + (1 - w(x))f,(x - b),

n > O.

It is immediately evident that if fo(x) satisfies the boundary conditions, the same will be true of all the succeeding functions fl(x), f2(x),-". The monotonicity and the positivity of the factors w(x) and 1 - w(x) are readily seen to imply that, if fo(x) is monotone, all the succeeding functions will also possess this property. Finally, the positive character of the factors w(x) and 1 - w(x) evidently guarantees that if fo(x) is so chosen that fl(x) > fo(x) holds everywhere, then, more generally, fn+l > f,(x) will also hold everywhere; similarly if the inequalities are reversed. Now suppose that we can find monotone functions, fo(x) and go(x), both satisfying the boundary conditions, such that the inequalities f o ( x ) < f l ( x ) , go(x) > gl(x) hold everywhere. Then it is immediately evident that the sequences fo(x),fl(x),"" and go(x),gl(x),.., converge pointwise to monotone functions, f(x) and g(x) respectively, which satisfy (1). If, furthermore, fo(x) < go(x) holds everywhere, it then follows immediately from (5) (and the analogous equation for the g's) that f,(x) < gn(x) everywhere. Thus, f(x) < go(X), and so (6a)

0
-

< go( - oo) = O.

1966]

DIFFERENCE EQUATION IN LEARNING-THEORY MODEL

147

Similarly, (6b)

1 = go( + ~ ) > f ( +

c~) >__f0( + ~ ) = 1.

Thus, f ( x ) satisfies (1) and the prescribed boundary conditions; similarly for g(x), and so, by the introductory remarks, f ( x ) = g(x). It remains to demonstrate a pair of functions {fo(x),go(x)} satisfying the conditions imposed in the preceding paragraph. Simple calculations show that the following choices will suffice, provided a > b:

-

(7a)

fo(x) =

{

2

k(k -

2k(k

1)

/

1)} '

na<x<(n+l)a,

f

_.4_a( x _ ~2), (7b)

x < --y,

n>l;

2a log 2 b

x > --7.

go(x) = l 1,

(The aforementioned restriction, a > b, is immediately set aside by the following observation: If the unique solution which has been shown to exist for a = b is denoted as f ( x ; a , b ) , then 1 - f ( x;b,a) satisfies (1) and the boundary conditions when b > a. Therefore, we may confine attention to the case a > b.) Thus, we have demonstrated the existence of a solution to the given problem, and it follows from (7a) and (7b) that the boundary values are approached with Gaussian rapidity: (8a)

f ( x ) = O(exp { - x2/4a}),

x ~ - oo,

(8b)

f ( x ) = 1 - O ( e x p { - 1 -e)x2/2a}),

x~ + ~.

3. We now proceed to obtain an analytical expression for the solution whose existence has been demonstrated in the preceding section. The rapid approach of f(x) to its limiting values at + ~ , as indicated by (8a) and (8b), guarantees the existence, for all values of the complex parameter s ( = o" + it), of the bilateral Laplace transform

(9)

f?ooe -~:'{f(x) - u(x)} dx.

148

[September

B. EPSTEIN

[Here u(x) denotes the "unit step-function" 1/2(1 + sgnx).] Therefore, the integral (10)

f(s) =

x)e-'Xdx

exists for a > 0 and may be analytically continued to the whole (finite) s-plane except for a simple pole of residue + I at the origin. From (1) one readily obtains, for a > O,

](s){e °~- 1} = f(s + 1){1 - e-a(~+l)}, or 1 -- e -b(s+l)

(11)

f(s) = f(s + 1)e -"~

1-

e -as

To eliminate the factor e -"~ , let

(12)

f(s)=exP { 2 (s2-s)} } g(x).

Then from (11) and (12) one obtains 1 -- e -b(s+l)

g(s) = g(s + 1)

1 -- e -as

'

and more generally, for any positive integer k,

(13)

1 -- e -b(s+k) g ( s + k --

1) = g ( s + k ) 1 - e - a ( s + k - t ) "

Writing out (13) for k = 1,2,...,n and then multiplying and simplifying, one obtains

g(s) = g(s + n)

1 -- e-a(s+n)

1 - e -o~

fi {1 - e -b("+k)}

k=l

fi

{I - e -"c~+~ }

k=l

(14) Letting n ~ oo, one easily sees that the two products appearing in (14) converge for all s, not only for tr > O; it follows that g(s + n) converges to an entire function, which will be denoted as h(s). Since h(s) = l i m ~ ~og(s + n) = lira..., oog(s + (n + 1)) = lim~_,® g((s + 1) + n), it follows that

1966]

DIFFERENCE EQUATION IN LEARNING-THEORY MODEL

(15)

h(s + 1) = h(s).

149

Thus, from (14) one obtains

(16)

f l {1 - e -~(~+k) }

h(s) g(s) = i -

k=l

e -~

f i {1 - e -°~+k) } k=l

and hence co

e ("/2)~2 h(s) k~l {1 - e-bO+k)} (17)

f(s) =

2 sinh as~2 f i {1 - e -"(s+~')} k=l

Now, the infinite product appearing in the numerator in (17), which we shall henceforth denote by b(s), possesses simple zeroes at the points s = - k + (2rmri/b) (k > 0, m ~ 0), while the other product, which will be denoted a(s), has simple zeros at s = - k + (2mzi/a) (k > 0, m ~ 0). Thus, the quotient of these products is analytic and different from zero at the negative integers, but possesses simple poles at the points s = - k + (2mrci/a) (k > 0, m # 0). (Actually, this assertion is justified only if b/a is irrational, for otherwise some of the non-real zeroes of the numerator and denominator cancel each other out; nevertheless, the final result is easily seen to be valid for rational as well as irrational values of b/a.) Since f(s) must be analytic everywhere except at the origin, the factor h(s) must possess zeroes at the non-real zeroes of a(s) and of sinha2/2; i.e., at the points - k + 2mrri/a (k > 0, m # 0). From the periodicity property (15) it then follows that h(s) must vanish at the points - k + ( 2 m r c i / a ) ( k ~ O , m ~0). Now, the theta-function 01(ns, exp ( - 21rZ/a)), which for brevity will henceforth be denoted O(s), is entire and possesses simple zeroes at the p o i n t s - k + 2mrci/a (k ~ O, m ~ 0); furthermore, (18)

O(s +

1) =

-

O(s).

Hence, the entire function O(s)/sin zs has simple zeroes at precisely the points which have been shown to be zeros of h(s), and it is periodic with period one. It follows that

O(s) (19)

h(s) = finns H(s),

where H(s) is also an entire function with period one. From (17) we now obtain

September

150

B. EPSTEIN

(20)

H(s) = a(s) 2 sinh as/2 sin zcs ~ b(s) exp(as2]2) ~(s) f ( s ) .

It will be shown next that H(s) must reduce to a constant. For all values of s, the inequality (21a)

12sinha"2sin '] _<_2exp {al- - 2 ~l - atr2 + --~a,2+ exp(as2[2)

I

}

is easily verified; furthermore, for some positive integer N, the inequality

[ b-N] o(s) <2

(21b)

holds whenever ~ > N, since each of the functions a(s), b(s) is readily seen to approach unity as a ~ + oo. Now, to obtain a suitable estimate on 10(s)I, one employs the identity (22)

0

s + 2~ia

-2~is+~

-0(s)exp

.

Using (22) repeatedly, one obtains for any positive integer n the equality -

-

°

a

Thus, by setting s = a + (gila), one obtains the following inequality, which is valid everywhere on the line t = (~(2n + 1)/a : (24)

]0(s)l > Cexp { 2n2n+a2n2n2 } '

10(s) l

where C denotes the positive minimum of on the line t = rc/a. Finally, since f(x) is everywhere positive, it follows from (10) that the inequality

lf(s)l

(25)

holds everywhere in the half-plane a > 0. Taking account of these several inequalities, one finds that, on the line segment N < a < N + 1, t = (n(2n + 1)/a), the inequality (26)

IH(s) I < =

C' exp 2rc2n a

holds, where C' = (4/C)exp(3rc2/2a). By periodicity, however, the above restriction N < tr < N + 1 may be dropped, so that (26) holds everywhere on the line t = n(2n + 1)/a. Now let s be confined momentarily to real values. As a smooth periodic function, H(s) can be expanded in a convergent Fourier series:

1966]

DIFFERENCE EQUATION IN LEARNING-THEORY MODEL

H(s) =

~, Cke 2*i*', Ck = ~l H(s)e

151

ds.

do

k=-oo

Employing the Cauehy integral theorem, one obtains (28)

Ck =

fr

H(s)e - 2,ak~ds,

where Fn denotes the path consisting of the vertical and upper sides of the rectangle with vertices at 0, 1, ~(2n + 1)i/a, and 1 + (~(2n + 1)i/a). By periodicity, the integrals along the vertical sides cancel, and one is left with the integral along the upper side of the rectangle. Taking account of (26), one obtains (29)

--a--/ + n(2k + 1)]

I ck I < C' exp

Since n may be chosen arbitrarily large, it follows from (2) that ck must vanish for negative values of k. For positive values of k the same result is obtained by integrating in the lower half-plane. Therefore, H(s) must reduce to a constant on the real axis, and hence, as asserted above, everywhere. The value of this constant is determined by the condition, stated after (10), that f(s) must have a residue of + 1 at the origin. In this way we are led to the formula

(30)

C" expas2/2 0(s) b(s) f(s) = 2 sinh as~2 sin rcs a(s) '

where (31)

rra Cp

__

_

a(0)

_

0'(0) b(0)"

Thus, the function f(x) whose existence was demonstrated in the previous section must admit the integral representation (32)

1 fo°+i~e~Xf(s)ds, y(x) = 2rri ~oo-,oo

ao>0.

Alternatively, one could have obtained (32), independently of the considerations of the preceding section, simply by assuming the existence of a solution to the given problem whose behavior near _+ oo permits the use of the analytic device which we have employed. It is then quite simple to justify this procedure a posteriori by showing directly that (32) defines a continuous functionf(x) satisfying (1) and the prescribed boundary conditions; as might be expected, the proof that the boundary conditions are satisfied involves a suitable change in the path of integration and an application of the Riemann-Lebesgue lemma. (In fact, f(x) is easily shown to be analytic in the complex variable x in a strip of width 2rr about the real axis, and to depend continuously on the parameters a and b.) However,

152

B. EPSTEIN

we have been unable to establish directly from (32) either the monotonicity off(x) (or even the inequalities 0
f(x)

(33)

~ exp{ = n=o ~=_~

exp

{

(x-na)22a + x2"-----na} (x - na) 2a

+

It may be remarked that no assumption of any sort, even of boundedness or measurability, is needed. It might be of interest to investigate whether uniqueness can be demonstrated in the case a # b under conditions appreciably weaker than boundedness. Finally, we remark that by taking account in (30) and (32) of the expansion

O(s)= constant.

(34)

~, ( -

1)kexp{-2~Zk(k + 1)/a} sin(2k +

1)zcs

k=O

we may obtain, in the case a ~ b, a series expansion off(x) which constitutes a generalization of (33), the rapid convergence of (34) will presumably be reflected, for "reasonable" values of a and b, in the rapid convergence of the expansion of

f(x). 5. I wish to express appreciation to Professor N.J. Fine, with whom I have discussed this problem. His solution, worked out quite independently, has some overlap with the one presented here. REFERENCES

1. L. Kanal, Psychometrika, 27, (1) (1962), 89-104. 2. L. Kanal, ibid, 105-109. UNIVERSITY O1~ NEW MEXICO AND OFFICE OF NAVAL RESEARCH

THE THEOREMS OF LOEWNER A N D PICK BY WILLIAM F. D O N O G H U E , JR.

ABSTRACT The study of the exact interpolation of quadratic norms in vector spaces depends in an essential way on the theory of monotone matrix functions developed by Loewner in 1934 [4]. This theory, in its turn, depends on Loewner's solution of a problem of interpolation by rational functions of a certain class. The discussion of this latter problem is necessarily complicated, and Loewner's text does not lend itself to ready reference. It has therefore seemed worthwhile to recast a portion of Loewner's results in a form more suited to the applications we have in view. Our work, however, is not wholly derivative; none of our theorems are explicitly stated by Loewner and our arguments, which are of a more geometric character, are essentially different. The knowledgeable reader will note that our hypotheses are slightly stronger than Loewner's and that our results are therefore also stronger. For the applications which we have in mind, Theorem 11I is the most important result; the proof of this theorem depends on all of the previously developed theory.

1. Introduction. We consider the class P of functions qS(0 analytic in the upper half-plane with positive imaginary part: ~b(0 -- U(0 + iV((), v(O >-_O. A convenient summary of the properties of this well-known class may be found in [1]. In particular, a function is in P if and only if it has a representation of the form (1)

~b(O = a ~ + f l +

2-(

22+1

where 0~ = 0, fl is real and p a positive Borel measure on the real axis for which j'(22 + 1)-ld/@~) is finite. The representation is unique. If (a, b) is the open interval a < x < b of the real axis, by P(a, b) we denote the class of those functions in P which are real and regular on the interval (a, b) and which therefore admit an analytic continuation into the lower half-plane which is given by reflection. It is not difficult to show that qS(0 belongs to P(a, b) if and only if the corresponding measure/~ has no mass in the interval a < 2 < b. It is important to note that the class P(a, b) has a certain compactness property: I f ~bn(O is a sequence in P(a, b) such that for a pair of distinct points z' and z" of the interval the sequences ~bn(z') and ~b~(z") are bounded, then there exists a Received May 20, 1966.

I53

154

W.F. DONOGHUE, JR.

[September

subsequence of those functions converging uniformly on closed subintervals of

(a, b) to a function ¢0(() in P(a, b). We do not give the proof in detail, but remark that the representation (1) for functions in P may equally well be written (2)

¢(()=~(+fl+

f

4(+1 4--(

d/~(4) 42+1

and if the coefficient ~ is thought of as a positive mass at infinity, we obtain a positive measure dr(4) on the compactified real axis consisting of that mass at infinity and the distribution (42 + 1)-ldp(4). This is a Borel measure of finite total mass, and to measures of this kind we may apply Helly's theorem. We will have ¢.(z') - ¢.(z") z' - z" =

: + J

din(4) (4 -

- z")

and since there exists a positive C such that C-1(2 2 + 1) < (2 - z')(2 - z") < C(2 2 + 1) for all 4 outside (a, b), the boundedness of the numbers ¢.(z') and ¢.(z") implies the boundedness of the total masses of the corresponding measures dr.(4). We also note that the subsequence of ¢ . ( 0 converges to ¢o(0 at all points which are bounded away from the supports of the measures dv.(4). In the special case when the functions of the sequence are all rational functions of degree at most N, the measures dv. will consist of at most N point masses, one of them possibly at infinity; the limiting measure dvo will have the same property and hence ¢0(0 will also be rational, of degree at most N. Suppose that ~b(0 belongs to P(a, b) and that N is an integer > 1; let ~1, ~z, Ca, "",~N be any set of N distinct points in (a, b) and let similarly r/i, ~/2, ~/3, "", r/N be another set of N distinct points in that interval. We make no hypothesis requiring that these two sets be disjoint. Form the matrix M of order N defined by M i j = ¢ ( ¢ i ) - COb) ~ - rb

i f ~i ~ tb

~b'(~)

if ~ = r/i.

Evidently the matrix elements Mii are non-negative. THEOREM 1. If ¢(~) belongs to P(a, b) and M is a corresponding matrix of order N then (i) det ( M ) = 0 if and only if ¢(0 is a rational function of degree at most N - 1. (ii) If the sequences {~} and {~b} are both monotone increasing then det(M) > 0. (iii) If det(M) = 0, the function ¢(0 is uniquely determined by the data, i.e.

1966]

THE THEOREMS OF LOEWNER AND PICK

155

the points {~,}, {t/j} and the values of ~(~) at those points, as well as the values of its derivative if such values occur in M. Proof. We first prove (i) and (ii) under the special hypothesis that ~ = 0 in the representation (1) for cP(O. Using that representation, we easily find for all i and j that

Mij =

f

d#(2)

(4 - ~i)(4 - t/j)

even when ~ = t/j. We then introduce the N functions f~(X) = I/(2 - ~ ) which are integrable square relative to the measure/~, and similarly the N functions gr( )

=

1 2 - qr

which have the same property. Clearly M~r = (fi, gr), the inner product being taken in L2Q2),and the determinant of M is a kind of Gramm's determinant. If q~(0 is rational, of degree at most N - 1, the measure/~ consists of point masses at the poles of qS(0, and there are at most N - 1 of them. Thus the space L2Q2)has dimension at most N - 1 and there exists a non-triviallinear dependance between the fl, whence d e t ( M ) = 0. Conversely, if the determinant vanishes, there must exist a linear combination f * ( 2 ) ' = 2~=1 c~fi(2) which is orthogonal to every gj(2). We may write f*(2) = p(2)/Q(2) where Q(2) = I-[~=1(2 - ~t) and p(2) is a polynomial of degree at most N - 1, since the rational f*(2) vanishes at infinity. If R(2) = N j N=1(,~-- t/j), the function g*(2) = p(2)/R(2) is a linear combination of the functions gr(2), and hence (f*, g * ) = 0. This may be written

f

PO') 2 d#(2) = 0 Q(;t)R(2)

and we deduce, since the denominator is bounded from below on the support of # by a positive constant, that the measure # is concentrated at the zeros of p(2), a set consisting of at most N - 1 points. Thus $(() is rational, of degree at most N - 1 . If we suppose that ~b(0 is not rational of degree at most N - 1 and that the points {~i, t/r} are given as monotone sequences, we vary the {t/r} by writing t/r(t) = t~r + (1 - t)t/j. As t varies over the unit interval, det M(t)is a continuous function of t. For any t there will be N distinct t/j(t) as well as ~, hence the determinant will never vanish and therefore keeps a constant sign. For t = 1, however, we have ~i = t/t, whence fi(2) = g/(2) for all i, and therefore M is a Gramm's determinant. Since it does not vanish, it is positive, whence det M(0) > 0. For 4~(~) rational of degree at most N - 1, det M(t) is of course identically zero.

156

w . F . DONOGHUE, JR.

[September

We next establish (i) and (ii) without the special hypothesis g = 0. From the formula (1) it is easy to deduce that the non-negative c¢is the limit as ~ approaches infinity along the imaginary axis of the ratio ~b(~)/~. Accordingly, if qg(~) in P(a, b) corresponds to a positive g and is positive in that interval, the function ~(~) = - 1/q~(~) is also in P(a, b) and corresponds to g = 0. If M* is the matrix corresponding to ~k(~), it is easy to see that d e t ( M * ) = C-ldet(M) where C = 1-]~=l~b(~z)~b(r/i) > 0. Since ~(~) is rational of degree k if and only if qg(~) is, we see that (i) and (ii) hold whenever the function qS(~) is positive on (a, b), or at least on a subinterval containing the points {~} and {r/i}. Since the addition of a constant to ~b(~) does not affect the matrix M at all, and similarly does not affect the rationality or degree of the function, the assertions (i) and (ii) are valid in any case. To establish (iii) we suppose that det(M) = 0 and therefore that ~b(~) is rational of degree at most N - 1. If there were two such functions ~b(~), their difference would be rational of degree at most 2N - 2, but the total order of its zeros would be 2N, thus the difference is identically zero. This completes the proof of Theorem I. REMARK 1. We suppose that ~b(~) is of degree exactly N - 1, the determinant, therefore, being zero. Supposing, in addition, that r/i coincides with none of the {~i} we expand the determinant along the first column, displaying the dependence of terms on r/i and ~b(r/1) to obtain det(M) =

~ i=1

¢(~i)mi ~ S~-~1

N ~b(r/l) 2~ i=i

~i -

mi r/1

where the ms are appropriate(non-zero)minors of the determinant. We introduce the rational functions F(~) =

N

N

I: ¢(~,)mi and G ( ~ ) =

]g

mi

and note that since r/i may be chosen almost arbitrarily in the interval 0 = det(M) = F(~) - ¢(~)G(~), whence ~(~) =

(a, b),

F(~)/G(~).

The determinant is obviously a linear function in the entry ¢(r/~) and our purpose here is to emphasize that this linear function is not identically zero. We have 0 = F(r/1) - ¢(r/1)G(r/i) and if this function vanished identically in the variable ¢(r/1) we would have 0 = F(r/1) = G(r/l); it would follow, since G(~) and F(~) have a common zero at ~ = r/1 as well as at infinity that their ratio was a rational function of degree smaller than N - 1, contradicting the hypothesis that ¢(~) is of degree exactly N - 1. It is not difficult to extend this argument to the case when r/~ happens to coincide with one of the points ¢~. We shall often make use of this remark in the sequel where we will have a function ¢(~) of degree N - 1

1966]

THE THEOREMS OF LOEWNER AND PICK

157

in the class P(a, b) and a set of 2N points of the interval, as well as another function g ( 0 defined on those 2N points and coinciding with ~ ( 0 at all but one of them. Then the hypothesis that the determinant of type M computed for g ( 0 vanishes will imply that g(() coincides with ~b(0 on all 2N points. REMARK 2. For the applications of Theorem I it is incovenient that the theorem requires that the points {~} and {t/j} be interior points of the interval (a, b). We therefore introduce the class P[a/b] consisting of those functions in P(a, b) which are continuous on the closed interval [a, b]. The proof of Theorem I carries over to functions in this class, where we permit a choice of the points {~i) and {~/j} which may include one or more of the end points. Here, however, we cannot admit that one of the end points be taken both as a ~ and an ~/, since the derivatives ~b'(a) and ~'(b) may be infinite for some functions in P[a, b]. The proof of this variant of Theorem I is virtually the same, except that if, say, 41 = a, the corresponding function f1(2) = (2 - a) -1 may no longer be in the space L2(/0; however, the integrals which we have written exist in any case. 2. The cone P(Z)).

We suppose given on the real axis a finite set Z of l points: Z1 <

Z2 <

Z3 <

...

<

Zl

and let C(Z) denote the space of all real functions f(z) defined on Z. C(Z) is a real/-dimensional vector space containing a convex cone, P(Z), the restrictions to Z of functions in the class P[zl, z~]. For any f i n C(Z) we introduce its Loewner determinants, which are defined as follows. For any subset S of Z consisting of an even number of points we write the points of S in the following fashion: ~1 < /71 < ~2 < /72 < "'" < ~N < ~N

and form the matrix M, where M~j = f ( ¢ i ) - f ( ~ j )

The determinant of this matrix is the Loewner determinant associated with f and S. Evidently there are as many Loewner determinants as there are nonempty subsets of Z of even cardinal, viz. 2 ~- 1 _ 1 of them. The following assertion is an immediate consequence of Theorem I. If f(z) in C(Z) belongs to P(Z), then I. All of the Loewner determinants o f f are non-negative and II. If a Loewner determinant o f f vanishes, so also do all other Loewner deter-

minants of the same or higher order. It is our purpose to show that these conditions are also sufficient for f in C(Z) to belong to P(Z). To establish this, it is necessary to study the convex cone P(Z) in greater detail. We consider it first under the auxiliary hypothesis that I is odd:

I = 2N + I, N ~_ I.

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LEMMA 1. If f belongs to P(Z), it is an interior point of that cone if and only if all of its Loewner determinants are positive. Proof. Suppose, first, that a Loewner determinant vanishes; from Theorem I it follows that f is the restriction to Z of a uniquely determined function in the class P[Zl, zd which is rational of degree at most N - 1. Suppose k is the degree of that function and consider a Loewner determinant of order k + 1. This determinant vanishes. From Remark 1 we see that the determinant is not identically zero in the variablef(zl). It follows that if the value o f f at zl is slightly changed in the appropriate direction, the Loewner determinant becomes negative and the perturbed function is not in the cone P(Z). Thus f i s the limit of elements in the complement of P(Z) and hence is a boundary point of the cone. We note also that this part of our argument does not depend on the parity of 1. Conversely, if all of the Loewner determinants are positive f is the restriction to Z of a function ~b(0 in P[zl, zd which is not a rational function of degree smaller than N. Suppose, first, that ~b(O is rational of degree N, and indeed of the form N

~b(~) = fl + X~

ms

Here the ms are positive and the poles 2 i are outside the interval [zl, zt]. We adjoin 2N + 1 ---- l real variables: c, al, a2, "', aN, bl, b2, "", bN to form

r(¢,a,,b~,c) =(13 + c) + [ s=l

(1 +

ai)ms 2i + b i - ~ "

For small values of al and bi this is a function in the class P[zl, zd; its restriction to Z then determines a mapping of a neighborhood of 0 in the space of I real variables into a neighborhood of f i n C(Z). We have only to show that the mapping is onto, i.e. that its Jacobian does not vanish at the origin. If we compute the partial derivatives of F at the origin we find 8F

~ai

_

m~

2~- ~

dF _

abj

-mj

a F _ 1.

(,~j - 0 2

Oc

If the Jacobian vanishes, there is a linear combination H(0=B+

N Ci Z ~ + i=1 ~i ~

N D~ Z (2i ~)2

S ffil

--

which vanishes on the set Z and hence has 2N + 1 zeros. Since the total order of the poles is at most 2N this is impossible. The same argument goes through if we suppose that the function ~b(0 is rational, of degree N, but with a pole at infinity, viz.

1966]

THE THEOREMS OF LOEWNER AND PICK N-1

159

ms

~(~) = ~ + ~+ ~_~ ,~- (The auxiliary functions are

F((,a. bi, c)

= ~(1 +

. (~ + bN) aN)([ -- ~-b-NN)+ ~ + c ) +

N-1 i ffil

(1 + ai)mt 2 i + b~ -

We remark next that if ~b(() can be decomposed in any way into a sum q~l(~) + ~b2(() of functions in P there exists a corresponding decomposition of f = f~ + fz in P(Z); if one of the terms is an interior point of P(Z) t h e n f i s too. It follows that if the measure/t associated with ~b(() has N distinct point masses t h e n f i s an interior point of P(Z). To complete the proof, therefore, we need only consider the case when the measure # has no point masses, and we may suppose that # is concentrated on an interval I lying wholly to one side of the interval [zl, z~]. On that interval we consider the 1 functions hk(,~) = 1 2 - Zk 1-1 which will be linearly independant in L2(p) since otherwise there would exist a linear combination of them which vanished for infinitely many points in I and was a rational function of 2. For a fixed small positive e we may write ~b(() in the form q~(0 = ~ ( 0 + e ~=l~bi(() where ~b(0 is in P[zl, zl] and ~bi(~) = fhi(2)d#(2)/(2 - ~) is in the same class. The functions ~bi(0, when restricted to Z, form a linearly independant set in C(Z) since the determinant of the matrix Hij = ~bi(zk) is essentially the Gramm's determinant (hi, hi) and therefore cannot vanish. Thus for small coefficients a~ the functions l

~k(~) + e ~

(1 + ai)~bi(O

iffil

map onto a neighborhood of f in C(Z). LEMMA 2. Every f in P(Z) is a restriction to Z of a function in P[zl, zl] which is rational and of degree at most N. This function is unique. Proof. The uniqueness is obvious, since if there were two such functions their difference would be rational of degree at most 2N but would have 2N + 1 zeros. To prove the lemma, we consider the set R(Z), the subset of P(Z) consisting of restrictions to Z of rational functions of degree at most N which are in P[zl, zd. We must show that R(Z) coincides with P(Z). Clearly R(Z) contains all of the boundary points of P(Z) which belong to that cone. Moreover, if f belongs to R(Z) and is an interior point of P(Z), the argument of the second part of the proof of the previous lemma shows t h a t f i s surrounded by a neighborhood which belongs itself to R(Z). Thus the interior points of P(Z) which belongs to R(Z) are

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interior points of the latter cone. To complete the proof, we must show that no interior point f of P(Z) can be a boundary point of R(Z). If such anfexists there is a sequencefn(z) in R(Z) converging tof(z) uniformly on Z. Using the compactness property mentioned in the Introduction we deduce that there exists a rational function ~o(~) of degree at most N which coincides with f(z) at all points z in Z bounded away from the supports of the measures #n occuring in the representation (l) for f~(0. Thus we have (~o(zk) --f(zk) for all k satisfying 2 -< k -< ! - I. We cannot have f(z,) = O0(zl) and f(z,) = ~o(Zl) simultaneously, since f by hypothesis is not in R(Z). Suppose, say, f(zl) ~ (~o(zl), but f(z,) = ~)0(zz). Since f(zl) = lim~fn(zl) it follows that the point z I is not bounded away from the supports of the measures g#; in more simple language, the functions f~ have poles ,~ near z, and with increasing n those poles converge to z,. Such poles correspond to terms of the form m~/(2n - z,) and it is easy to infer from the boundedness of the numbers f~(zl) that these terms are uniformly bounded in absolute value. Thus, since the denominators tend to zero, the numerators m~ also do, and we infer that the function (~0(0 has not so many poles as thef~(~); i.e. degree ~o(~) is at most N - I. It follows that the Loewner determinant computed for ~o(~) and the set z2, z3, z3, "", zt vanishes, and since this is also a Loewner determinant for f, that function has a Loewner determinant which vanishes. This contradicts the hypothesis that f was an interior point of P(Z). In the case thatf(z) differs from ~bo(Z)at both endpoints, the function fro(Z) is of degree at most N - 2 since at least two poles have disappeared in the limiting process. We then argue as before using a Loewner determinant of lower order. The uniquely determined rational function associated with f in P(Z) by the previous lemma will be called the canonical representation off. Note that it exists only when l is odd. LEMMA 3.

A boundary point f of P(Z) belongs to that cone if and only if H is

satisfied. Proof. We suppose that f is a boundary point of P(Z) which does not belong to that cone but which does satisfy II and deduce a contradiction. Since f is a limit of a sequence of interior points of P(Z), each representable as a rational function in P[zl, z,] of degree at most N, we argue as in the previous lemma to find a rational ~b0(~) in P[z 1, zz] of degree k < N - 1 which coincides with f at all points of Z except perhaps the end points. Since f is not in P(Z), f cannot coincide with ~b0(~) at both end points. If k < N - 2 there exists a subset S of Z consisting of 2k + 2 points and not containing the end points z~ and z~; the corresponding Loewner determinant of ~bo(~), and therefore off, is zero. Since f satisfies II, all Loewner determinants of that order for f vanish, in particular the one computed for the system of points zl, z2, z~,..., z2k+2. Because f coincides with ~b0(~) at all of th (points except

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THE THEOREMS OF LOEWNER AND PICK

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perhaps zl, the Remark following Theorem I guarantees that f ( z l ) = qbo(Zl) as well. We argue similarly to show f(zl) = $0(zl), whence f coincides with tko, a contradiction. If k = N - 1 the function ~bo(0 coincides with f for at least one of the end points, because if not, then in the limiting process fn(O converging to ~b0(0 at least two masses would be destroyed, whence degree $0 < N - 1. We suppose f(zl) = ~bo(Zl) and deduce that the Loewner determinant o f f associated with the set of 2N points obtained by omitting zl vanishes. From II then, it follows that the Loewner determinant o f f associated with the first 2N points of Z vanishes, and again, by the remark following Theorem I, f(zl) = ~b0(z~), a contradiction. We pass next to the more complicated case when l is even: l = 2N, N > 1. The results of Lemmas 1 and 2 can be brought over to this case by the following device. We select a point ~ in the interval zx < $ < z2 and adjoin it to Z to obtain a set Z of l + 1 points. Every f in P(Z) is the restriction to Z of an element in P(g). The projection mapping of C(Z) onto C(Z) carries P(Z) onto P(Z) and maps interior points of P(Z) into interior points of P(Z). Thus from property II we obtain most of the following lemma. LEMMA 4. A point f of P(Z) is an interior point of that cone if and only if all of its Loewner determinants are positive. Every f in P(Z) is the restriction to that cone of a rational function in P[zl, z~] of degree at most N; this rational function is unique if and only if f is a boundary point of P(Z). Proof. Since an interior point of P(Z) possesses an extension to P(2~) which is an interior point of that cone, it is clear that there exist infinitely many choices for the value of the extension at ~ each of which corresponds to a different canonical representation of the extended function. Thus the representation cannot be unique for an interior point, however, Theorem I guarantees that it is unique when f is a boundary point, since then a Loewner determinant vanishes. The assertion of Lemma 3 is also valid when l is even, its demonstration however, is difficult. This is the content of Lemma 6 of the next section; we will assume it now and pass to the proof of our principal theorem. TrmOl~M II.

A function f in C(Z) belongs to P(Z) if and only if I and H are

satisfied. Proof. We know that the conditions are necessary for f to belong to P(Z). Our argument is by induction, the assertion for l = 2 and l = 3 being trivial. Supposing the theorem true for 1 - l, and that f in C(Z) satisfies I and II, there exists a g(z) in P(Z) so thatf(zk) = g(Zk) for all k ~ 2 by the inductive hypothesis. We form ht(z ) = tg(z) + (1 - Of(z), where 0 < t < 1. I f g = hl is an interior point of P(Z) all of its Loewner determinants are positive. Since the Loewner determinants for h t are linear in t, all positive for t = 1 and

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non-negative for t = 0, it follows that no ht for t > 0 can be a boundary point of P(Z); from continuity considerations, then, either f =/lo is in P(Z) itself, or is at least a boundary point of that cone. Because II is satisfied, it follows t h a t f is an element of P(Z). I f g = hi is only a boundary point of P(Z) it must be the restriction to Z of a uniquely determined function ~b(0 in the class P[zl, zd which is rational of degree k. If we were to suppose that no Loewner determinant o f f were zero, we choose any non-rational function $ ( 0 in the class P(zl - 1, z~ + 1) and form f + 85 for small positive 5. If 8 is sufficiently small, f + e5 has all of its Loewner determinants positive, these determinants being continuous functions of their arguments. The function g + ~5 belongs to P(Z) and is not the restriction to Z of a rational function in P[zt, z~], hence is an interior point of P(Z). Since it coincides with f + t5 at all points of Z except z2, the argument which we have just given shows t h a t f + 85 is in P(Z); the ~ being arbitrary, it follows as before that either f is in the cone P(Z) or it is a boundary point of that cone. Since f satisfies II, it must then be an element of that cone. Thus, for the balance of the proof, we may suppose that g is rational of degree k and that both f and g have Loewner determinants which vanish. We must have k < N - 1 where 1 = 2N or l = 2N + 1. There exists therefore, when l is odd, or when k < N - 2 a proper subset of Z consisting of2k + 2 points not containing z2 so that the corresponding Loewner determinant ofg (and therefore off) vanishes. Since both f and g satisfy II, the Loewner determinants computed for 2"1, Z2, "", Z2k+2

vanish, whence f(z2) = g(zz) by Remark 1. If l = 2N and k = N - 1, only the largest possible Loewner determinant vanishes, but it vanishes for both functions; we obtain, as before, f(z2)= g(z2) completing the proof. 3. Representations for even I. In this section we suppose that f is an interior point of P(Z) where l = 2N is even. Because f is interior there exist a variety of extensions of f to P(~) where Z i s obtained from Z by the adjunction of a point. Thus there exist a variety of functions, rational of degree N belonging to P[zl, z~] which coincide with f on Z. Our purpose is to study these representations. Let fo(z) and f~o(z) be two such functions; we consider their difference h(z) = fo(z) -f~o(z) which is rational of degree at most 2N. Since this function does not vanish identically and has 2N zeros as the points of Z, it follows that it has degree exactly 2N, and since its poles are simple, both fo and f® are of degree exactly N and have N distinct poles. We see that there is no loss of generality if we suppose that the poles of fo and fo0 are all finite. We write these functions as ratios of relatively prime polynomials with real coefficients:

1966]

THE THEOREMS OF LOEWNER AND PICK fo(z)

_Oo(Z) Zo(Z)

f

163

(z) =

where the denominators are polynomials of degree exactly N with real distinct and simple zeros. For the difference we then have h(z) =

-

o

(z)ro(z) _

•

T=(z)z(z)

The numerator, ~(z) cannot vanish identically and is of degree at most 2N; since it vanishes at the points of Z it has degree exactly 2N and has simple zeros at those points. We consider any two adjacent poles of h(z) in the interval z < Zl; since the zeros of h are all in the set Z, there is no zero between these poles, and hence the residues at these poles are of opposite sign. Because h is the difference of two functions in P, and such functions always have negative residues, it follows that one of this pair of poles is a pole offo and the other a pole offo0. The same argument holds if we consider a pair of adjacent poles of h in the interval z~ < z or in a projective neighborhood of the point at infinity. We conclude that the zeros of Zo(Z) and T~o(z) separate one another on the projective real axis. From the foregoing it follows that the function T(z) = - Zo(Z)/Z®(z) has residues of the same sign at all of its poles; there is no loss of generality in supposing that these residues are all negative, and therefore that T(z) itself is in P. Next we introduce the family of polynomials zt(z) = zo(z) + tz®(z) where t varies over the real axis with the convention that t = oo corresponds to zoo(z). All of these polynomials, save one, are of degree N, and the exceptional one will be supposed to have a zero at infinity. With this convention it is immediate that the zeros of zt(z) are exactly the set of points on which T(z) = t, and since T(z) belongs to P, it is monotone increasing between its poles and assumes every real value just once in any such interval. Thus for any pair of values, t, s the zeros of zt(z) and T~(z) separate one another. Moreover, all of these zeros are real and simple, and there are exactly N of them. From this circumstance it follows that the Wronskian W(z) =

-

never vanishes on the real axis since no z,(z) has a multiple zero. We may therefore suppose that this real polynomial is positive on the real axis. The values taken by the function T(z) on the set Z are called exceptional values of t; since T belongs to P[zl, z,] these values are distinct, and there are therefore 2N of them. We write t k = T(Zk) and note that ztk(z) has a zero at z = Zk but does not vanish at any other point of Z. For values of t which are not exceptional the rational functions

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A(z) = ~o(Z) + ta~(z) _ ~,(z) •co(Z) + t ~ ( z ) ~,(z) coincide withf(z) on the points of Z and are of degree at most N. It will presently become clear that they are of degree exactly N. For exceptional values of t, say t = tk, the denominator zt(z) vanishes at the point zk, however, the numerator also does, since %(z)=

a(zO = 0 [z~(zOl-*[z~(z~),ro(Z,) - 'co(Zk)~(z~)] = "coo(zk)

Hence for the corresponding ft,(z) we must understand the rational function obtained after the common factor ( z - zk) is cancelled from numerator and denominator. This function coincides with f at all points of Z other than Zk; it is of degree N - 1 and is the canonical representation for the restriction of f to the set of 2N - 1 points obtained by omitting Zk from Z. That function therefore belongs to P, however, f(zk)cannot equal ftk(Zk) since f is an interior point of P(Z). If x is a real point not contained in Z, then as t varies over the real axis the quantity ft(x) is a linear fractional transformation in t which is non-degenerate its determinant, ~(x) being non-zero. We deduce that ifg(z) is any rational function of degree at most N which coincides with f on the set Z, then g is a member of the family ft; to show this, we remark that g cannot coincide with ft when t is exceptional; thus there exists a real x not in Z for which g(x) v~ftk(x) for all k. We then select t so that f,(x) = g(x) and note that the difference g(z) - f,(z) is of degree 2N and has 2N + 1 zeros. Again, when x is real and not in Z there exists exactly one value of t such that ft(z) has a pole at x, viz. t = T(x). We compute r(x), the residue of that function at that pole. We choose a small circle C with center at x and radius 8 and write

r(x) = 1 fc aoCz) + T(x)tr~(z) 2=-~ "Co(Z)+ T(x)'c,o(z) azl~ If we suppose that T(x) is finite the integrand may be written z®(X)Oo(Z)

-

Zo(X)~(z)

•coo(X)*o(Z)- *o(X)*®(z) and for small values of 8 the numerator approaches -r®(x)o'o(X) -

"co(X)a~(x)

=

~(x)

=

(z - x ) w ( z ) .

while the denominator is approximated by (z - x ) [ ' c . ( x ) ~ ( x )

-

"co(x)e(x)]

1966]

THE THEOREMS OF LOEWNER AND PICK

165

Since e may be arbitrarily small, it follows that r ( x ) = 6(x)l(W(x)) and this rational function has no poles on the real axis and has exactly 2N simple zeros, all of which are at the points of Z. By continuity, the same formula is valid for the points x which appear as poles of T(z). Since there exists a value of t for which ft(z) is in P[z~, zz] we see that r(x) is negative for x outside the closed interval [z~, zt] as well as for x inside intervals of the form Z2k < X < Z2k+,, while r(x) is non-negative elsewhere and zero only at the points of Z. Since Z is contained in an interval between two adjacent zeros of zoo(z), an interval which we may call (a, b), every ft has precisely one pole in [a, b). Thus ft is in P[zl, zt] if and only if it has no pole in [zl, zz]. We pass to the study of the function for exceptional values of t. We have already remarked that ftk(Zk) is not equal to f(Zk);it is possible to compute the difference of these numbers. For the sake of simplicity we avoid the special value of t for whichft(z) has a pole at infinity if that value is exceptional; the functionsft may then be written in the form n

mi(t)

ft(z) = ~(t) + ~=I Y~ 2~-'~ -- z where m~(t) = - 6(2~(t))/W(2i(t)) and the 21(t) are the roots of T(z) = t. The quantity ~(t) is the value off,(z) at infinity and is a linear fractional function of t. We will suppose that 2x(t) is the root which varies from a to b. As t approaches tk = T(Zk), Ix(t) approaches zk and the first term in the sum converges to lira ~,~

- 6(x) IV(x) ( x - z , )

_ - 6'(zk) W(zk)

"

The other terms in ft(z) depend continuously on t and converge to the corresponding terms offtk(z); no term corresponding to the first appears inftk(z) even though the limit above is non-zero. Thus

~'(z,) Z~(z,) - f(zk) =

W(z,)

and the sign of this quantity depends on the parity of k. In particular, one verifies that ft,(zt) > f ( z l ) as well as f,(zz)
LEMMA 5. I f l is odd, f in P(Z) and dp(O the canonical representation off, and i f ~(~) is any function in P[z~, zt] which coincides with f at the points of Z, then ~(~) is regular in any interval containing [zx, z~] in which ~k(~) is.

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Proof. Suppose $(~) regular in an interval [z', zz] containing [zl, zl]; we adjoin z' to Z to obtain a set of 2N + 2 = 1 + 1 points Z'. The restriction of ~/(~) to Z' determines a function g in P(Z') which coincides w i t h f at the points of Z. We may suppose that g is an interior point of P(Z'), since otherwise the functions ~,(~) and $(~) coincide. It follows that $(~) is one of the exceptional functions in the family gt(~) associated with g in P(Z'), hence is regular analytic in the interval [z', zl]. We can argue similarly with a point z" to the right of z v In addition, we will have g(z') = ~b(z') < ~b(z') and g(z") = ~b(z") >- ~p(z"). In conclusion, we establish a lemma which we used in the proof of Theorem II.

LEMMA 6. When l is even, a boundary point f of P(Z) belongs to that cone if and only if H is satisfied. Proof. We suppose l = 2N. As before, we deduce a contradiction from the hypothesis that f is a boundary point of P(Z) which does not belong to that cone but which does satisfy II. Our argument is much the same as before; f is the limit of a sequence f~ in the interior of P(Z), and these functions may be represented by rational functions of degree N in P[zl, zl]. Since there is a choice for these representatives, we take them in such a fashion that the nearest poles are distributed symmetrically about [zl, zt]; more precisely, we select each time the representativefttn)for f~ so that if 2' is the nearest pole offt°)to the left of zl and if2" is the nearest pole to the right ofz~, then z~ - 2' = 2" - z,. A subsequence of the sequence of rational functions so determined then converges to a rational ~b([) which coincides with f at all points of Z except, perhaps, the end points. The representatives have been chosen in such a way that at least two poles are destroyed, i.e. ¢(() is necessarily of degree at most N - 2. Thus the Loewner determinant for ¢(~) associated with the set z2, z3, z3,'", z~_ ~ vanishes and therefore the corresponding Loewner determinant o f f does. We now argue exactly as in the proof of Lemma 3 to infer thatf(zt) = ~b(zl) andf(zl) = ~b(zs), hence t h a t f is in P(Z), a contradiction. 4. The cone P'(Z). By P' we denote the subclass of P consisting of functions which are regular and positive on the open right half-axis. These functions admit the canonical representation obtained from (1) which follows:

(3)

0o

where • _~ 0, /~ = ~b(0) > 0 and j'_°~(1 + ~2)-1d/~(2) is finite. It is easy to see that if q~(~) belongs to P', so also does 6(~)= [~b(1/~)]-1 as well as ¢*(~) = ~¢(1/£). In this section we shall suppose that Z is a subset of the open right half-axis and shall seek necessary and sufficient conditions that a function f i n C(Z) should

1966]

THE THEOREMS OF LOEWNER AND HCK

167

belong not just to P(Z) but to P'(Z), the cone consisting of restrictions to Z of functions in P'. It is clear that the cone P'(Z) is closed, for a sequence in P' which converges on the points of Z has a subsequence converging on all points of the positive real axis, those points being bounded away from the supports of the measures; moreover, the limiting function is non-negative on the right half-axis, hence is in P'. It is also evident that i f f belongs to P'(Z) there exists a non-negative value C such that i f f is extended to the origin by f(0) = C, the extended function is in P(Z u 0). Unfortunately, we cannot always take C = 0. We introduce the set Z* consisting of reciprocals of points in Z

1/zl < 1/z1-1

< 1/zt-2

< ""

< 1/Zl

which may also be written z* < z* < z* < ... < z* and consider the following conditions, concerning f in C(Z). III. f may be extended to a non-negative function in P(Z u O) IV. The functionf* defined on Z* byf*(z~) -----z~f(1/z*) may be extended to a non-negative function in P(Z* u O) V. The function J~defined on Z* by j~z*)= [f(1/z*)] -I may be extended to a non-negative function in P(Z* u 0). We then have TrmOREM III. A function f in C(Z) belongs to P'(Z) if and only if (a) when l is odd, 111 and V are valid (b) when l is even, 11I and IV are valid. Proof. The necessity is an immediate consequence of our comment concerning the functions ~ ( 0 and ~b*(O when ~b(() is in P'. For the sufficiency we must give different arguments depending on the parity of L We remark that we possess examples showing that the state of affairs is essentially different when I is odd and when l is even. When l = 2N + 1 is odd, we pass from f in C(Z) satisfying III and V to its canonical representation ~b(0, a rational function of degree at most N belonging to P[zl, zd. Since there exists a non-negative function in P[0, zl] which coincides with f on Z, it follows from Lemma 5 that ~b(0 is regular and non-negative in [0, zd, and we have only to show that this function has no poles to the right of z~, thereby putting it in P' and therefore putting f i n P'(Z). However, we may argue similarly with the functionfdefined on Z* to find that its canonical representation ~ ( 0 is non-negative and regular in [0, z*]. Since both ~b(0 and ~k(0 are rational and of degree at most N and satisfy the equations ~b(z*)= [~b(1/z*)]-1 for all 2N + 1 values of k, it follows that identically in ~ we have

~#(0 -- [~(l/O] -1.

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W.F. DONOGHUE, JR.

[September

The regularity of ~(~) in [0, z*] therefore implies the regularity of qS(~) in z z < z < + oo. Hence ~(~) is in P'. When l = 2N is even our argument is somewhat more complicated. Since III and IV surely imply that f is in P(Z) and f * is in P(Z*) we will suppose at first that each of these functions is an interior point of the corresponding cone and make use of the representation theory developed in the previous section. Let C be so chosen that w h e n f i s extended to 0 by the definition f(0) = C the extended function is in P(Z u 0); because of III there exists such a C which is non-negative. In the familyft(z) associated w i t h f w e select the functionf,(z) for which f~(0) = C; this function is rational and of degree at most N and is the canonical representation of the extended function considered on the l + 1 points of Z u 0. It follows that f~(z) is in P[0, zt]. It is not difficult to see that if t is varied so that the poles o f f t move to the right, the numberft(0) diminishes; it follows that we can pass continuously to that member of the family for whichft(0) = 0 without departing from the class P[0, zt]. We let t = 0 correspond to the rational function so determined; fo(z) is of degree at most N and belongs to P[0, zz] and satisfies fo(0) = 0. Since fo(z) is non-negative in [0, zl] we have only to show that it has no poles to the right of z~ to make sure that it belongs to P'. For this purpose we pass to the ratonal function g(~) = ~fo(1/~) which is also of degree at most N and which coincides with f * on the points of Z*. It follows that g(£) is a member of the familyft*and therefore that the residue ofg(~) at any pole to the left of z* = 1/z~ is negative. If, now, fo(~) had a pole to the right ofzz, g(~) would have one in the interval 0 < z < z* and the residue there would be negative. However, if we make the explicit computation we will have fo(~) = h(~) + ~

m

where m > 0 and ~ < z t with h(~)

regular near 2, and g(¢) = CY(ll¢) +

-

which has a positive residue at the pole 1/2. Thus fo(~) had no pole to the right of z, and was therefore in P'. Finally, if the functions f and f * are not both interior points of their respective cones, we have only to pass to f + e x/z which corresponds to f * + e x/z for small positive ~. The perturbed functions are interior points of those cones and also satisfy III and IV. From the fact that P'(Z) is closed we infer that f i s inP'(Z), completing the proof of Theorem III. We do not give the easy proof that an element f i n P'(Z) may be extended to the origin by f(0) = 0 if f is an interior point of P(Z).

1966]

THE THEOREMS OF LOEWNER AND PICK

169

5. The theorems of Pick and Carath~dory. The problem considered in Section 2 can equally well be studied under the hypothesis that the finite set Z is a subset of the open upper half-plane; we would then seek conditions for a function in C(Z) to belong to the cone P(Z), the restrictions to Z of functions in P. The solution has been given by Pick. [5]

THEOREM. A function f(z) in C(Z) which is not a real constant is the restriction to Z of a function c~(~) in P if and only if the imaginary part off(z) is positive and the matrix of order l

Pij

--

f(z~) -- f(zj) z i -- ~j

is a positive matrix. This matrix has the eigenvalue 0 with multiplicity k > 0 if and only if qb(~) is a rational function of degree I - k and in this case ~(¢) is determined uniquely by the data. We do not give a proof of this theorem which can be established by the same arguments which we have used to prove Theorems I and II, the proof, however, is substantially easier since in the present case the cone P(Z) is closed and we may also always argue with positive matrices rather than with determinants. A completely analogous theorem is due to Carath6odory who considered the convex cone of functions u(z) harmonic and positive in the unit circle; such functions admit a Fourier expansion

u(r e i°) = Z

ckrlkl eikO

the summation being taken over all integers. The following theorem is due to Carath6odory. TI-mOREM. A system of numbers {el} - N <- i <- N form the Fourier coefficients o f order < N of a positive harmonic function u(z) if and only if the matrix of order N + 1 defined by

Ctj = c~_y is a positive matrix; 0 is an eigenvalue of Cii with multiplicity k > 0 if and only if u(z) is the real part of a rational function of degree N + 1 - k, and in this case u(z) is uniquely determined by the coefficients. Since the class of positive harmonic functions in the circle is in a one-to-one correspondence with the positive harmonic functions in the half-plane and since the latter are the imaginary parts of the functions in P, the similarity between the foregoing theorems is to be expected. It is important to note a geometric fact: in both cases one studies the projections on a linear space of dimension I of a certain cone in an infinite dimensional space, and the projections are cones which

170

W. F. DONOGHUE, JR.

inherit a certain property of the original cone, viz. the set of (suitably normalized) extreme points forms a skew curve. The cones thus have an extraordinary multiplicity of faces of lower dimension. This geometric situation is more conveniently studied if a suitable normalization condition reduces the study to one of a convex body; we then obtain a convex polytope and the study reduces to the study o f "neighborliness" introduced in recent years. [3] REFERENCES 1. N. Aronszajn, and W. F. Donoghue, On exponential representations of analytic functions, Journal d'Analyse Mathrmatique 5 (1957), 321-388. 2. C. Carathrodory, tJber den Variabilitiitsbereich der Fourierschen Konstanten yon positiven harraonischen Funktionen, Rendiconti del Circolo Mathematico di Palermo 32 (1911), 193-217. 3. D. Gale, Neighborly and Cyclic Polytopes, Seattle Convexity Symposium, Amer. Math. Soc. (1963). 4. K. LSwner, tJber monotone Matrixfunktionen, Math. Zeit. 38 (1934), 177-216. 5. G. Pick, ~ber die Beschri~nkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden, Math. Ann. 77 (1916), 7-23. MICHIGANSTATEUNIVERSITY, EAST LANSING,MICHIGAN

FARTHEST POINTS OF SETS IN U N I F O R M L Y CONVEX BANACH

SPACES BY

MICHAEL EDELSTEIN ABSTRACT

Let S be a closed and bounded set in a uniformly convex Banach space X. It is shown that the set of all points in X which have a farthest point in S is dense. Let b(S) denote the set of all farthest points of S, then a sufficient condition for ~6 S = ~6 b(S) to hold is that X have the following property (I): Every closed and bounded convex set is the intersection of a family of closed balls. 1. Let S be a subset of a normed linear space and let s e S for which an element c exists such that

b(S)

denote the set of all

Ils-cll =sup{llx-clllx s)

(*)

i.e. the set of all farthest points in S. In [3] we proved that if S is a closed and bounded set in Hilbert space then b(S) # ;?J. Asplund [1] proved independently that in the case of a convex closed and bounded S in a Hilbert space H, S is identical with the closed convex hull, ~-6 b(S), of the set of farthest points; in addition he showed [2]* that the set C o f all points in H satisfying (*) for some s ~ S is dense (in H). In the present note we show that the last result is true for any closed and bounded set in a uniformly convex Banach space X. If, in addition X has a certain smoothness property (I) (of. section 3), known to hold for all reflexive spaces having a strongly differentiable norm, then 6-6 S = U6 b(S). 2. In a normed linear space X let V = {x ]]] x ]J < 1}. F o r any real e, 0 < e < 2, define the function 6(e), called the modulus of convexity of X, by setting (1)

6(e)=inf

1-~-

I l x + y l l Ix , y e V , I l x - y l l

>--.

The space X is called uniformly convex if 6(e) > 0 for all e in the domain of definition o f 6. Clearly (2)

e >_-e' :~ 6(e) >= 6(s')

Also, as is readily verified, Received June 13, 1966. * I am indebted to Dr. Micha Perles for these references. 171

i72

M. EDELSTEIN

(3)

[September

6(e)< T

LEMMA X. Let x, y e V, x # y, and suppose 0 < # < ½ ; (4)

then

X - I ] #x + (X - #)y It > 2#6(11 x - y [I)

Proof.

Let z = #x + (1 - #)y. It clearly suffices to show that all w ~ X with II w - z II --< 2#6(11 x - y I1~ are in V. Set v = ½ #(w - (1 - 2#)y). Then w = 2#v + (1 - 2#)y is a convex combination o f v and y and it suffices to show that v is within distance 6(I [ x - y I[) from ½(x + y). Now X IIv - -~ (x + y)ll

=

1 [ l w - ( 1 - 2 # ) y - #(x + 2---#-

y)[t

X 2# IIw - #x - (x - #)y II 1 2# II w - z LEM_V~ 2. L e t 0 < ~ < X , the following conditions

0
and suppose x, y E X and f E X* satisfy

(5)

Ilxll---- x = Ilyll

(6)

f ( x ) < 1 - ot

(7)

II --< 6(11 x - y II)-

=f(y)=

VII

11x - ay II --< x -

Then

(8)

II x II --< 1 -

2,6'6(c0

Proof. Let u = ( 1 / ( 1 - f l ) ) ( x - f l y ) ; t h e n ]u ~ 1 and x = f l y + ( 1 - f l ) u . It follows from L e m m a 1 that 1 - [I x [ > 2fl6( u - y ][) > 2fl6(11 x - y [1). N o w

~11---llfll

[lYas asserted.

Ily-

xlI _-__s=f~y~-s<x)>~. Thus

x

~ t -2,6<~)

THEOREM 1. Let S be a nonempty closed and bounded set in a uniformly convex Banach space X. Then the set C, of all points c in X for which there is a point s e S with IIs-cll=sup{ltx-clllx~s}, is dense (in X). Proof. (91)

Given Co ~ X let r 1 = sup

~ll x - eo 11 Ix ~ s l

We m a y d e a r l y assume that rl > 0. T o prove the theorem it suffices to show that for an arbitrary p, 0 < p < r 1, there is a c e X, as required, with IIc - co II z p.

1966]

173

FARTHEST POINTS OF SETS

To this end we define inductively sequences {c,} and {x,}, n = 1, 2,..., converging to c and s respectively. Let, then, xl e S be chosen so that (101)

HXl-Coll>=rl

1-2-~-rl

Next, let

(111)

Cl = C O +

Co - xl IlCo-Xxl[

P 2

Assuming r.-a, x.-1 and c.-1 already defined set (9.)

r. = sup {ll x - c._, I1 x e S}

and choose x. e S so that

(lO.)

p IIx.-c.-,II =>r. 1 - ~7~.

6.+1(1) )

Finally, let (11.)

C n - 1 - - Xn

c. = c . - , + It c . - i - x . U

p

2"

Of the sequences {r.}, {x.} and {c.} thus defined the last one is clearly a Cauchy sequence by (11.). We proceed to show that so is {x.}. For each positive integer n let then R. > 0, ½ >/~. > 0 and f. E X* be defined as follows. (12.)

R. = r. + 2 - " p

(13.)

ft. = 2-" p

x. - c.-1

R-7 ". = rFx.-c._l~

and (14.)

f .(u.) = Ilf . 11 = 1

From (9.+0, (11.), (10.) and (12.) we get

(15.)

r.+ a > R . - p

t5"+1(1)

It follows from (9.), (11.) and (12.) that both x. and x.+l satisfy the inequality (16.) Further,

I ?ll

=< 1

M. EDELSTEIN

174 f . ( x . - c.)

= f.(x. - ~._1) + f.(c.-,

[September

5"+t(1) + 2-"p= R.(1

=> r. - -~.

c._, [[ +

llx.-

- c.) =

2-"p

2"R.P 6"+1(1))

(17.) > R.(1 - 6"(1)) To complete the proof that {x.} is a Cauchy sequence it suffices now to show that

(18.)

f.(x.+ 1 - c.) > R.(1 - 6"(1))

Indeed (16.), (17.) and (18.) are easily seen to imply

IIx.÷~ - x. II ~ R.'~"-*(1) ~ (r, + v)V-'(1) ~ 2-"+'(, ", + p)(*) To establish (18.) we make use of Lemma 2. We note, then, that

x.+,-c.

I[

R.

p.u. <=l-p.

For [Ix.+ , - c._, [I < r. = R. -- 2 - " p = R.(1-- fl.) and Xn+ 1 - - Cn_ 1

R.

X n + l - - Cn

-

Cn - - Cn_ 1

a.

+

Xn+ l - - Cn

a----7----

]

p.u..

a.

and using (10.+1) and (15.)

[1~.÷,-c.II a ~.÷,

(

1

P

2.+ 1/'n+ 1

6.+2(1))

R. - p 6"+1(1)- ~ P

> R.

6"+2(1)

2 P l 6"+1(1) = R . ( 1 - 2fl.6"+1(1))

(*) For 1

X n + I - - Cn

x. - c. > 1 . (.x.+ l _ - c.

> 1 - 6"(1) and, it clearly follows from (1), that [[ x , + t -- Xn[[ < R,5 -1 (1).

Xn - - Cn

1966]

FARTHEST POINTS OF SETS

175

Thus (setting e~= 6 n+l (1)) we obtain

f.(x.+ 1 - c.) > Rn(1 - 6~(1)) as asserted. Let now s = lim..oox, and suppose c = lim._.oo c.. We clearly have

sup {lie - x II Ix ~ s} -- lira (sup {11e,- x II Ix ~ S}) n--~ O0

lira r~+l = lim [Jc.-x~+ll]=l[ c - s l l M-~O0

N --~ O0

concluding the proof of the theorem. REMARKS. In [5] Lindenstrauss defined the notion of a strongly exposed point as follows: A point s e S is said to be a strongly exposed point of S if there is a n f e X * such that f ( y ) < f ( s ) for y 4 s and whenever {xn} c S is such that f(xn) ~ f ( s ) t h e n Since every point on the boundary of the unit ball of a uniformly convex Banach space is known to be strongly exposed it follows from the above theorem that every closed and bounded set in a uniformly convex Banach space has strongly exposed points.

IIx,- sll-.0

3. DEFINITION. A normed linear space X is said to have property (I) if every closed and bounded convex set in X can be represented as the intersection of a family of closed balls. This property was introduced by Mazur I'6] and shown to hold for all reflexive Banach spaces having a strongly differentiable norm (cf. also Phelps 1'7, p. 976]). THEOREM 2. Let X and S be as in Theorem 1 and suppose, in addition, that X has property (I). Then S = 66 b(S). Proof. Clearly ~-6b(S)c~-6S. To prove the reverse inclusion suppose x (~c-6b(s). Then, by property (I) there is a closed ball

B(co, r) = {yl IIy-coil ~ r}, where c o e x and r > 0 , such that ~-6b(S) cB(co, r) and x - c o I l - r > 0 . By Theorem 1 there i s a c e X such that [ c - c o < X - C o - r with cEC. If s e S is farthest from c then I I s - c l l < s - c o l + Co " c < [[x - c o showing that S c B(co, r). Thus x ¢ c-6S and ~-6S c ~"6b(S) completing the proof.

176

M. EDELSTEIN REFERENCES

1. E. Asplund, ,4 direct proof of Straszewicz' theorem in Hilbert space (to appear). 2. - - , The potential of projections in Hilbert space (to appear). 3. M. Edelstein, On some special types of exposed points of closed and bounded sets in Banach spaces, Indag. Math. 28 (1966), 360---363. 4. V. L. Klee, Extremal structure of convex sets II, Math. Z. 69 (1958), 90-104. 5. J. Lindenstrauss, On operators which attain their norm., Israel J. Math. 1 (1963), 139-148. 6. S. Mazur, Ober schwacheKonvergenz inden Raumen (LP), Studia Math. 4 (1933), 128-133. 7. R.R. Phelps, A representation theorem for bounded convex sets, Proc. Am. Math. Soc. 11 (1960), 976-983. SUMMER RESEARCHINSTITUTE, CANADIAN MATHEMATICALCONGRESS DALHOUSIE UNIVERSITY, HALIFAX, NOVA SCOTIA

PROPERTIES OF GENERALISED JUXTAPOLYNOMIALS BY

YEHORAM GORDON*

ABSTRACT

Given F(z),fl(z),f2(z) . . . . fn(z) defined on a finite point set E, and given B - - the set of generalised polynomials ~,= 1 a~fk(z) -- the definition of a juxtapolynomial is extended in the following manner: for a fixed 2(0 < ). __<1), f(z) ~ B is called a generalized 2-weak juxtapolynomial to F(z) on E if and only if there exists no g(z) E B for which g(z) = F(z) whenever f(z) = F(z) and Ig(z) -- F(z) ]< ~ If(z) -- F(z)I whenever f(z) ~ F(z). The properties of such f(z) are investigated with particular attention given to the real case. 1. Preliminaries. Throughout this paper, we assume E is a finite point set in the complex domain, consisting of at least n + 1 points, F ( z ) , f l ( z ), ...,f,(z) are given complex functions defined on E , and ]E~=I akfk(Z)= 0 on any n points o f E ~ - a 1 = a 2 . . . . . a n = 0, this condition which we denote by F , is obviously fulfilled by the functions fk(z)--Z "-k k = 1 , 2 , . . . , n . Let B be the class of all functions of the f o r m X~=la~fk(z) where a l , a 2 , . . . , a n are constants. For any fixed 4, 0 < X < 1, and any set E ' _ E , we define the following classes of functions: f ( z ) e J I ( X , E ' ) , ~ f ( z ) e B and there is no g(z) e B which for every z ¢ E ' satisfies the inequality I g(z) - F(z){ ~ X [f(z) - F(z) (the symbol ~ means g ( z ) = F ( z ) when f ( z ) = F ( z ) and I g ( z ) - f ( z ) I < x f ( z ) - F ( z ) w h e n f ( z ) # F ( z ) ) . f ( z ) e J2(X, E ' ) ~ . f ( z ) e B and there is no g(z) e B which for every z e E' satisfies the inequality I g ( z ) - F(z)] < X ] f ( z ) - f(z)]. J I ( X , E ' ) [Jx(X,E')3 is called the class of generalised X-weak [k-strong] juxtapolynomials to F(z) on E'. Let Ja(X,E') = JI(X,E') - J2(X,E').

I

2. When fk(z)=_z "-k k = 1,2, ...,n, J I ( 1 , E ) is defined by T. S. Motzkin and J. L. Walsh as the class o f j u x t a p o l y n o m i a l s of degree n - 1 over E , namely " n e a r e s t " polynomials to F(z) on E - - and if in addition F ( z ) - - z ' : {z ~ - f ( z ) ; f ( z ) ~ Jl(1,E)} is known as the class of infrapolynomials of degree Received July I1, 1966. * This note is an extension of a part of the author's M.Sc. Thesis under the supervision of Prof. B. Griinbaum to whom the author wishes to express his sin~rest appreciation. The "author also wishes to thank Dr. J. Lindenstrauss for his valuable remarks in the preparation of this Imper. 177

178

Y. GORDON

[September

n over E. The importance of the class JI(1,E) is in the fact that only for such functions f(z), a minimum deviation of a monotone norm is obtained: M = M(f(z), F(z), E) is called a monotone norm if M is a positive function defined for f(z)E B, which decreases when f(z) is replaced by g(z)~ B for which l g(z)- F(z)[ ~ If(z)- F(z) I on E. Two well known examples of monotone norms are: Tchebycheff norm: M = max {[f(z) - F(z) l; z ~ E} Least pth power norm: M = ~,~eelf(z)- F(z)[P where p > 0 . 3. The basis of this work are papers [1], [2] by the late Prof. M. Fekete. By similar methods a generalization of his results is obtained, giving in the end the structure of a function f(z) in Ja(1, E) without the restriction f(z) ~ F(z) throughout E. We shall also see (Theorems 1,4), that if F(z) C B then B=J2(1,E) Ua
J3(1,E) = ~ .

Proof. From the definitions J2(1,E) _~ JI(1,E). B # J~(1,E), because due to condition F we may construct fo(z)~ B distinct from zero on E, and cfo(z) ¢ Jl(1, E) for sutficiently large constants c. Then if f(z) e B - J2(1, EO, there exists (f(z) ~)g(z) ~ B such that [ g(z) - F(z)[ < If(z) - F(z)[ on E. Let h(z) e B be any function which assumes the values g ( z ) - F(z) on Eo = {z e E; f(z) = g(z)}; such h(z) exist due to condition F and the fact that Eo contains less than n points (otherwise g(z)-f(z)!). Let g~(z)-½g(z)+ ½ f ( z ) - ,h(z) where 0 < 8 < 1 will be determined later. For any z 0 e E we have one of the following cases: (i) z o n e o. Then clearly [g,(Zo)- F(zo)[ ~ I f ( z o ) - F(zo)l. (ii) I g(zo) - F(zo) l < If(zo) - F(zo) I • Then

] ½g(zo) + ½f(zo) - F(zo) ] =< ½]g(Zo) - F(zo)l + ½ ]f(zo) - F(zo)] < ]f(zo)-F(zol, and since E finite, there exists ,1 (0 < ,1 < 1) such that for every 0 < e < '1 and every Zo as above I g,(Zo) - F(zo)[ < If(zo) - F(zo)l.

1966]

PROPERTIESOF GENERALISED JUXTAPOLYNOMIALS

(iii) Zo ~ Eo and

179

]g(zo) - F(zo) l = If(zo) - F(zo) I • Then

1½g(zo) + ½S(Zo)- F(zo) l < ½Ig(zo)- r(.o)I + ½lf(zo) - r(:o)I = IS(zo) - e(zo) I, and here too there is 82 ( 0 < 8 2 <81) such that for any z 0 as above

Ig,,(Zo)- e(z0)l < IS(z0)- F(zo)l. We proved that in all cases I g,,(z) 4.

THEOREM1.

e(z) I ,~ If(z) - e(=) I , that i s f (z)~ J~(1, E).

If F(z)~B then B = J 2 ( 1 , E ) U ~ < ~ J 3 ( 2 , E ) .

Proof. Let f ( z ) e B - J l ( 1 , E ) and let 2'=Sup{2;f(z)eJl(2,E)}, then 4 ' > 0, since 4 ' = 0 implies that for every 2 < 1, there is ga(z)eB such that Ig~(z)- e(z)l ~ 2If(z)-F(=) I on E; taking a sequence 2,,--,0, we obtain a sequence ga.,(z) --) F(z) on E, a suitable subsequence g~,,, (z) will converge to a limit function g ( z ) e B ; this is due to condition F , since if M = Max {If(z) - F(~) I + l e(=)l; z ~ E), the set gx.,(z) is uniformly bounded by M on E, and if a ( ' ) = ~,,lt"(,,),,2.,(') , --., a,(')) is the coefficients' vector of gain(z) then due to condition F for any set of distinct points{zl,z2, ..., z,} ___E, the vectors f(zi)= (fl(zi),f2(z~), ...,f,(zi)) i = 1 , ..., n form a basis to the n-dimensional complex vector space, and as the scalar product g~,,(z~)= (at"),f(zi)) is uniformly bounded, it follows that the set {a (-) } is uniformly bounded and therefore has a subsequence {a ("~')} which converges to a limit point a'= (a~,a'2,...,a',), for which

F(z) = lim g~,,, (z) = g(z) (=- ~ a;fk(z)) for every z e E, i'-*~

k=l

a contradiction!

f(z)eJl(2',E),

since

assuming

otherwise,

there

exists

g(z)eB

with

]g(z)- F(z)l ~: 4' If(z)- t(:) I on E, E being finite, the inequality holds for some 2, 0 < 2 < 2 ' , contrary to the definition of 4'. Assume f(z)eJ2(2',E), then there is (contrary to the definition of 2'!) e > 0, 8 + 4' < 1, such that f(z) E J1(2' + 8,E), because otherwise for every e > 0, e + 4' < 1, exists g,(z) e B with [ g,(z) - F(z) l < (4' + 8)If(z) - F(z)[ on E, and similarly as above taking a sequence 8m~ 0 with g,,,(z) converging to a limit function g(z) e B on E, we obtain the inequality ] g ( z ) - F(z)[ < 4' If(z)- F(:)[ on 6, impossible by our assumption. Hence f(z)~ Jz(2',E). Note that the classes which compose B are all distinct. We see now that f ( z ) e Jz(2,E)¢~. 2 < 4'. We shall show later how 4' may be calculated under certain conditions (Theorem 4). Lemmas 2, 3 are required in the proofs of some later results. 5.

LEMMA2.

the inequality

Let f ( z ) e B and 0 < 2 <

1. f ( z ) e J l ( 2 , E ) ~

for no h(z)~lB

180

Y. GORDON

[September

(D

IF(z) -f(z) + h(z)I ~:AIF(z) - f(z) - h(z) l

holds throughout E. Proof.

If 2 = 1, and if for some g(z) ~ B,

IF(z)-g(z)l ~ IF(~)-f(z)l

(2)

throughout E, then let h(z)-f(z)- g(z). If, for Zo e E, f(zo)= F(zo) then

g(zo) = F(zo), hence h(zo)--0 which implies (1) for z o. If f(zo) ~ F(zo), then

IF(go) -f(zo)-h(zo)! = lF(zo)-2Y(Zo) + g(zo)[ = IF(zo)-S(Zo)l t 2- (F(z°)- g(zo)~ ] W(Zo)-/(Zo)}

> IF(g°) -f(z°)] (2 - I F(zo) -

g('o)

IF(zo)-:(zo)l

IF(zo)- g(zo)[ = IF(go) -f(zo) + h(zo) l,

thus (1) holds for every z e E. / K[~ ~ Conversely,(1) (with,~ = 1)impliesthatwhenF(zo)~f(zo) , Re '(')f(J~-~---°)F(zo) ._v.-

\

>0.

Let a__sup{I

h(z).12/2

:(z)--F(z)

and put g(z) -f(z)

(h(z))

Re,:(z) - r(z) ;~E,:(z)~ r(z),,

Ih(z) + a' then inequality(2) easilyfollows.When, for Zo ~ E,

F(zo) =f(zo), then h(zo)= 0, hence g(zo)= F(zo), and again (2) is satisfied. i If 2 < I, let h(z)- 1.d1__~(f(z )- g(z)) be the relation between g(z)EB and h(z)~ B. Obviously,

]r(~o)-g(zo)] ~: ~lF(zo)-f(Zo)] ,~ (I) holds

for go.

Note that I.emma 2 holds if 0 < 2 < 1 and we replace above "JI(2,E)" with "J2(~,E)", and replace " < " with "_-<".

L~MA 3. Letf(z)~B, E o = {zeE;f(z)=F(z)}. For every E' c _ E - E o , define the set R(E') in the Euclidean space 82": R(E,)ffi~(rl,sl, r2,s2,...,r~,s,);r~=Re L[ f(z)~(z). ] - F(z) ]l ' s~ = I m [If(z)ft(z) - F(z) J

l <_i<_n, z~E'}. For any a=(al, bl,...,a,,b,)~ 2" let n(a,1)={x¢82~; (x,a)>O}, and for 0<#
1966]

PROPERTIESOF GENERALISED JUXTAPOLYNOMIALS

n(a,#) = { x = (xl,Yl,...,x.,y.)e82";

181

~ (x k + iy~)(a~--ibk)-- 11 < #}. I

k=l

Given 0 < 2 ~_ 1, f(z) e JI(2,E) ~ for every a = (al, bl, "", a., b.) e 82" which satisfies ~,~=~(ak -- ibk)fk(z) = 0 on Eo,R(E - Eo) ~ ~(a,2). Proof.

~=

:

Suppose f ( z ) ¢ J~(2, E), by Lemma 2 there exists

h(z) -- ~ (a t -ibi)fk(z) which satisfies (1) on E , let a = (as, bl, "", a,, bs). If 2 = 1, (1) means: h(z) = 0

[ h ),1

o n E o , and Re i f ( z ) - F ( z ) J

>0

on E - E o

If 2 < 1, (1) means: h(z) = 0 on Eo, and

that is R ( E - E o ) cn(a,1). x/1 - 22 f(z) - F ( z ) - 1

<

on E - E o that is R ( E - Eo)~_ ~(a', 2) where a ' = ~/=J- 22 a . =~: This is proved similarly by working backwards. TImOI~M 2. Given 0 < 2 < 1 and f(z) e B, suppose E o = {el, c2, '-', ci} = ={zeE; f(z)=F(z)} where O < l < n . f ( z ) e J l ( 2 , E ) ~ t h e r e is a set E , + t ~_ E - E o of m + l points, n - l ~ m ~_ 2(n - l), such that

f(z) e J1 (2, Era+1 U Eo). Proof. ~ : Obvious. =~: Assuming to the contrary f(z) ~ J1(2, E2,- 21+ l U Eo) for every E2,_2z+l={Zo, Zl,...,z2,_21}~_E-Eo, there is by Lemma 3 a = ( a l , b l , . . . , a , , b , ) e 8 2" such that Y ~ = l ( a k - i b k ) f i ( z ) = O on E 0 and R(E2s_2,+l) ~_n(a,2). Let x j = ( R e f l ( c l ) , Imfl(cl),...,Ref,(cl) , Imf,(e/)), Yt = (Imfl (cj),-Refl(cj), ..., I m f , ( c j ) , - Ref,(cj)),j = 1,2, ..., l, and for any b e 82" let H(b) = {x; (b, x) = 0}. Since the vectors x 2, Y1,'",xt, Yzare linearly in dependent, H = N J = I H ( x j ) n H(yj) is 2 n - 21 dimensional. Let x ( z ) e R ( E - Eo) be the point associated with z e E - Eo. By our assumption a e HI"1 ~ o 2~n(x(zi),2). Using the well known Helly's theorem on the intersection of convex sets in 82"-2~ (the sets n(x(z),2) are convex), there exists a'=(a~,b~,...,a',b'~) in Hf'),,E-Eon(X(Z),2), that is ~,~.-_l(a~-ib~)f~(z)=O on Eo and R ( E - Eo)~ n(a',2), meaning by Lemma 3 f(z)~J~(2,E) - - a contradiction! We proved the existence of Em+~ with m ~ 2 (n - / ) , deafly m > n - l, since otherwise, by condition r , there exists g ( z ) e B equal to F(z) on Em+l U E o . ~ . al) Obviously if Eo has n points at least, then f(z)eJl(2,E'o) where E~( __. Eo) contains n or more points. bl) A similar theorem holds for J2(2,E). el) If F(z), f ( z ) , fk(z) (1 ~ k < n) are real we obtain m = n - l (since in the proof R ( E - Eo) can be embedded in 8~). This is also true for J2(2,E).

182

Y. GORDON

[September

6. The following theorem is established in [2] for the case f ~ ( z ) - z " - t (1 <- k < n) and Eo = ~f. Trmom~M 3. Suppose f ( z ) ¢ B , Eo = {c~,...,c~} = { z ¢ E ; f ( z ) = F ( z ) } and O<-l 0 (0 < i < m) and wj (1 < j < l), such that (3)

m ! ]~ 2if~(zi)(F(zi)--f(z~)) + ~, wJk(c~) = 0 i=o

j=t

for every 1 < !¢ < n. (ii) I f (3) holds for some such Era+l, 2~, wj, then f(z)¢Jl(1,Em+ 1 UEo).

Proof. We retain the notations of Theorem 2. (i) By Theorem 2, f ( z ) ~ J l ( 1 , E m + t UEo) for some such Era+ 1. By Lemma 3, a ~ H ~ R ( E m + t ) ~(a, 1), that is CR(Em+I)t~ H(a) # ~ , meaning CR(Em+ 1 ) n H J # ~ (where CR(Em+I) denotes the convex hull of R(Em+I), and H ± the space orthogonal to H). Therefore there exist constants Z t ~ 0 X t % o ~ q = l , real constants g~,]~j (1 ~ j _- 0. (ii) Conversely, (3) means C R ( E = + O N H ± # ~ , that is a~H

=~ R(E,,,+I) ¢; n(a, 1),

and by Lemma 3 f ( z ) ~ J t (1,Em+l UEo). REMARK. Theorem 3 with fk(z) = z ~-k , F(z) = z" and E o = ~ supplies us with the structure of an infrapolynomial [1]: let p(z) = zn+ a l z ~ - l + .-. + a~ # 0 on E. p(z) is an infrapolynomial on E¢~ there exists Em+l = {zo, zl,-..,zm} __GE with n < m < 2 n for which p(z) is an infrapolynomial .,~ there exist 21 > 0 i = 0,1,.-.,m ~i=_-oXi = 1 such that ~i~o2~I-I~=o:~,~ (z - zj) is divisible by p(z). 7. We shall find in this section the structure of f ( z ) ~ J a ( 2 , E ) when F(z),A(z),f(z) are real and A < 1. Given 0 < 2 ~ 1, an integer n, and a set E, denote by Ii(2,n,E) [I2(2, n, E)] the set of all polynomials p(z) -- z n + alz "-1 + ... + a, with the property - - for no polynomial q(z) = z" + ... the inequality Iq(z) l 2lp(z) l [I q(z) l =<21p(z)l] holds throughout E. Let Ia(2, n, E) = 11(2, n, E) - 12(4, n, E). REMARKS. a2) It is shown in [3] that when E is compact I3(1, n , E ) = IZ (compare with Lemma 1), also by similar methods to Theorem 1

1966]

PROPERTIES OF GENERALISED JUXTAPOLYNOMIALS

I 2 ( 1 , n , E ) U ~ < I I 3 ( 2 , n , E ) consists z n + a l z n-I + "" + an (for finite E).

b2)

of

every

polynomial t(z)

Given zo ~ E, t(z) ~ Ik(2, n, E) and t(Zo) = 0 ~ Z

--

of

183 the

form

e Ik(2, n-- 1 , E - {Zo}).

Z0

c2) Presently we shall need the following result (see [4]): If F(z), fk(Z) (1 < /C< n) are defined on E = {Zo, Z l , ' . ' , z n } , and if condition F holds and F(z) ~ B , let f ( ° ( z ) e B be the function for which F ( z ) - f C ° ( z ) vanishes on E - {zt}, i = 0,1,..., n. Given any f ( z ) E B there is a polynomial p(z) = z ~ + ... (or alternately, given any p(z) = z ~ + ... there is f ( z ) e B) such that (4)

p(zi)

(z i - zj) = (F(z~) - f ( z t ) ) / ( F ( z i ) - f ( ° ( z t ) ) ,

0 <_ i < n.

j#!

d2) When f ( z ) and p(z) are related by (4) on such a set E, y(z) ~ J~(a,E) o I1(1, n,E) = { ~ o ~

p(z) ~ Ik(,~, n, P,).

rI

j=0

=

j

1

so it follows from (4) that Jr(l, E) = {f(z) ~ B; 2, = (F(z,) - f(z,))/(F(z,) - f(0(z,)), 2, _~ 0, ~ 2, = 1}. lffiO

e2) When F(z),fk(z) (1
2(En+t) --

~=o F ( x i ) - f ( ° ( x i )

where f ( ° ( x ) is defined f o r En+ ~ as in c2 (define 2(En+l) = 0 if F(xi. ) =f(~')(x~,) for some (i'). Let 2' = sup{~(En+I);En+ t - E}, then 0 < 2' < 1 and

184

Y. GORDON

[September

(i) 2' = 1 o f(x) ~ JI(1,E), (ii) 2' < 1 o f(x) ~ J3(2',E). Proof. Assume 2' > 1 and suppose 2' = 2(E'+1) where E',+I = {X'o,X'x,'",x'}. F(x)~ ~.,7=lajfl(x) on E ' + I , since otherwise 2(E'~+1)=0. Let p(x)---xe+ box, - 1 +... + b~ be related tof(x) by (4) for the set E~+ 1. Let to(x) - IIT=o(X - x[) then

--

|=0

to'(x'~)

i=0

F(x'~) - f(')(x|)

i=O

= 2(E• + 1"--"-"~

which contradicts 2(E'+1) > 1. 2' > 0, since 2' = O=~F(x)~Bt (i) =~: Let 2 ' = 2 ( E ' + 1 ) = 1 , using (4) and (5) we have p(x;)/to'(x't)~_O, and from d2 f(x)~Jx(1, E'+I) c_ J1(1, E). (i) ~=: By cl there exists E~+ 1 such that f(x) ~ Jx(1, E'+ 1), using (4) and d2, (5) follows with equality everywhere, giving 2(E'+1)= 1. (ii) =~ : Suppose 2' = 2(E'+ 1) andf(x) ~ J1(2', E'+ i), then there exists g(x)~ B such that I g(x;) - F(x;) l ~ 2' If(x;) - t(x;) I i = 0,1,.--, n. Let (4) relate r(x) =-x'+ ... and g(x) for E'+I, then

1=

r(,:,)

I

to'(xO -< i~= 0

i=0 N

f

,=o

to'(x9

=

!

F-~--f(-~(~)

< 2(E~+1) ,=o

F(xl)-f(O(x't)

this contradiction implies f(x) ~ Jl(2', E'+ 1) _c J1(2', E). For any E,+l={xo,...,x,}, f(x)~J2(A(E~+l),E~+l). This is obvious if A(E,+ 1) = 0. If 2(E,+ 1) > 0, let (4) relate p(x) - x n+... andf(x) on E,+ 1, define

r(x)-

~ 2(E.+,)IP~X') l=0

] to(x)

~

X ~

Xl ~

where t o ( x ) - f i ( x - x , ) /=0

then I r(x,) I = 2(E, + 1) I P(x~)l, meaning f(x) ~ J2(2(E~ + 1), En + 1) ( ~ J2(2', E,+ 1)). Let

c(x) = {(al,

8";

i=l

a,f,(x)- r(x) I rli(x)- r(x)l},

_n we have [ 7 C(x~)~ ~ for every En+l ~ E, and by Helly's theorem (the sets l=O

C(x) are convex) N ~eE C(x) ~ ~ , so f(x) ~ J2(2', E). Therefore f(x) ~ Jz(2', E). (ii) ~ : JI(1,E) = J2(1,E), therefore 2' < 1. COROLLARY 1. Suppose f(x) = F(x) only on Ek = {Xo, Xl, "",x~-l} where 0
1966]

PROPERTIES O F G E N E R A L I S E D J U X T A P O L Y N O M I A L S

185

otj,pj ~ 0 (j = k , k + 1, ...,n) with ~,]=kO~j= ~'~=kflj = 1 and ~Jfli = 0 such that (6)

f(xj) - F(xj) = (1 + 2)~j - (1 24

-

2)rj(f(j)(Xj)R.

-- F(Xj)) (k < j < n) = "

(ii) I f (6) is satisfied for such En+1, ~j, flj then f ( x ) ~ Ja(2,En+l). Proof.

(i) The existence of such En+ 1 for which f ( x ) e J3(2, En+ 1) is assured

by c~, and by Theorem 4: ; l = - ] -~ I F(x')-f(x') i=o F(xi) - ftO(x~) L

~1 = 1 + 2

define flj--

F(xj) -f(~)(x~)

-

] ~I - t . "

Define

.J

when this ratio is positive, and when it is negative

22 [ F ( x j ) - f ( x j ) ] (take e j = 0 1 - 2 F(xj)-f(J)(xj)

[~j=0]

when p j # 0

• [~j # 0]), representation (6) then follows.

(ii) Calculating 2(En+~) of Theorem 4 from (6) we have 2(E.+1) = 2 < 1, the rest follows from Theorem 4(ii). 8. We aim to prove here a supplement to Corollary 1: COROLLARY 2. With the assumptions of Corollary 1, (i) f ( x ) ~ J 2 ( 2 , E ) ~ there exist: E~+I=Ek U{XR,Xk+x,...,X~}, 2~j > O (k < i # j < n) ~,2q = l such that (7)

constants

f(xj) - F(xj) = ~, (1 + 2)2ij - (1 - 2)2ji (f(J)(xj) -- F(Xj)) (k < j ~ n) 22 i=k

(ii) I f (7) is satisfied for such En+~,2ij, then f ( x ) ~ Jz(2,E,+l). The proof of Corollary 2 depends on the following two lemmas of which the first is obvious: LEMMA 4. I f C ~_ ~ is a compact convex set, let n(a, 2) = (x ~ e";] (x, a ) - 1 1< 2} where 0 < 2 < 1 and a e an. C ~ n(a,2) for every a ~ ~ ¢~ the orthogonal projection of C on any line I through 0 is a closed interval [(x(l),y(1)] with the prod(y(/), O) O) > 1----2-22 1 + 2 (d denotes the Euclidean perty: either 0 ~ [x(l),y(l)] or d(x(1), metric function). LEMMA 5. Let A = { x o , X~,...,x~} c_~ n be an affine independent set. For (1 + 2)xj - (1 - 2)x~ 0 < 2 < 1 and ~(a,2) as above, define xtj = 22 (O
186

Y. GORDON

[September

Proof. For any G _~ e" denote by C(G) its convex hull. STATEMENT a: Let F ' = (xo;0 < i ~ j < n}, the relative interior of C(F') is contained in int C(F). This is due to X,o and Xo, being strictly separated by n - the plane containing F ' - - w h e n c e relint C(F') ~_ int C(F' U {X,o,Xo,}) - int C(F). Moreover A _ intC(F). STATEMENT b: Let 7c1,~2 be parallel distinct planes supporting C(A)at x.i,x t respectively. Let 7r3 support C(F) so that rh separates n3 and ~2, then x u ~ ns. This is easily verified for n < 3. Suppose n > 3, let xu ~ C(F) n n3 and assume that A' = {Xo, x~,..., Xn- 1} contains x , x j, Xk, X~, let the plane n contain A', and let F' = {x~j;0 < i ~ j < n}, n ' = lr, n n r = 1,2,3. Reducing the problem to the (n - 1) dimensional "space" n, the proof is carried out by induction. STATEMENT C: With the notations of statement b, dora, n2) d(na, nl)

d(xtj , x~) d(x+j, x j)

1+ 2 1 2"

<=: Assume O~intC(F). Take n, ( r = 1,2,3) as in statement b so that n3 separates O from C(F) and C(F) ~ rc3 is an n - 1 dimensional face of C(F). Pass a line I orthogonal to % through O, let l t3 ~, = x' (r = 1,2,3), then [x~,x~] is the orthogonal projection of C(A) on l, and by statement c d(na, re2)

1+ 2

d(na, ~1)

1- 2

d(x'3, x'2) d(x'a, x'~)

d(O, x'2) d(O,x'~)

which contradicts Lemma 4. ~ : Let l be any line through O, let n~ (r = 1,2,3) be as in statement b and orthogonal to l, let x" = I n n, (r = 1,2,3) and suppose O~(x'~,x'3), then

d(O, x'2) d(x'a, x'2) . 1+ 2 d(O, x'~) > d(x'3,x'~) (by c ) = i----- 2' and according to Lemma 4, a ~ ~" =~ n(a, 4) ~ C(A). Proof of Corollary 2.

(ii)

2(E.+l)

1=k

i=k

I F(x+)-f(x,) F(xi) _ f(O(xi) I = t=o j=k t=k

24

2 '

Theorem 4 gives f(x) ~ J 1(2(E, + 0, E, + 1) -~ J2(2, E, + 1). (i) By Remark cl f(x)eJ2(2,E,+O for a suitable E,+I ---Ek. Let (4) relate " t k - l t X - X~), by d2 p,(x) = x"+ "" and f(x) for E,+I, let p,_k(X)=p,(x)/ rllj=O~ and b2 P,-k(X)eI2(2,n-- k,E,+ 1 --Ek). Let X~=(p,_k(X,))-I(XT-k-1,.",X, 1) ~ e "-k i = k,k + 1, ..-,n, and let tr = {Xk, X~+~,"',X,}. As in Lemma 3:

1966]

PROPERTIESOF GENERALISED JUXTAPOLYNOMIALS

187

pn_h(x) e I2(;~,n -- k, En+ 1 - E k ) o e , x(a,2) = { x e s " - t ; l ( a , x ) -- 1 1 <=;~} for every a e 8"-~. is ( n - k)-dimensional, for assume it is not so, then the linear hull of a, L(o contains the origin; since otherwise the orthogonal projection of L(a) on a line through the origin and orthogonal to L(a) is a single point; and this by Lemma 4, contradicts the result: ~ r , 7r(a,~) for every a e ~,-k. Therefore, there exist n - r (__
y. j=k

(

(1 +

2);tu ~-)~(1 -

t=k

)

"

H (x-

x,).

i=k

(7) follows immediately. 9. For every f ( x ) - ~ , ~ f a a i f ( x ) e B ( a l , a 2 , . . . , a , ) e e", then

we

associate

here

the

point

THEOREM 5. Under the preliminary conditions of Theorem 4, JI(2,E) is compact and connected. Proof. By cl JI(~,E)= uJl(X, En+l), it is sufficient therefore to establish compactness of JI(2,E,+l): E,+I = {Xo, .-.,x,}. According to Theorem 4, ,=o

I Fr(x,)-f(x,) ] =<1~ ~ ( - ~ )---f(O-~-~)

f(x)eJ~(2,E.+~),

and compactness is now obvious, Let f(x) e Jx(2, E.+ ~) and consider the nontrivial ease: F(x) ~ Z"~ = ~a~f~(x) on E.+~. Let g.(x) = o~f(x) + (1 - ~)f(°)(x) (0 < a < 1), then

i=o

~

--f~

I< 1 -

~z + oc ,=o

F(x,) - f(O(x,)

I

2 < "~

that is g,,(x)edx(2,E,,+:). We finish by showing that J~(1,E) is connected. Let gx(x), g2(x)eJ~(1,E), there are .~,~°)+~ = {X(ol),..., x(O} (i = 1,2) such that g,(x) e d~(1, E~l)+~). Suppose n o w -~,+ ~'(~)1 c~E~2)~={x~,x2,...,x,,} let g ( x ) e B be the function for which F(x) -- g(x) vanishes on ..,. + ~ ~ .~. + ~, then g(x)eJi(1,E(~° 0 (i = 1,2), and ~(1) )~(2) since Jt(1,E.(~x) is convex (deduced from d2), the proof is established for this ease. In general, construct sets .~.+~'°)-.-~,.2,~" ...,F, = ~.+x~'(') such that F~ (1 _-
188

Y. G O R D O N

has n + i points exactly, and Fi n F~+ 1 has n points exactly, we may n o w connect g1(x), g2(x) through the intermediate sets Ft. REMARKS. Also it may be verified that J~(2, E) (0 < A < I) is open, connected, and its closure is JI(A,E). In [3] it is shown that l~(1,n,E) is convex if and only if n = 0 or n = 1 or E has n + I points. This is not true in the general case as shown by the following counter example:

E = {xt,x2,xa,x+} (xl+l > xi), F(xl) = F(x2)

---- F(xa)

= 0 and F(x+,) = 1,

and fl(x) - x, f2(x) - 1. We obtain from Remark d2: 4

J~(I,E) = ,=V~ JI(I,E - {x,}) = JI(I,E - {x2}) which is convex. REFERENCES 1. M. Fekete, On the structure of extremal polynomials, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 95-103. 2. M. Fekete, On the structure of polynomials of least deviation, Bull. Res. Counc. Israel 5A (1955), 11-19. 3. T.S. Motzkin and J. L. Walsh, Underpolynomialsand infrapolynomials,IllinoisJ. Math. 1 (1957), 406--426. 4. T.S. M o t z k i n a n d J.L. Walsh, On the derivative of a polynomial and Tchebycheff approximation, Proc. Amer. Math. Soc. 4 (1953), 76-87. THE HEBREW UNIVERSITY OF JERUSALEM

SEPARABLE

BANACH

SPACES

WHICH

ADMIT

l:

APPROXIMATIONS BY

t

E. MICHAEL* A N D A. PELCZYNSKI** ABffFRACT In this paper we study a class of separable Banach spaces which can be approximated by certain special finite-dimensional subspaces. This class is characterized in Theorem 1.1, from which it follows that the space of continuous scalar-valued functions on a compact metric space always belongs to this class, and that every member of this class has a monotone basis.

1. Introduction. In several recent papers [7, p. 25], [2], [5], [8], the concept of a hi-space has been found useful: A Banach space B is a 7rl-space if it contains a directed (by inclusion) family (E~)~ ~A of finite dimensional subspaces, whose union is dense in B, such that each E~ is the range of a projection of norm one from B. It is rather easy to show that LP-spaces (1 < p < + oo) are hi-spaces. The problem becomes more difficult for the space C(S) of all continuous scalar (i.e. real or complex) valued functions on a compact Hausdorff space S, and there it is only solved (affirmatively), as far as we know, for metrizable S (cf. [9]). If C(S) is a ~ ~-space, then (of. the footnote on p. 197) each of the spaces E~ appearing in the definition of a hi-space is isometrically isomorphic to some 1~), where l~ denotes the space of n-tuples of scalars with norm IIx II -- m a x ~ , ~ , l This suggests the following concept. DEFINITION. A Banach space B is a n~°-space provided B has a directed (by inclusion) family of subspaces (E,)~ ~A, whose union is dense in B, such that each E, is isometrically isomorphic to some I~). It is well known that there always exists a projection from a Banach space onto any subspace which is isometrically isomorphic to some I f (see Lemma 2.1 for a proof). Hence every z~°-space is a lh-space. However, there are rhspaces which are not n~°-spaces, such as any Hilbert space (of dimension > 1). In fact, every infinite-dimensional n~-space is non-reflexive [7; p. 66 Corollary 1, and Theorem 6.1 (2)]. Our principal result asserts that, for separable Banach spaces, the property of being a n~°-space is equivalent to a property (a ®) which is formally weaker

x(i)l"

Received July 22, 1966 * Supported in part by N.S.F. Grant 11-5020. ** Supported in part by N.S.F. Grant GP-3579.

189

190

P

E. MICHAEL AND A. PELCZYNSKI

[September

and much easier to verify. (If z • B and E c B, then d(z, E) will denote infe ~E [I z - eli ). (a ~°) I f Z is a finite subset of B and ~/> 0, then there is an integer n and a linear map T:I~ - } B such that d(z, Tl~) < t/for z • Z , and

(1 +

IIx II --< II Tx II (1 + t/)II x II

for x • l ~ . THEOREM 1.1. Let B be a separable Banach space. Then the following conditions are equivalent. (i) B has property (a~°). (ii) B is a It~-space. (iii) B has an increasing sequence of subspaces El c E 2 ~ ..., whose union is dense in B, such that each E n is isometrically isomorphic to l~. This result easily implies (cf. section 6). COROLLARY 1.2. Every separable 7c~-space has a monotone basis. Corollary 1.2 improves a result of GurariI [6, Theorem 81, who proved that a separable Banach space with property (a °°) has a basis(*). Using the standard technique of "peaked partitions of unity" (see Section 5), one can easily verify that C(S) has property (a ~o) for arbitrary compact S (cf. [7, pp. 28-29], [91, [11]). Therefore Theorem 1.1 gives an alternative proof of the result [91 that, if S is compact metric, then C(S) is a 7t~°-space. Combining this result with Corollary 1.2 we get the following corollary, which was asserted by Gurarff in [6; footnote on p. 2981, and which strengthens some results of Vaher [121 and Bessaga [1]. COROLLARY 1.3. I f S is compact metric, then C(S) has a monotone basis. In conclusion, let us list, without proof, some further examples of lt~-spaces: The space Co(S)= ( f • C(S): f ( x o ) = 0}, with S compact metric and Xo• S. The spaces C~(K), with K compact metric, of Day [3, p. 891. The weak tensor product (cf. [3, p. 651]) of any two ~t~°-spaces. 2. Preliminaries. We commit the notational abuse of writing i • n instead of i = 1, 2,..., n. If x = (x(i))~ ~n • l~, then

N(x)={i•n:lx(i)l

= II ll }"

By u~") we denote the k-th unit vector of

I~, i.e. u~*)(i) = 6~ (i • n, k • n). In the sequel we shall need the following three lemmas. Lemma 2.2 follows from [4, p. 74, problem 34]. Lemmas 2.1 and 2.3 are probably also known, but we include their proofs for completeness.

(*) Actually, Guraril"assumed that B has a property ("Bis a space of class (~")which is formally stronger than (a~). In fact, his property is equivalent to (a~)(cf. [7, p. 221).

1966]

BANACH SPACES WHICH ADMIT In~ APPROXIMATIONS

191

LEMMA 2.1. Let R be a linear map from l~ into a Banach space B, and let

R have a bounded inverse R -1. Then there exists a projection P from B onto Rl~ with It P t} < IIR II II R-1II • In particular, if R is an isometry, then IIP [I = 1. Proof. By the Hahn-Banach theorem, there exist linear functionals ~bt on B ( i = 1, ..., n) such that q~,(y)=(R-ly)(i) for all y ~ g l ~ , and l] ~bill < IIR -1 ]1. Now let Px = ~,~ ~,~bi(x)Ru~~) for all x e B. Then P is the required projection. Let T: Im ~ ~ l~ be a linear map, and let e.i = TuJm)forj ~ m. LEMMA 2.2. ]l T II = max, ~, I ~.i~me.i (i) 1" LEMMA 2.3. T is an isometric embedding if and only if (a) E.i~,~le.i(i)l < l for i e m , (b) II e.i [I = 1 for j ~ m, (c) if i ~ N(ej) and s # j, then e~(i) = 0 (s e m, j e m). Proof. Necessity: The necessity of (a) (by Lemma 2.2) and (b) is trivial. If T is an isometry, i e N(e.i) and s # j, then, choosing 2 such that [ 2 ] = 1 and

le.i(i) + 2es(i)I = [ej(i)[ + I ;tes(i)[ = 1 + I e~(i)[, we get 1 = I1e.i + 2es [I > 1 + lea(i)I" Hence e~(i) = 0. Sufficiency: Choose i v ~ N(e~) for v e m. Let x = ~tju~ m) E l~. By (a) and Lemma 2,2, II Tx II Z I1x I1 On the other hand,

HTxll --max[ ~ t.iej(i)l ~_max I Y~ tje.i(iv)l =maxltvl = [Ix[I, icn

j~m

vem

jem

because, by Cb)and ¢c), I~.i..t~e.i¢iv)l-lt.I

vera

for ~ m

Thus IITxll=llxll

for x e Im ~. 3. Isometric and almost isometric subspaces of l ~ .

LEMMA 3.1. Let 0 < rl < 1. Then, for each linear map T: l ~

(~')

IIx !1--< II T ~ II --<( I + n)II x II

l~ such that

for x ~ I : ,

there exists a linear isometric embedding S: l~ --* l~ such that IIs-

T

II < ~

Proof. For each j e m, let fj = Tu~m) and

N j = {iEn:lfj(i) I ~ 1). By the left-hand side of ( ~ ) , 1 = u~=) H<- [lfJ II" Therefore for each j ~ m there is an ij ~ n such that f.i(ij) = f.i ~ 1. Thus the sets Nj are non-empty. They are also mutually disjoint: Indeed, by the right-hand side o f ( ~ - ) a n d by Lemma 2.2,

J+,>__llrll~=~JGm If.i(i)l

for i~n.

192

E. MICHAEL AND A. PELCZYlqSKI

[September

Thus if s # j (s ~ m,j • m), and i • Nj (or equivalently [fj(i) ] _->1), then

[f.(i) I =< t + ,1 -If/i)[ _~ ,1 < L Hence i ¢ N~. For all j e m and i • n, let

I

if ieNj, if i • N s for s # j

oS~(i)I (f/i)l -~

ej(i) = L fj(i) (1 + 17)-t

(S E m),

otherwise.

Clearly X~.le/i)l ~ ~ for i•n, Ile, satisfy condition (c) of Lemma 2.3. Define the linear map S: l ~ ~ l ~ by

ll =

1 for

j•m

and the sets N(ej)= Nj

Sx = • x(j)ej. jem

Clearly S is an isometric embedding. Since TuJ=)=fj and SuJ m)= ej, the inequality z - s II < ~ will follow f r o m

II

X [fj(i) - e~(i)[ < t/

for i • n.

jem

We consider two cases. If i ~ [.3 Ns, then

E [fj(i)-e~(i)l= Z

jem

jem

If~(i)l

,s -~-~

<~vi"

If i e N, for some s • m, then

Z [fj(i)-ej(i)[=

jem

~, if~(i)[+if~(i)[-l<=[[T]I-l
jerm

j~s

That completes the proof. LEgMA 3.2. If the subspace E of l~ is isometrically isomorphic to l~ (m < n), then there is a subspace F D E of l~ which is isometrically isomorphic to lm~°+l. Proof. For j • m, let ej be the image of the jth unit vector under some fixed isometric isomorphism from l ~ onto E. Therefore the elements ej satisfy conditions (a), (b) and (c) of Lemma 2.3. Since m < n, (c) implies that either one of the sets N(ej) (j • m) contains at least two indices, or there is an index i ~ n which does not belong to any N(ej). In both cases it is easy to choose io • n such that the sets N(ej)~{io} are non-empty.Let F be the linear subspace of I~ spanned by elements f~ (j •_m + 1), where

1966]

BANACH SPACES WHICH ADMIT 1~ APPROXIMATIONS f1 =

fm+t

=

ej --

el(io)u}"o)

193

for j ~ m,

U~)"

Define T: l~ ~ l~ by T(x)= ~j .mx(j)fj. Then T satisfies conditions (a)-(c) of Lemma 2.3. (We have N(fj)= N(ej)~{io} for j ~ m, and N ( f ~ + t ) = {io}.) Thus F is isometrically isomorphic to l~+ ,. Finally F DE because ej = f j + ej(io)fm+ forj~m. Ra~t~K 3.3. In the language of affme geometry, Lemma 3.2 can be restated as follows: Let W be an n-dimensional parallelopiped in n-dimensional Euclidean space, and let the origin be the center of symmetry of W. Let Lm be an m-dimensional hyperplane passing through the origin and such that L,, n W is an m-dimensional parallelepiped. Then there is an (m + 1)-dimensional hyperplane L,+ 1 such that Lm÷ 1 D Lm and Lm+ 1 r~ W is a parallelepiped. Some interesting and far-reaching improvements of this result have recently been obtained by Perles. 4. Proof of Theorem 1.1.

LEMMA 4.1. Let B be a Banach space and let 0 < r/< 1/6. Then there is a ( T / ) = a > 0 such that, if Q:l~m-->B and R : I ~ - - , B are linear maps, and if

<1 ÷ ~)- 111x II ~ IIQx II ~ <1 ÷ ~)II x II

for xel~.,


= IIRyll ~< (l÷~)llyll

for y e l~,

11x 11

for O # x e l~,

dCQ~,RIT)<

~

then there is an isometric embedding S:l~--, l~ such that

IIRS-QI[ < 12t/.

Proof. Choose a < 1/8r/. Let P:B--,RI~ be the projection defined in 1.3. Then IIP[ < IIR II R - ' _< <1 + ~)2 < 1 + 3~ Fix x ~ l~ and choose b in R l ~ such that Q x - b < a x . S i n c e P b = b, w e g e t

IIe~

- ee~

II =< IIe~ - b II + IIeb

- eex

II =< <1 ÷ IIe II)~ II~ II <- 4~ II~ II

Thus

Ilee,, II >= 11ex II- 4~11~11=>[(1 +,~) -'

- ~,7] Ilxll >--

(1- },l)llx II.

Hence, using the inequalities 8a < ~/and 0 < ~/< 1/4, we get

IIR - ' e Q x II ~ <1 + ~)-' IIe e x II ~ <1 - 2.)II ~ I[ Let us set

t

194

E. MICHAEL AND A. PELCZYNSKI T = (1 -

Then, since I1R-'

[September

2.)-tR-1PQ.

II I1P 11IIe II < (J + a)3(1 + ,) < 1 + 2,. <1+2,

II~ II --< II T~ II -- 1 - 2. I1x II z (x + 6r/)11~ II Thus. by I.emma 3.1. there is an isometric embedding IIs - r II < 6,. Hence

S:l~.-'. l~

such that

URs - Q II ~- IIR tl l! s - T It + ItRT - ~ I1 <(1 +,)6, + IIPQ-QII + IIPQ III1 - (1- 2,)-11 < 7. + 4a + (1 + 3a)(1 + ")1 -2, 2. < = 12.. That completes the proof.

Proof of Theorem 1.1. (i) ~ (ii). First observe that, if B has property (a ~ , then, for any finite dimensional subspace E of B a n d , > 0, there is a linear map T: l~ -+ B such that (1 ÷ .)-111 x II --< IITx II < (1 ÷ ,)II x II for x ~ i : and d(e.Tl~) < , II e II for 0 # e • E. To see that. let Z be an ,/3-net for the unit sphere of E. and take T : l ~ B as in the definition of (a °°) w i t h , replaced by ,/3. Let (b.) be a countable dense subset of B. By the preceding observation and by Lemma 4.1. we can inductively define, for v = 1.2.-... linear maps T.. l~. ~ B and linear isometrics S~: l~ ~ l~+. such that •

OO

IIT,+ is. - T~ II < 12" 2 -3', max(ll T~ I1. UT.-'II) < 1 + 2-3". and

d(f,Tv+ll~,,)<2-3"IIfll

for f~F,,

where F, is the smallest linear subspace which contains T~l~ and the elements bl, b2,'", b,. For k--1,2,..• and v = k + 1,k + 2,..., let V..~= T." S.-1"

$.-2""S~.: I:--',B.

Since all S, are isometric embeddings,

II U.÷l..-u..~ II _~ IIr.÷:. - T.II <12" 2 - " . Therefore. for all k. the sequence (U..0.~.~+I satisfies the Cauchy condition in the operator norm. Let us set

1966]

BANACH SPACES WHICH ADMIT 1~ APPROXIMATIONS U~.k = lim, U,.~, E~ = U~o.d,~

195

(k = 1,2, ...).

Since the S, are isometric embeddings and since we have

lim, IIr, II--lira, l{r~-'lI = 1.

lim, It U,., II = lim, ll v~ll = IIv~.,ll = IIU=.~ II = 1. Hence the U~,,k are isometric embeddings, and the E~ are isometrically isomorphic to l ~ ( k = 1,2,...). Clearly E/,=E'~+I, because oo U,.k+11,~+~ =U,,~l,koo

(k = 1,2,---; v = k + 1,k + 2, ..-).

I ~°= 1E 'k is dense in B. Take an arbitrary element b,, Finally we will show that ik.;k of the sequence (b,)~= 1, fix k > m, choose x e l~ such that I] Ti,x - b.II-< 2- ~llb.ll, and put e = Uoo.k x. Clearly

I1T :

- uoo.:

II =

lira, I1T : - u , , :

I1.

Furthermore v-k-1

II T : - ~,, : It

=< IIr:-v,÷,.:[l+

x

II(v,÷,÷,.,-v~÷,.,),,ll

j=l v-1

00

=< i =Xk 12.2-~'llxll -< i =x k 12.2-3'11111. By the definition of x,

Ilxll-< II z;lll" IIz:ll <= IIz:lll( 1 + 2-~)11 b- I1• Hence

lib.-ell--< [I r: - b. I1+ IIr: - v~o,,x II o0

__< 2-*'llb.II + 117:111<2-3'+ 1) ~: 12.3-'llb.ll. i=k

Since limkll rClll = 1, the last inequality implies limkd(bm, E~)= 0. Since the sequence (b,) is dense in B, this completes the proof that the union I,.J~ 1E~ is dense in B. (ii) ~ (iii). If B is a separable n~' space, then there is an increasing sequence (E~')n~ 1 of subspaces of B such that ~.Jn~ 1E" is dense in B, and E~ is isometrically isomorphic to l~ for some nk (k = 1,2,...). (Indeed, let E~ be as in the definition of a ~°-space, and let {b,},oo= 1 be a dense subset of B. By induction, choose E', = E, such that d(b~,, E'~) < n - I for m = 1, ..., n, and E" = E',_ 1.) Therefore, to complete the proof of the implication ( i i ) ~ (iii), it is enough "fill in the gaps", i.e. to show that, if k = 1,2,... and nk+ 1 - n ~ > 1, then there is a chain of subspaces E~ = Fo ~ F t c . . . F,~. 1-,~ = E~÷ 1 such that F, is isometrically isomorphic to

196

E. MICHAEL AND A. PELCYZNSKI

[September

1~+, (v = 0, 1,2, ..., nk+X- nk). But the existence of such a chain of subspaces follows immediately from Lemma 3.2. (iii) ~ (i). This implication is trivial. 5. A refinement of Theorem 1.1 for B = C(S). In the case of a C(S) space, S compact metric, one can prove a slightly stronger result than Theorem 1.1 (cf. Corollary 5.2 below). We recall that a finite-dimensional subspace E of C(S) is called a peaked partition subspace provided it is spanned by a peaked partition of unity, i.e. by non-negative functions fx,f2,'",f, such that Zfi = 1 and I]f~ll = 1 (i = 1,2, .-. n), where 1 denotes the function which is identically one on S. PROPOSITION 5.1. Let E be a linear subspace of C(S). Then the following conditions are equivalent. (o) E is a peaked partition subspace. (oo) E is isometrically isomorphic to l~ for some n, and 1 e E. Proof. (o)--*(oo). This implication is well known (cf. e.g. [9], [11]). (oo)-~ (o). Let f~'e E correspond under some isometric isomorphism to the k-th unit vector u~n) of l~ for k e n . Then clearly [[fkll = 1. Let us set k : [/[(s~)] t - l v j'~, where sk e S is chosen in such a way that [f~(sk)[ = 1. Clearly (cf. Lemma 2.3 (c))f~(sz)= 0 for l # k, l e n. Since l e E , there are scalars (t°)k , , such that 1 = ]~=lt~f~.Thus 1 = l(sz) = Y.~=lt°fk(s~)=t ° for l e n . Thus oo S ~ s n t ' Zk=lfk() 1 for s e S . Since [] Zk=lt~fk]l=]] Zk=lkf~]]=maxk~,]tk] for arbitrary scalars tl, t2, ..', t,, we get ~=l]fk(s)] < 1 for s e S. Thus all fk are non-negative. Hence {A,A,'",fn} is a peaked partition of unity. Combining Proposition 5.1 with the main result of [9] and Lemma 3.2 (cf. the proof of implication (ii)-, Off)) we get COROLLARY 5.2. Let E be an m-dimensional peaked partition subspace in C(S), S compact metric. Then there exists an increasing sequence of peaked partition subspaees E 1 ~ E2 ~ ... of C(S), whose union is dense in C(S), such that dimEn=n for all n and Em= E. 6. Monotone bases in separable ~]o -spaces. We recall (cf. [3, p. 67]) that a sequence (e,),~ t is called a (monotone) basis for a Banach space B provided each b in B has a unique expansion b = ] ~ , t~ei(and ]] b ]}~ [} Z=x t,e~ ][for n = 1,2,...). If (e,)~=x is a monotone basis for B, then the operator P~b = Y~--_ltiei, for b = Y-~=ltie~ e B, is a projection of norm one from B onto the subspace En spanned by et,e2,-.',er Therefore the existence of a monotone basis in B implies the existence of projections P,: B ~ E~ such that (a) liP-I[-- 1, (fl) each range P,B = E, is an n-dimensional subspace of B, (~) E, cE,+x (n = 1,2,...), (6) E =U,~xE, is dense in B.

19661

BANACH SPACES WHICH ADMIT l~ APPROXIMATIONS

197

Conversely, the following observation is due to S. M a z u r (cf. Bessaga [2]). PROPOSmON 6.1.1/f in Banach space B there exists a sequence of projections p oo ( . ) . = t satisfying conditions (~)-(tS), then B has a monotone basis.

Ile.ll

Proof. Define (e,)~=l inductively such that = 1 and eneE, N k e r P , - x for n = 1, 2, .... (For convenience, we set Po = 0.) Since the range o f P~-x is (n - 1)-dimensional, the kernel K e r P , - 1 has codimension = n - 1. Therefore the intersection E, ~ Ker P , _ i is non-empty. T o prove that ( e ,)~Qo= 1 is a m o n o t o n e basis for B, observe first that, since 11P, [[ = 1, e , + l e Ker Ps and e~ ~ E n for v = 1,2, ...,n.

t.e.ll ~_ IP~C~at, e.) [[ = I1~ 1 t.e.[ n+m

ll

n

for arbitrary scalars tl,t2,...,t,+ 1. Thus by induction II >--II L ~ t , e , II for arbitrary scalars t~, t2,-.-, t,+~ (n, m = 1,2,...). But the last inequality, together with (6), implies that ( e~ ) ,®= i is a m o n o t o n e basis for B. (c. f. [10])

Proof of Corollary 1.2.

This follows f r o m T h e o r e m 1.1, L e m m a 2.1 and

Proposition 6.1. It follows f r o m a result o f Lindenstrauss I'7] that, if B is a n~-space and if E is the range o f a projection of n o r m one f r o m B with dim E = n < + oo, then E is isometrically isomorphic to l~(*). Hence we can complete Corollary 1.2 as follows: COROLLARY 6.2. Let B together with a sequence of subspaces (E,) satisfy condition Oii) of Theorem 1.1. Then there exists in B a monotone basis (en)~=l such that (et, e2, ... , e,} spans E~ (n = 1, 2,...). Conversely, if (e~)~=t is a monotone basis for a rc~-space B, then B and the subspaces E n spanned by { e l , e z , . . . , e , } satisfy condition (iii) of Theorem 1.1. REFERENCES 1. C. Bessaga, Bases in certain spaces of continuousfunctions, Bull. Acid. Polon. SCi., Cl. III, 5 (1957), 11-14. 2. F. E. Browder and D.G. de Figueiredo, J-monotone non-linear operators in Bausch spaces, to appear. 3. M.M. Day, Normed linear spaces, New York, 1962. 4. N. Dunford and J. T. Schwartz, Linear Operators, I, New York, 1958. 5. D. G. de Figueiredo, Fixedpoints theoremfor non-linearGalerkin approximable operators, to appear. (*) Indeed, ifB is a n~-space, then, by Corollary 1 of [7, p. 66], B satisfies conditions (1)-(13) of Theorem 6.1 of [7, p. 62]. Since E is the range of a projection of norm one from B, the subspace E also satisfies the same conditions. In particular, E** is a ~l-space. Since E is finite-dimensional, E** = E. Thus E is isometrically isomorphic to In~.

198

E. MICHAEL AND A, PELCZYNSKI

6. V. I. Gursri~, Bases in spaces of continuous functions on compacta and some geometric problems, Izv. Akad. Nauk SSSR Sex. Mat. 30 0966), 289-306 (Russian). 7. J. Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964), 8. E. Michael and A. Pe~czy~ski, ,4 linear extension theorem, to appear in Illinois J. Math. 9. ~ , Peaked partition subspaces of C(X) to appear in Illinois J. Math. 10. V. N. Nikol'skii, Best approximation and basis in a Fr~chet space, Dokl. Akad.| Nauk SSSR 59 (1948), 639-642 (Russian). 11. A. Pelczyfiski, On simultaneous extension of continuousfunctions, Studia Math. 24 (1964), 285-304. 12. F. S. Vaher, On a basis in the space of continuous functions defined on a compactum, Dold. Akad. Nauk SSSR 101 (1955), 589-592 (Russian).

U

~

oF WASHINGTON,

AND UbriwP.srrY oF WARSAW, AND UNIVERSITYOF WASmNGZON

ON A CERTAIN BASIS INto BY

M. ZIPPIN* A B S I T ,A C T

A basis {Xn}~__1 is constructed in co such that there exists no bounded linear projection of co onto the subspace spanned by a certain subsequence oo ¢~o {X,~}t=1 of {Xn}n=1.

I. In~odu¢fion. A. Pelczyfiskiraised the following question ([3],Problem 4): Let {xn}n%I be a basis of a Banach space X. Is each subspace of X spanned by some subsequence {x.~}~°=l of {x.}~=l complemented in X? In this paper we show that the answer is negative by constructing a suitable example in co. Our main tools are the following two propositions: PROPOSmON 1. (See [1] Theorem 3.) l] +1 can be isometrically imbedded into l~" and every linear projection P of 12" onto l~+1 has norm

IIe I1 (. + 1)2-" [n/2

(In/2]

•

denotes the greatest integer ~_ n ]2.)

PROPOSmON 2. (See [2] p. 16, Corollary 3.) IrE is a finite dimensional subspace of a Banach space X for which X** is a P~ space and there exists a projection with norm c from X onto E, then E is a PT¢ space. (X is called a Py space if for every Banach space Z containing X there is a linear projection P from Z onto

x with IIP II If {x~}l~l is a s e t o f elements of a Banach space X then [x~]~,1 denotes the ~2~ the usual basis closed linear space spanned by {xi}i~1. We denote by tYets/--x 2n of 12" and by {f~}l = l the corresponding biorthogonal functionals in 12"= (l~')*. 2. Preliminary lemmas.

Denote by AI the matrix

:) Received Aug. 5, 1966 and in revised form Sept. 8, 1966. * This is part of the author's Ph.D. thesis prepared at the Hebrew University of Jerusalem under the suppervision of Professor A. Dvoretzky and Dr. J. Lindenstrauss. The author wishes to thank Dr. Lindenstrauss for his helpful advice. 199

200

M. ZIPPIN

[Septembcx

and let A k (k > 1) be the matrix obtained from Ak_ 1 by substituting A1 for + 1 and - A~ for - 1. It is easily proved that 2-nAn is a 2 n x 2n symmetric orthogonal matrix. Denote by a~,~ the elements of An and 2 ~ by g(n). LV_~VtMA 1. A~ is obtained from A 1 by substituting + A k _ l f o r 1 and - A~_ t for - 1, k = 2, 3,4,....

The proof follows by induction from the definition of An. As a consequence of Lemma 1 we get LEmc~A2.

For 2 ~ n ,

l
and l ~ _ j ~ _ g ( n - 1 )

an _ _ n - 1 n nl,J -- aid , al,j+g(n-l) ~- atd n

al+s(n_l),j

n-1

aid

=

t

n ai+g(n-l),j+g(n-1)

and

=

n-1 -- aLj •

3. A basis in le~n). Denote by En and F n the subspaces [e~ + ,,n

"le(n- 1)

©l+g(n-1)Jl=l

r~,n

" l g ( n - 1)

,,~n

a n d i..©i - © i + g ( n - 1 ) l i = l

of/~(")respectively, and let Tn be the transformation from l~(n- 1) to E,, defined by e(n- 1)

T,

Z

)

cle]-'

e ( , - 1)

=

Z

1=1

c~(e]+ e~+,Cn_l)).

1=1

It is obvious that T~ is a linear isometry onto E n. Let us denote x ; = g.dj ve(n)..n : l ~ l , j ©~, j. L ~

Proof.

3.

For 2 ~_ n and 1 < i < g(n - 1), T,(xT-1) = xT. g(n-1) n - I

T~(x;-i) = T,

Z a~d ~

g ( . - t) =

-1 \

)

j=l

e(n)

ae--1

aid

j=l

n

n

(ej + ej+e(s_l)

n B a|,jej.

J=l

The last equality follows from Lemma 2. Let y~a= x~~ for i = 1,2 and define 1 ~_ i < g(k -

~-e(k-x) - ~

g(k-1)+l
for k > l .

1)

T~y~-1

)

1966]

201

O N A C E R T A I N BASIS I N co

Denote by l(n) the set {i: i = g(k), 0 <- k < n}. LEMMA 4. For n >=1 and i e I(n), x~ = y~. Proof. The case n = 1 is clear. Suppose that xik = yk for k < n and i e I(k). By the definition y~(.)=x~(.). Since I ( n ) = { g ( n ) } w I ( n - 1 ) , if i ~ I ( n ) and i < g(n) then i ~ I ( n - 1); therefore, by the induction hypothesis y~ = r, y7 - 1 = T. x 7-1 = x I. " (The last equality follows from Lemma 3.) LF~VIA 5. For n > 1, g(n -- 1) < k < m < g(n) and every sequence of scalars

Z cl(e~ + e~+e(s_l) ) H II

i=l

g(n- 1)

~_

Z

t 11

c~(e'2+ e~+z(._ 1)) +

Z

B

i=g(n-1)+1

|=1

<

n

ci(e~_g(,_ 1) - ei

c,(e? + e,~+g~,_1)) +

~

i=l

ci(e~-gc,- 1) - e7 )

i=g(n-1)+l

We omit the trivial proof. LEMMA 6. For k > 1 ,

1 < n < q
and every sequence of scalars

,. "t~(k) ~lit

= 1

(1)

,~1 c~yk <21t~1 ciyk

Proof. The ease k = 1 is obvious. Suppose (1) holds for k < m and let us prove the assertion for k = m + 1. We discuss separately the following four cases: (a) q < g(m) In this case (1) follows from the definition of y~+t, the fact that Tin+1 is a linear isometry from l~ m) onto Era+ 1 and from the induction hypothesis. (b) g(m) < n < q < g(m + 1) Bythe definitions of Tin+l, and y~,+t, ~+1 ~Em+1 for 1 < i < g(m); Therefore s,(~n)

Z

~'(rn)

c~y'~+I

i=l

for

y~,+1

some =

bl,b2,...,be(m).

•

~-

bi(e~,+i_t. ~ e_~,+1, i+g(ra))

f=l

On

the

m+1 ~+I e~_gc.) -e~ , hence, by L e m m a

other 5

hand,

for

g(m)
202 (2)

M. ZIPPIN

II~

c,y,~+t

II~

=

1=1

b~(et.+1 +

[September

ef~,+1. + ~ (m))

1=1

Cit,ei-g(m) -- e'~+ t ) t=g(m)+l

< =

itg_m+l--_m+l "t t"tkei "~ ¢i+g(m)] ~L

=

~

/ m+l c~[e~-e(m)

e~+t

i=g(m)+ l

I=1

c'Y?+'

(c) n ~ g(m) < q < g(m + l) y~+leFm+l for g ( m ) < i ~ g ( m + l )

,,+1 ~) ~ F,.+ 1 by Lemma 2), therefore (Yi(m+

q

g(m)

ciy~+1

=

i =g(m)+ 1

di(er +1 - - ei+g(m)) .+i.

2 i=1

for some dl, dz,'",dg(m). If g(m)

e(m) ciy~+l=

~

i=1

m + l ", b~(e'~+t+el+g(,.)),

i=l

(as in (b)) then, by Lemma 5, c,y~ +1

=

l=

II~== b , ( e ~ + l + e , + g t m ) ) =1

'

_

i=l

bi(e,~+t ~- e~+:(,.)) ,,,+1 . + ~

.+ 1 d~(e~ +1 - el+g(.,

l=l

From case (a) it follows that

I=I

/=I

=

(d) g(m) < n < q = g(m + 1) Denote by Pm+a the projection of l~ t'+l) onto the one-dimensional subspace m+l xg(m+ 1)JI defined by

1966]

ON A CERTAIN BASIS IN co

203

P m + t X = ( - 1)=( G '+1 =+1 ,)(x))XgCm+ , + l 1)" ) (x) - f;¢m+ (According to §1 {f~=+i} denotes the usual basis of I[ ('~+~).) It is easy to see that [IPm+t [I= 1 and that Pro+ ly~m+1=0 for 1 < i < g(m + 1) - 1. Hence, I - Pm+t is a projection of I~ m+ 1)onto [yi m+13~(=~+ 1)- 1along LYg(m+I)J ~ m+1 ~ a n d I I I - P . ÷ 1 1 1 _ < 2 Since n > g(m), it follows from (2) that

~ ciy~+l II~_

f=l

c,y,.÷~ U

i1~,-~ ,-, i=1

I1¢, ~., (=~q

~l

~

+1

|

Ci

1

Y~+I) II< 2 Ii~

ciym+lll

i=t

This concludes the proof of Lemma 6. 4. A non-complemented subspace of Co. Denote by {ei}i~ 1 the usual basis n Co and let U, be the natural linear isometry from l~(n)onto

~ai=g(.)

e "lg(n+ 1 ) - I n

= 0, 1, 2, ....

U.

cte7

= ~

\i=1

)

ciei+g(n)-I •

f=l

Put z ° = et and z~'= U.(y~) for n _>_1 and 1 <- i <- g(n).

The sequence f~,n'tg(n) t,,~si=l n=o,1,2 .... in its natural order

LEMMA 7.

0 1 1 2 2 2 2 Z 1, ZI~ Z2~ Z 1, Z 2 , Z3~ g4~ . . .

forms a basis in Co.

Proof. Obviously [z~]~)l .=o,1.2 .... = CoIf q =< r =< g(m + 1) then by Lemma 6

,3,

II,o -----

c,z, max

< max

_

+

iz, cll.

max

X c:zf

k~m

/=1

,

~ c~+'z: +1 =I

max

Y~ c~z~

k_Nm

i=l

, 2 t=l

x c,~z,~ + :~ k=O

/=I

i=l

Similarly, for m > n, q < g(n + 1) and r < g(m + 1) it follows from (3) that

204 (4)

M. ZIPPIN =o~=

c~z~

+ ,=1 ~ c,~+l-'+lz,

g(k) 2 max

~

k~n+l

2 max

~

1=1

{

max

k~_rn

Zcz

2

=0 \ i = 1

} ¢i~Zl~

~,=

c~z~

u/Ii ,

~., c~ ÷1z'÷1 ~1

ul

+ ~=1

The last inequalities show that the sequence {zT} in its natural order forms a basis in Co and Lcmma 7 is proved. By [1], p. 459, [xT]~ ~1(n) is isometrically isomorphic to /~+1 and since U, is a linear isometry, we get by Lemma 4 that [z~.]~ ,z(n) is also isometrically isommorphic to l~÷1. Suppose that P is a bounded linear projection from Co onto the subspace Y spanned by the sequence {z i }, ~i(~) n = 0,1,2, .... It is obvious that the sequence {z~}, ~ t(,) n = 1,2,... forms a basis in Z From the proof of Lemma 7 it follows that there exists a sequence of projections {Qn} from Y onto <[z']'/~'(') IIQ- I[ 211hp IIQn2...__ It < NOW,. Q,P is a projection from Co onto [z~l ~,(, ) and Co = m IS a P1 space; it follows from Proposition 2 that I~+1 is a P~ space, n = 1,2,3,..., where ~ = 211PII. This contradicts Proposition 1; therefore, there exists no bounded linear projection from Co onto Y. Since Y is spanned by a subsequence of the basis [~n'(g(n ~,.i s~= 1 n = 0, 1, 2, ... of co we have constructed the desired example. As J. Lindenstrauss has remarked, a similar example can be constructed in the reflexive space ~ ~o= 1 @ p l~ (*)• The proof will be almost the same.

It;

REFERENCES

1. B. Grtinbaum, Projection constants, Trans. Amer. Math. Soc. 95 0960), 441. 2. J. Lindenstrauss, Extension of compact operators, Mere. Amer. Math. Soc. 48 (1964). 3. A. Pelczyfiski,Some open questiorLsin functional analysis, a lecture given to Louisiana State University (dittoed notes). THe I - I ~ w UNzwasr~ OF JERUSALEM

ON BANACH SPACES WHOSE DUALS ARE L1 SPACES BY

A. J. LAZAR AND J. LINDENSTRAUSS*

ABSTRACT

A structure theorem for Banach spaces whose duals are L1 spaces, is proved.

The purpose of this note is to settle a question left open in [2, p. 66] as well as a related problem contained implicitly in [3]. A Banach space X is called an ,/Vz space [2] if there is a net {B~} of finite-dimensional subspaces of X directed by inclusion such that X = U~B~ and every B~ is a ~ space. It was proved in [2, p. 66] that if a Banach space is an ,W'~ space for every 2 > 1 then X* is an LI(/~) space for some measure/1. Here we shall prove that also the converse is true. In [3] Michael and Petczyfiski studied Banach spaces X which have the following property: For every e > 0 and every finite set A in X there is an integer n and an operator T : I ~ ~ X such that for every y E 1~ and such that the distance of x from T l~ is < e for every x s A. Here ! ~ denotes the space of all the n-tuples o f real numbers y = (21,22,..., 2n) with [] y [] = max, ]2i ]. These spaces were called in [3] a o~spaces. Since i : is a ~1 space for every n it follows easily that an a °° is an ,W~ space for every 2 > 1. Here we show that the class of a ~o spaces coincides with the class of the spaces which are ~ x for every 2 > 1. We consider only Banach spaces over the reals, but our result and its p r o o f are valid also in the complex case. We state now our main result. T , ~ O ~ M 1. L e t X be a Banach space. Then the following three statements are equivalent, (i) X * is isometric to the space Ll(Iz) f o r some measure IZ. (ii) X is an ,/V~ space f o r every ;t > 1. (iii) X is an a ~° space. F o r spaces X whose unit cell has at least one extreme point Theorem 1 can be also easily deduced from the results of [1]. The p r o o f of Theorem 1 presented

Received August 19, 1966. * The research of the second named author has been sponsored by the Air Force Otfice of scientific Research under Grant AF EOAR 66-18 through the European OffSet of Aerospace Research (OAR) United States Air Force. 205

206

A.J. LAZAR AND J. LINDENSTRAUSS

[September

here is, however, shorter than the arguments given in [1] from which the special case of Theorem 1 follows. A list of other properties equivalent to property (i) of Theorem 1 is given in [2, Theorem 6.1]. By combining the results of 1"3] with Theorem 1 we get immediately the following stronger version of Theorem 1 for separable spaces. THEOREM 2. Let X be a separable Banach space. Then the following two statements are equivalent. O) X* is isometric to the space Ll(lO for some measure g. (ii) X has a monotone basis {e~}i~l such that for every n the subspace of X spanned by {ei}~%1 is isometric to 1~ We pass to the proof of Theorem 1. As we have already remarked we need only to show that (i) ~ (iii). Let X satisfy (i) of Theorem 1, let A be a finite subset of X and let 0 < e < 1. In the definition of an a °° space it is clearly enough to consider sets A with It xl/-- 1 for every x e A (otherwise replace x e A by x/ll x II a n d ~ by e/maxx~a IIx ]l). so we assume that II~ II-- 1 for every x ~ A and let B be the subspace of X spanned by A. Let E o be the set of exposed points of the unit cell of B*. Let J~o be the set obtained from Eo by identifying every f with - f , and let ff be the quotient map ~b:Eo ~/~o. We metrize E o by putting d(dpf, dpg) = rain (llf- g II, Ilf+ Since B is finite-dimensional the metric space/~o is totally bounded. Hence, there is a finite number of subsets {Gl}~ffii of /~o such that G~n Gj = ~ for i ~ j , E o = 1,.J7=1G~ and Gl has for every i a non empty interior and a diameter < e. Since e < 1 there is for every i a subset G~ of Eo such that c~- l Gi = Gi U - G,, G, n - G, = ~ and < ~ f o r every f, g e G,. For every i pick an f~ E G~ such that fffi is an interior point of G~ and let x~ e B be such that A(x,)= II x, II = IIf, II = ~ and f ( x , ) < 1 for every f ~f~ in B* with

gll).

IIf-gll

Ilfl[ = Let E = [,.J~=1 Gt and let l °°(E) be the Banach space of all real-valued bounded functions on E with the sup norm. Let the operator U:B--. l°°(E) be defined by U b ( f ) = f ( b ) , b e B, f e E. Since the unit cell of B* is the closed convex hull of E u - E we get that U is an isometry. From our choice of the x~ and f~ it follows that there is a c5 > 0 such that If(x,) I < 1- a for every i and every f ~ E ~ G~. We assume as we may that fi < min (2]3, 1 - t). Let y~ e l°°(E), 1 < i < n, be defined by Yt(f) = 1 if f e Gl and Yl(f) = 0 if f e E ~ G,. By our choice of fi we get that 11--< ~-Xn fact, if f ~ E N G~ then

II -Wx,-y,

] ~ - ~ U x , ( f ) _ y,(f)] =

la-~f(x,)l

_< ~ - 1

--

1,

while for f ~ G~ we get (since ~-~f(x~) ~ (1 - ~)/~ ~_ 1)

Ia-lUx,(f)-

Y,(f)I = I~-' f (x,) - 11~ ~-~ - 1.

1966]

BANACH SPACES WHOSE DUALS ARE L~ SPACES

207

Since X* is an LI space there is (see e.g. [2, Theorem 6.1 (3)]) an operator T from l~(E) into X whose restriction to UB is equal to U-1 and with norm II T II < (1 - ~ + ~/2)/(1 - ~) We have, in particular, that for every 1 < i < n

Ill-'x,- Ty, ll =

II~-ITUx, -

Ty,[I---IlTllll~-'ux,-y,l[

<=~-I_

1 +el2,

and hence since IIx, II-- 1 we get that II Zy, II >-- 1 - ~ / 2 Let Y be the subspace of I~(E) spanned by (Yt}~'=l. Clearly, Y is isometric to 1~ We claim that for every y e Y

(1 - 2n)II y

II<

II T y I[ < (1 + e)H Y I[.

That II Z II < 1 + ~ follows from our choice of II Z 11 2 - ~ and hence since I1Zz II < 1 + ~ we get

that 11Ty II > 1 - 2~ In order to conclude the proof that X is an a ~ space it is now enough to show that for every x e B with fix I!--1 there is a y e Y with II T y - x II < 2~ Take y = ~7=~f~(x)y, E Y. Then I1Y - Ux II < ~ (recall that the diameter of each G, is < ~) and hence

11r y -

xll < 11TII H Y - U~II z~(1 + 8 ) < 2 ~

and this concludes the proof. REFERENCES 1. A.J. Lazar, Spaces of affine continuous functions on simplexes (to appear). 2. J. Lindenstrauss, Extension of compact operators, Mere. Amer. Math. Soe. 48 (1964), 3. E. Michael and A. Pelezy~ski, Separable Banach spaces which admit In~ approximations, Israel J. Math. 4 (1966), 189-198. THB HEBP~W UNIVERSITY OF JERUSALEM

RECURRENCE OF SUMS OF MULTIPLE MARKOV SEQUENCES BY

P. HOLGATE ABSTRACT

A d-dimensional random walk on a lattice is studied in which each step is bounded, and may depend on the previous m steps. It is proved that if trivial cases are excluded, there are no recurrent points for d _~ 3, and conditions are given for the existence of sets, recurrent conditional on the first ra steps, for d = 1, 2. Let X t , X 2 , " - , be a sequence o f d-dimensional random vectors with integer valued components, and let S , , = X 1 + ...+X,,. In the case where the X~ are mutually independent, each taking each of the values + e~ with probability 1/(2d), where ej is the vector with unity in itsj-th component and zeros elsewhere, P61ya proved in [8] that Pr{Sn = 0 for an infinity of values of n} (la)

=

1, if d = 1,2,

(lb)

= 0, if d > 3.

The recurrence problem when the Xi form a Markov chain has been studied in generality only by Gillis [5], although other writers have dealt with the distribution of Sn in this case for d = 1, and Seth [9] has obtained some further results on recurrence for this value of d. Gillis proved that P61ya's result still holds if the Xi have the same range as above and satisfy symmetry conditions of the form (2)

Pr{Xn = ¢ [ Xn-I = ¢'} = Pr{X, = - ¢[ X , _ l = - ¢'},

and simplifying restrictions which he conjectured to be inessential, although his p r o o f o f (lb) did not cover the odd integers ~ 3. In this note I suppose that the Xi form a multiple Markov chain o f arbitrary dependence, and are bounded, and by means of a different method based on the recurrence properties o f finite Markov chains and generalisations of P61ya's result, [1,4], give a necessary and sufficient condition for (1) to hold essentially. (Complete enumeration o f the particular cases corresponding to degeneracies in the transition structure would be tedious). However, unlike Gillis's analysis of the Received June 9, 1966 and in revised form Sept 26, 1966. 208

R.F.CURRENCE OF SUMS OF MULTIPLE MARKOV SEQUENCES

209

backward equations of the Sn process, this method does not readily provide information on the distribution of Sn. Let us suppose that the X~ form a multiple Markov sequence of order m, [2, pp. 89, 185], subject to the condition that if X[ j)is the jth component ofXtthen with probability one d

(3)

IIx, II =

I "1

B.

Let Z~= (Z~1), ..., Z~m)) = (Xi,... , Xi-m +i)be defined for i ~ m as the m dimensional vector whose components are consecutive vectors of the sequence {Xt}. The sequence {Z~} is a simple Marker chain M, [2], and by (3) it has a finite state space C. To avoid the uninteresting complications referred to above it will be assumed that M consists of a single ergodic class, (assumption E). Now choose an arbitrary state ~oeC, and denote by 7"1,T2, ..., the random values of n for which Z,,= ~o and define the following random variables, k

U|= T[+I- Ti, Vi=ST,+I--ST,, Wk= ~a V[ i=l

(4)

Xt(~)= 1 if Zt = [, = 0 otherwise,

Tl+1 v,(O = ~ x,(O. /=TI+I

It is known from recurrence theory, [3, Chapter 15, in particular Exercises 19,25] that the Ul are independent, and for i =>2 are identically distributed with Pr{U i = k} = a t where for some positive A < 1, ak < 2~. The V, are also independent and for i > 2 are identically distributed. (This approach and similar notation has been used by Katz and Thomasian [7] to obtain probability bounds for sums of functions defined on Marker chains.) Further, since M has a finite state space it admits a stationary distribution p([) with the property that for some constant A, #v,(O = Aft(O, ( = v(O). Let ~(~) be the marginal distribution induced by fl on the first component of ~.

If conditions (3) and (E) hold, then for the sequence V~ defined above, (i)

,V~=A

Y~ ~ ( 0 ,

ill v, II < oo.

210

P. HOLGATE

[September

Proof. (i) TI+t

ce, =

~

~*Xj

j=Tl+l

=

2

t;°)v(O

=

A ~ ~")P(O

=

A

~(0.

E

oO

(ii)

llv, ll _<

ak(kB) 2 k=l OD

< B2 ~

k22k

k=l

=

B~(1

+ ~ ) / ( 1 - ~)3.

Let p be a possible value of W~, and let Q1, Q2, "", be the finite or infinite subset of {T~} for which W~ =p. Then since Wk =P is equivalent to STk+,--ST, =p, Zrk+l = (o, we have the result (5)

Pr{St = a I('; Q~ < 1 < Qi+l} is independent of i.

Suppose now that the initial state is chosen arbitrarily, Zm=~', i.e. the first m steps XI, .-.,Xm are specified. A value a (or a pair a,O, will be called a possible value (or pair), given (', if respectively Pr{Sn = , [ Z , = ¢'} > 0, Pr{S, = a, Z~ = ~ [Zm = ¢'} > 0. Trmo~M 1. Under assumptions (3) and (E), for d = 1, 2; (i) if ?E~(¢)= O, every point a that is possible given ~' is also recurrent given ~', (ii) if X ~ ( ¢ ) # O, no point is recurrent. Proof. (i) Consider the sequence Wk defined in (4), taking ~o = ~'. By Lemma 1, (i) it satisfies the conditions for the existence of a recurrent set, given in 1"1,4]. Let us choose a point p belonging to this set and suppose that a is a point, possible for S, given ~', not belonging to it. Then Pr{St = e l i ' ; Qt < l < O,+a} > 0 for some i. But by (5) this holds for all i, and hence a is recurrent. The set of possible values given ~' will in general consist of a subset of the cosets of a subgroup of the additive group of one or two dimensional integers respectively.

1966]

RECURRENCE OF SUMS OF MULTIPLE MARKOV SEQUENCES

211

(ii) The points which are possible for Wk given ( ' are transient for Wk given ('. Let p be such a point. Now suppose that some point a is recurrent for S.. Then by the finiteness of C, some pair (a, (") must be recurrent. Let T[, T~,..., be the values of n for which S. = a, Z, = (". Now Wk=P for some k if and only if Sl = p, Zl = ( ' for some I. Since p is possible for Wk we have Pr{S, = p,Z, = ( ' l ~ ' ; T ( < l < T[+,] > 0 for some i, and hence for all i since (S, = tr, Z,, = [") is a regenerative event in the sense of (5). This implies the recurrence of p and the contradiction establishes the result. If the symmetry condition Pr{X.

= ¢.1x._1

Pr(X.

=

= ¢._....,x._.

= ¢._.}

=

(6) -

=

-

¢.-1,-..,x.-.

=

-

n > m,

is imposed on the transition probabilities, it is easy to see that the stationary distribution on m must be symmetric, f l ( [ ) = / 3 ( - O, hence ~(~)= c t ( - ~) and condition (i) of Theorem 1 is satisfied. It can readily be seen that (6) is satisfied by the correlated random walk studied by Gillis in [5], mentioned earlier, and also by the two dimensional process which he discussed in [6]. In both these eases all points are recurrent whatever the initial step. TrlEOREM 2. For d >=3, every point is a transient point of S,. Proof. Suppose there exists a point a such that Pr (S. = a for an infinity of n) = n > 0. Then since the state space C is finite there must exist a ~o such that S. = a , Z , = (o for an infinity of n, with probability re. Using this [o, define a sequence V~as in (4). Then we have a sequence of independent random variables, for d => 3, for which it is not true that every point is transient, which contradicts the assertions of [1, 4]. I am most grateful to the referee for pointing out an error in my discussion o f Lemma 1, (i), and showing that Theorem 1 held under wider conditions than I had originally imposed. REFERENCES

1. K. L. Chung and W. H. J. Fuchs, On the distribution of values of sums of random variables, Mere. Amcr. Math. Soc. 6 (1951), 1-12. 2. J.L. Doob, Stochastic Processes, Wiley, New York, 1953. 3. W. Feller, Probability Theory and its Applications, I (2rid Ed.), Wiley, New York, 1957. 4. F. G. Foster and I. J. Good, On a generalisation of Pdlya's random walk theorem, Quart. J. Math. (2) 4 (1953), 120-126. 5. J. Gillis, Correlated random walk, Proc. Camb. Phil. So¢. 51 (1955), 639-651.

212

p. HOLGATB

6. L Gillis, A random walk problem, Proc. Camb. Phil. Soc. 56 (1960), 390-392. 7. M.L. Katz and A. J. Thomasian, An exponential bound for functions of a Markov chain, Ann. Math. Statist. 31 (1960), 470-474. 8. G. P61ya, ~)ber eine Aufgabe der Wahrseheinlichkeitsrechnung betreffend die Irrfahrt im Strassennetz, Math. Ann. 84 (1921), 149-160. 9. A. Seth, The correlated unrestricted random walk, J. Roy. Statist. Soc. B. 25 (1963), 3 ~ A.O0.

]BIOld]~rRIC~SECTION, T ~ NA~tnte CONSnVANCY, LONDON

APOLOGY The Editors apologize to the author of the article: "On the mean length of the chords of a closed curve", which appeared in Vol. 4, No. l, for printing his name incorrectly. The author's name should read G~bor LOK(~.