Fourier Series in Orthogonal Polynomials

tourier ueries in Orthogonal Polynomials

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tourierueriesin Orthogonal L olynomials

Woiis Osilenker Moscow State Civil Engineering University

World Scientific Singapore • NewJersey•London* Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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FOURIER SERIES IN ORTHOGONAL POLYNOMIALS Copyright © 1999 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-02-3787-1

This book is printed on acid-free paper.

Printed in Singapore by Uto-Print

Contents 1

Preliminaries 1.1 Notations 1.2 Some Topics from Function Theory of a Real Variable 1.3 Some Topics from Functional Analysis 1.4 Interpolation Theorems. Estimates of the Hardy-Littlewood maximal function

1 1 3 24 33

2

Orthogonal polynomials and their properties 2.1 Orthogonal systems in a Hilbert space 2.2 Elementary properties of the orthogonal polynomials 2.3 Jacobi polynomials 2.4 Some estimates of the orthogonal polynomials

49 49 59 80 105

3

Convergence and summability of Fourier series in L^ 3.1 Fourier series in an abstract Hilbert space 3.2 Fundamental Theorems on a Convergence of the Fourier series in O.N.S 3.3 Behaviour of the partial sums of Fourier series in orthogonal polynomials 3.4 (C, l)-summability almost everywhere

121 121 124 134 143

4

Fourier orthogonal series in L£ (1 < r < oo) and C 155 4.1 On a divergent Fourier series of the continuous functions 155 4.2 Estimates of the Lebesgue A-function 161 4.3 Strong summability of Fourier polynomial series in L£ (r > 1) . . . 179 4.4 Linear methods of summability of Fourier polynomial series in L^(r > 1) and C 184

5

Fourier polynomial series in L^. Analogs of Fatou T h e o r e m s 5.1 On an almost everywhere divergence of orthogonal expansions . . . 5.2 On linear methods of /f / fdfi. lim inf / fndp,> d\x. n ->°° JE JJE E T h e o r e m 2.3 ( B . Levi's T h e o r e m ) . If {fn} is a sequence of increasing functions L^-integrable on the set E such that

IJ>"

\ j fndix\O >(

((00 l ; l / r + l / r ' = l).

(2.10)

b) Minkowski's inequality OO

+v

*i

OO

-. i

OO

rr

d>* *r) l), Theorem 2.15. Let F e Z£ (r > 1), then llim i mi i j^f ++

r \F(t) -\F(t)-F(x)\ F(x)\r d»(t) =0 d»(t)=0

(2.15)

holds for almost every x. Proof. We divide the interval I over which / | F | r d/z is to be carried out into three subsets: let M\ be the set of the points x with /JL'(X) ^ 0, Mi the set at the points of which //(a?) = 0 and Ms the set of the points x for which //(#) does not exist. 1. By Lemma 2.14 and the Remark 1, (2.14) holds - in the sense of Lebesgue - almost everywhere for x G M\. In view of Theorem 2.5 lim lim

n(x HJx u(x ++ h) h) -— - fi(x) fi(x) u(x) ,, ; = = /a a/ (Ix) Ix) < 00 oo #) < h

holds almost everywhere on J D Mi, consequently (2.15) is satisfied almost everywhere on Mi (see remark 2). 2. F(x) is finite at almost all points and by Theorem 2.6 Hm \1 j f +

\F(t)\r d»(t) = J ; j f \F(t)\r d»{t) dp(t) = |F(z)|V(z) |F(*)|V(*) = 0

holds almost everywhere on Mi\ thus almost everywhere I

0oh,Jx r < 2 lim - / \F(t)\r dfi + 2r\F(x)\>*-*>Jx lim / dfi(t) r r x + h X) r r = 2 \F(x)\ lim rt l ^ = 2 \F{x)\ n'{x) = 0, = 2 r |F(x)| r lim M» + fe)-M») = 2r\F(x)\r»'(x) = 0, whence (2.15) is valid almost everywhere on M2, too. whence (2.15) is valid 2.5, almost M2,oftoo. 3. By Theorem theeverywhere set M3 is on a set measure zero in the sense of Lebesgue. 3. By Theorem 2.5, the set M3 is a set of measure zero in the sense of ^From 1,2 and 3 follows the assertion of Theorem 2.15. Lebesgue. ^From 1,2 and 3 follows the assertion of Theorem 2.15.

Chapter 1. Preliminaries

12 A p p r o x i m a t i o n by polynomials.

T h e o r e m 2.16 ( T h e o r e m of Weierstrass). A junction, continuous in a finite closed interval, can be approximated with a preassigned accuracy by polynomials. A function of a real variable which is continuous and has the period 2TT, can be approximated by trigonometric polynomials. For information concerning this theorem we refer to Zygmund, v.I, [1959]. Let u{5) be the modulus of continuity of a given function / ( x ) , continuous in the finite interval I = (a, 6): a;(/;S) := m a x | / ( x i ) - f(x 7, |xi | m -- xx22\\ 0, u(S) < C5 C6a, where C doesn't depend on < 6,S, then we shall say that the function / satisfies Lipschitz's condition of order a in the interval (a, b) and denote / G A a . Only the case 0 < a < 1 is interesting, since if / G A a , a > 1, then cj(*)/tf u(6)/6 tends to zero as 6 -> 0, and d'(x) existst and d\x) = = for all x, i.e. / = const. We shall say that the function satisfies Dini-Lipschitz condition and write / 6 DL(7), D L ( 7 ) , if

«(/,*)
0). (*-+0).

For an arbitrary a € (0,1], 7 > 1: DL(j) DL{-y) D A a . The following two theorems are due to Jackson D. T h e o r e m 2.17. If f(x) is a In-periodic continuous function with the continuity modulus CJ(/, S), then there exists a trigonometrical polynomial Tn(x) of order equal u(f, 5), at most to n with the aworoximation approximation nronertii property max \f(x)-T | / ( x ) -nT (x)\=0L r nn ( x ) | = o L ( // ;;; M M - J| . o<x = 2xn(2n2 + +1)1)J_LT"wX /( *+ +l{x+t) >*> {'^m') \~^m~) 2<xn(2n* {^tJW) 2^^TT)J_J

MX) Jn(X

f o r n = 1 , 2 . . . . Since /sinn(*/2)\2 /sinn(t/2)V S=i^ fc cc [( sin(i/2) sinft/2) j) == nn + 22]^L( (nn- -f c* )0ccoo ss ff cctt

n#fJ = " + 22>~ )

(2.16) (2-16)

is a cosine polynomial of order ( n - 1 ) , the kernel of the integral Jn(x) and therefore Jn(x) itself too is a trigonometrical polynomial of order (2n - 2). Using (2.16), we

1.2. Some Topics from Function Theory of a Real Variable

13

obtain

£ ( W ) 4 *-£["♦'£«-*]'* n-l

\2n22 + 4 ^ ( n - fc) k)22]l = = -ir[2n 2/3(n- - l)n(2n l)n(2n- - 1)]. 1)]. = 7T [2n 7r[2n2 2++2/3(n k=i fc=i

So

J_ r / s i n n ( t / 2 ) \ 4 2TT / _. W„ V sin(i/2) sin(*/2) J

_ n(2n 2 + 1) ~ " 3

and thus

l/{+ +f) f{x)] 4 dt 5$?)' * 'J« - '" * s s ^ n y f" -"'' ( ?(W£) IJ» - /wi * S ^ T T ) r

follows. Let us cover the interval [x-\t\,x+\t\] [x — \t\,x + \t\] by the minimal number N of congruent intervals of length < 1/n, having at most end points in common. Since the total length of the intervals is Nn'1 > 2\t\ and at the same time Nn'1 < 2\t\ + n _ 1 , N satisfies the inequality 2n|t| (f; 1/n) < (2n|t| + l)u,(/; 1/n). \f(x +1) - f(x)\ < J2 W Vkif) N«>(f\ Vn) fc=i

Thus we obtain the estimate .r / x

*/ M ./ x. ^^ M 6w(/;l/n) 6w(/;l/n) /"* / /ssinn(t/2)> si innnn( (t t/ /22) )\ \4 4 ^^ / ; i / n ) /"* 2 7r(2n + 1) , sin(ty2)

/ r/ r

, ,, ,x x

Let us divide the integral on the right-side into two parts, viz. into those extended over [0,7r/n] and [7r/n,7r]. Then it follows that

rr 7o

/sinn(t/2)\\ 44 __ ^4 4r7r" 7" /sinn(t/2 V sin(t/2) J ~ Jo

44

r d^ t 3

2 222

J*/n t

Hence | Jnn(x)-f{x)\(f;l/n) (x)-/(x)|