Functional Analysis, Holomorphy and Approximation Theory, Proceedings of the Seminario de Analise Functional, Holomorfia e Teoria da Aproximaçāo

481 Pages·1982·6.19 MB·English

by  Jorge Alberto Barroso (Eds.)

NORTH-HOLLAND
MATHEMATICS STUDIES 71
Notas de Matematica (88)
Editor: Leopoldo Nachbin
Universidade Federal do Rio de Janeiro
and University of Rochester
Functional Analysis, Holomorphy
and Approximation Theory
Proceedings of the Seminario de Analise Functional, Holomorfia
e Teoria da AproximaGGo, Universidade Federal do Rio de Janeiro,
August 4-8,1980
Edited by
Jorge Alberto BARROSO
lnstituto de MatemBtica
Universidade Federal do Rio de Janeiro
1982
NORT-HOLLAND PUBLISHING COMPANY-AMSTERDAM NEW YORK OXFORD
North-Holland Publishing Company, 1982
All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem,
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ISBN: 0 444 86527 6
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Lihrary of Congress Cataloging in Publication Data
Semingrio de Anslise Funcional, Holomorfia e Teoria
da Aproximaf % o (1980 : Universidade Federal do Rio
de Janeiro)
Functional analysis, holomorphy, and approximation
theory.
(North-Holland mathematics studies ; 71) (Notas de
matematica ; 88)
1. Functional analysis--Congresses. 2. Holomorphic
function--Congresses. 3. Domains of holomorphy--
Congresses. 4. Approximation theory--Congresses.
I. Barroso, Jorge Alberto. 11. Title. 111. Series.
IV. Series: Notas de matematica (North-Holland Publishing
Company) ; 88.
QAl.N86 no. 88 [QA320] 510s L515.71 82-18908
ISBN 0-444-86527-6
PRINTED IN THE NETHERLANDS
FOREWORD
This volume is the Proceedings of the Semindrio de Andlise
Funcional, Holomorfia e Teoria da Aproximapzo, held at the Insti-
tuto de MatemBtica, Universidade Federal do Rio de Janeiro (UFRJ)
in August 4-8, 1980. It includes papers of a research or of an
advanced expository nature. Some of them were presented at the
Seminar. Others are contributions of prospective participants,
that, for one or another reason, could not attend the Seminar.
The participant mathematicians are from Brazil, Chile, England,
France, Spain, United States, Uruguay, West Germany and Yugoslavia.
The members of the organizing committee were J.A. Barroso
(Coordinator), S. Machado, M.C. Matos, L. Nachbin, D. Pisanelli,
J.B. Prolla and G. Zapata.
Our warmest thanks are due to the support of the Conselho
de Ensino para Graduados e Pesquisa (CEPG) of UFRJ, Mainly through
its President, Profeseor SQrgio Neves Monteiro, and to the I.B.M.
do Brasil.
We are happy to thank Professor Paulo Emidio de Freitas Bar-
bosa, Dean of the Centro de CiGncias Matemgticas e da Natureza
(CCMN) of UFRJ, in whose facilities the Seminar was very comforta-
bly held.
Our gratitude and admiration to our friend Professor Leopoldo
Nachbin, whose experience and support made the task of preparing
this volume easier.
We also tbank Wilson Goes for a competent typing job.
Jorge Albert0 Barroso
Functional Analysis, Holomotphy and
Approximation Theory, JA. Barroso led.)
0N orth-Hollond hblishing Company. 1982
In memory of A. MONTEIRO, an extraordinary man and teacher
ON A LIFTING THEOREM AND ITS RELATION
TO SOME APPROXIMATION PROBmMS
Rodrigo Arocena and Mischa Cotlar
SUMMARY
We point out that there is a close relation between some
approximation problems and a lifting theorem studied in previous
papers. A new simplified proof and an improved version of the
theorem, more adequated to our aim, are given.
1. INTRODUCTION AND NOTATIONS
In this self-contained paper we continue the study of some
questions considered in [2] and related to a lifting theorem. We
show that this lifting theorem allows to approach some classical
approximation problems. Conversely, these approximation problems
yield a natural motivation of the lifting and suggest the corres-
ponding theorem. Thus, we give a new simplified proof and an im-
proved version of that lifting theorem, adapted to and motivated by
approximation questions.
We shall work in the unit circle TN [0,2rr] and use the
following notations: en(t) = exp(int), P+ = {analytic polynomials,
‘kn=O ckek(t), n ;r 03, P- = Ex,, -1 ck e k(t ), n=1,2 ,...3 , eJP+ =
= {x;+n ckek( t), n t 0) , p = P+ + P- , dt = the Lebesgue measure
-n
in T, P - =~ { c ckek(t), n > 03, HP = {f 6 LP(T) : 2.(n)= 0,
k=-1
2 R. AROCENA and M. COTLAR
Y n < 01, where 2. is the Fourier transform of f and p E [ l,m],
so that f E Hp has an analytic continuation f(z) in { IzI < 11,
and C(T) = {all continuous functions in TI.
Since H" is a subspace of Loo, given g Lm(T), a clas-
sical approximation problem is to characterize the distance of g
to H", describe the set of all best approximations of g by ele-
ments of H" and give a condition for unicity of the best approxi-
mat ion.
More generally, given g E Lm(T) and d > 0 we want:
(i) a condition for the existence of a h E H" such that
/(g-h((<m d, that is Ig(t)-h(t)( S d a.e.; (1)
(ii) a parametrized description of the set 3(g,d) = {h 6 ff
satisfying (I)] ;
(iii) a condition for the unicity of h E 5(g,d).
Condition (1) is equivalent to saying that the matrix
1
N = N =N(t)=
n h
d
is positive definite for almost all t. It is easy to see that
this is eJq uivalent to (cfr. (2b) bellow)
(Nf,f) = fl?, d dt +[f,?,(g-h)dt +JFlf2(E-fi)dt +(f2f2 d dt 2 0,
for all f = (fl,f2) E C(T)xC(T), and in this case we write N 2 0
and say that N is positive. Thus (1) is equivalent to:
3 h E H" such that Nh 5 0. (la)
Since only g and d are given, it is natural to consider the
matrix
ON A LIFTING THEOREM 3
and try to replace (la) by some similar condition on M. We cannot
assert that (la) implies M B 0. However, since h is analytic,
it is immediate that (Mf,f) = (Nf,f), Y- f = (fl,f2) E p, x 63- ,
and we indicate this fact by writting M N. In particular, (la)
implies that (Mf,f) > 0 Y f = (fl,f2) E P, x p- , which is de-
noted by M > 0 or by saying that M is weakly positive.
Thus (la) says that there exists h E H" such that the
forms (Mf,f) and (Nhf,f) coincide on p, x p- and the form Nh
is positive on the whole of C(T) x C(T), while M is positive
.
only on 63, x 63- We express this property by saying that Nh is
a positive lifting of (the weakly positive form) M. We shall see
that M t 0 implies, and therefore is equivalent to (la), so that
M > 0 is the desired condition on M.
In other words, if M > 0 then M has a positive lifting
This fact is a special case of the following general lifting
Nh.
theorem. Consider 2x2 matrices M = (mug), a,@ = 1,2, whose
elements are (complex) Radon measures in T; we suppose that
m21 = i12. With each such matrix M we associate the form (M-,*)
in C(T) x C(T) defined by
(fbfe
(Mf,f) = "up ' f = (flf2) E C(T)XC(T) (2)
a ,8=1
It is easy to see that (Mf,f) = (Nf,f), V f E C(T)xC(T), iff
M = N.
- .
We write M N if (Mf,f) = (Nf,f), V f E p, x p- By a
theorem of Fejer-Riesz if g E P and g B 0 then g L flPl where
.
fl E P, And by classical theorem of F. and M. Riesz,
4 R. AROCENA and M. COTLAR
-
implies n12 = m12 h, with h 6 H1. From these facts it follows
easily that
-
M N iff mll = nll, m22 = n22, n12 = m12 -h, hcH1. (2a)
We write M 2 0 if (Mf,f) 2 0 V f E C(T), and M > 0 if
.
(Mf,f) 2 0 V f E P+ x P- If dm = gUB(t)dt, g E L1,
a% a%
a,% = 1,2, then it is easy to see (by letting fl = X1u, f2 = X2u,
u(t) -+ (l/8)lE(t), E = (to,to+8)) that, in this case,
M z 0 iff the matrix (g (t ) is positive definite, V a.e.t. (2b)
Though M > 0 doesn t imply M 2 0, however the following
lifting theorem is true:
M > 0 c, 3 N with N - M and N 2 0.
That is, if the restriction of (PI.,.) to 6, x P- is
positive then this restriction can be lifted to a positive form
(N*,-) on C(T) x C(T). In other words
(1" (mil -
m12 hdt
-
-
Y2)> c1 mZ1 hdt m22 2 0 (4)
21 m22
for some h E H1. Let 3(M) = {h E H1; h satisfies (k)]. Then
M > 0 iff 3(M) f 0, and the h E 3(M) furnish all the positive
liftings of M. (4a)
This lifting theorem was proved in [4] and studied im more
detail in [ 2 ] . In section 2 of the present paper we give a new
simplified proof and a more precise version of (4) which leads to
an improved description of 5(M), more adecuated to our aims.
Using this version we give in section 3 a condition for the unicity
ON A LIFTING THEOREM 5
of the lifting (that is for cardinal 3(M) = 1). In the special
case where M has the form (lb), M > 0 is, as seen above, equi-
valent to (la), and the set 3(M) is in 1-1 correspondence with
the set 3(g,d). Therefore the results of sections 2 and 3 furnish
in particular a solution of the above approximation problems (i),
(ii), (iii). Moreover they also apply to the classical case
d = distance of g to H". In fact, in this case we have by de-
c; >.
finition that Z(g,d+e) f Q, V e > 0. Hence by the above remark
0, Y E > 0, and letting E + 0 we get M 3 Mo>O.
M = d+E
By the above lifting result we get that 3(g,d) f 0. We have thus
proved the existence of a best approximation of g by elements of
H" (which is well known)and the results of section 3 give a con-
dition for the unicity of this best approximation.
Moreover, an explicit expression of the unique best appro-
ximation h is given in terms of its Fourier coefficients, so that
a condition for h to belong to a smooth class Cn+a can be
written down. These solutions of the above approximation problems
are somewhat different from the well known results due to Adamjan,
Arov and Krein.
In section 4 the general results of sectionss 2, 3, are
applied to and motivated by a general approximation problem.
Finally in section 6 a similar procedure is used to study the
balyage of generalized Carleson measures, whose characterization
is also related to approximation questions.
We are pleased to acknowledge our gratitude to Prof. Jorge
Albert0 Barroso for inviting us to contribute to this volume.
6 R. AROOENA and M. COTLAR
2. DESCRIPTION OF 3(M)
Let us fix the matrix M = (map), a,B = l,2, where the m
112.
are Radon measures in T satisfying: mll 2 0, mZ2 2 0, mZ1 =
We associate with M a seminorm p in C(T), defined as
.
follows
Let R = {w E C(T): 3 a constant c > 0 with
V t], so that w E R implies l/w E R, and set (for
p(f) = inff 1I f I (wdmll + 1 dmZ2) : w E R] . (5)
It is clear that if dm = gu (t)dt, 0 < gu(t) E C(T),
aa
a = 1,2, then
(5a)
and that if
"11 = m22 then p(f) = 2 (5b)
Thus in both cases it is evident that p is a seminorm. In order
to show that this is true in the general case, it is enough to
verify the inequality.
+[I
(jllllfldmll 1 Ifldmz2)
2 p(lfl+lgl), v f
Consider first the case
define
(If
w, w' by wllfl + w21g( =
Since w2/w1 + w1/w2 2 2 we have w' 2 1 so that the left
[$
side of (5c) is 2 [w(lfl+lgl)dmll + (Ifl+lgl)dmZ2 p(lfl+lgl),
since w E R. The general case is proved by fixing wl, w2 and
ON A LIFTING THEOREM 7
applying the result just proved to If1 + E and g, using
P(lfl + E + lgl) 5 P(lfl+lgl) and letting E -b 0.
Thus p is a seminorm, and it is easy to verify (tcking
w = lf112/(lfl+e) and lefting E + 0+) that
[Ifl12 (lf212
if f = flf2 then p(f) S dmll + dm22. (5d)
We also associate with M a real linear functional I. in
C(T), defined by
-
-
I (f) = -2 Re f dm12 = -fdm1 2 dm21 (6)
for f E C(T), where C(T) is considered as a real vector space.
IXMMA 1. If M = (ma B), a,p = 1,2, is a matrix measure with
mll 2 0, mZ2 2 0, m21 = i12 , then:
where I. and p are the functionals associated with M by (6)
and (5), respectively, and "f+= the closure of eP+ in C(T).
PROOF. The condition M > 0 can be rewritten as
where 6 is the closure of p* in C(T). Assume that M > 0,
f
that is that (#-) holds, and let us prove that
Every I E ef+ can be written as I = Il h where h is
a finite Blaschke product, h(O)=O,and Il E k3+ with I1(z) f 0