Methods of Approximation Theory [Reprint 2012 ed.] 9783110195286, 9783110630022

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Methods of Approximation Theory

Methods of Approximation Theory A.I. Stepanets

ISBN: 90 6764 427 7 © Copyright 2005 by Koninklijke Brill NV, Leiden, The Netherlands. Koninklijke Brill NV incorporates the imprints Brill Academic Publishers, Martinus Nijhoff Publishers and VSP A C.I.P. record for this book is available from the Library of Congress All rights reserved. No part of this publication may be reproduced, translated, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission from the publisher. Authorization to photocopy items for internal or personal use is granted by Brill provided that the appropriate fees are paid directly to the Copyright Clearance Center, 222 Rosewood Drive, Suite 910, Danvers, MA 01923, USA. Fees are subject to change.

Contents

PREFACE

xi PART I

1. REGULARITY OF LINEAR METHODS OF SUMMATION OF FOURIER SERIES 1. Introduction 2. Nikol'skii and Nagy Theorems 3. Lebesgue Constants of Classical Linear Methods 4. Lower Bounds for Lebesgue Constants 5. Linear Methods Determined by Rectangular Matrices 6. Estimates for Integrals of Moduli of Functions Defined by Cosine and Sine Series 7. Asymptotic Equality for Integrals of Moduli of Functions Defined by Trigonometric Series. Telyakovskii Theorem 8. Corollaries of Theorem 7.1. Regularity of Linear Methods of Summation of Fourier Series

1 1 6 15 21 23 28 43 66

2. SATURATION OF LINEAR METHODS 1. Statement of the Problem 2. Sufficient Conditions for Saturation 3. Saturation Classes 4. Criterion for Uniform Boundedness of Multipliers 5. Saturation of Classical Linear Methods

79 79 81 84 90 98

3. CLASSES OF PERIODIC FUNCTIONS 1. Sets of Summable Functions. Moduli of Continuity 2. Classes Ηω[α,0] and Ηω 3. Moduli of Continuity in Spaces Lp. Classes Ηωρ

101 101 108 110

ν

vi

Contents

4. 5. 6. 7.

Classes of Differentiable Functions Conjugate Functions and Their Classes Weyl-Nagy Classes Classes L p f l

112 116 119 120

8.

Classes

126

9.

Classes L ^ f l

130

10. 11. 12. 13. 14. 15. 16.

Order Relation for (ψ, /3)-Derivatives ^-Integrals of Periodic Functions Sets aJio,9JIoo, and m c Set F Two Counterexamples Function ηα(ί) and Sets Defined by It Sets Β and 2Ji0

133 137 147 153 156 160 162

4. INTEGRAL REPRESENTATIONS OF DEVIATIONS OF POLYNOMIALS GENERATED BY LINEAR PROCESSES OF SUMMATION OF FOURIER SERIES 1. First Integral Representation 2. Second Integral Representation 3. Representation of Deviations of Fourier Sums on Sets C^ Μ and i ß

165 165 167 173

5. APPROXIMATION BY FOURIER SUMS IN SPACES C AND Li 187 1. Simplest Extremal Problems in Space C 189 2. Simplest Extremal Problems in Space L\ 198 3. Approximations of Functions of Small Smoothness by Fourier Sums 203 4. Auxiliary Statements 207 5. Proofs of Theorems 3.1-3.3' 225 6. Approximation by Fourier Sums on Classes Ηω 235 7. Approximation by Fourier Sums on Classes Ηω 239 8. Analogs of Theorems 3.1-3.3'in Integral Metric 243 9. Analogs of Theorems 6.1 and 7.1 in Integral Metric 252 10. Approximations of Functions of High Smoothness by Fourier Sums in Uniform Metric 253 11. Auxiliary Statements 259 12. Proofs of Theorems 10.1-10.3' 271

Contents 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

24.

vii

Analogs of Theorems 10.1-10.3'in Integral Metrie 278 Remarks on the Solution of Kolmogorov-Nikol'skii Problem 279 Approximation of ^-Integrals That Generate Entire Functions by Fourier Sums 284 Approximation of Poisson Integrals by Fourier Sums 294 Corollaries of Telyakovskii Theorem 303 Solution of Kolmogorov-Nikol'skii Problem for Poisson Integrals of Continuous Functions 310 Lebesgue Inequalities for Poisson Integrals 338 Approximation by Fourier Sums on Classes of Analytic Functions . . 345 Convergence Rate of Group of Deviations 363 Corollaries of Theorems 21.1 and 21.2. Orders of Best Approximations374 Analogs of Theorems 21.1 and 21.2 and Best Approximations in Integral Metric 378 Strong Summability of Fourier Series

383

BIBLIOGRAPHICAL NOTES (Part I)

393

REFERENCES (Part I)

399 PART Π

6. CONVERGENCE RATE OF FOURIER SERIES AND THE BEST APPROXIMATIONS IN THE SPACES Lp 0. Introduction 1. Approximations in the Space L2 2. Direct and Inverse Theorems in the Space L2 3. Extension to the Case of Complete Orthonormal Systems 4. Jackson Inequalities in the Space L2 5. Marcinkiewicz, Riesz, and Hardy-Littlewood Theorems 6. Imbedding Theorems for the Sets L^L P . . . _ 7. Approximations of Functions from the Sets U p L p by Fourier Sums 8. Best Approximations of Infinitely Differentiable Functions 9. Jackson Inequalities in the Spaces C and Lp 7. BEST APPROXIMATIONS IN THE SPACES C AND L 1. Chebyshev and de la Vallee Poussin Theorems 2. Polynomial of the Best Approximation in the Space L 3. General Facts on the Approximations of Classes of Convolutions . . .

429 429 432 437 439 444 448 452 455 466 481 489 490 492 495

viii 4. 5. 6. 7. 8. 9.

Contents Orders of the Best Approximations Exact Values of the Upper Bounds of Best Approximations Dzyadyk-Stechkin-Xiung Yungshen Theorem. Korneichuk Theorem Serdyuk Theorem Bernstein Inequalities for Polynomials Inverse Theorems

8. INTERPOLATION 1. Interpolation Trigonometric Polynomials 2. Lebesgue Constants and Nikol'skii Theorems 3. Approximation by Interpolation Polynomials in the Classes of Infinitely Differentiable Functions 4. Approximation by Interpolation Polynomials on the Classes of Analytic Functions 5. Summable Analog of the Favard Method 9. APPROXIMATIONS IN THE SPACES OF LOCALLY SUMMABLE FUNCTIONS 1. Spaces Lp 2. Order Relation for (φ, /3)-Derivatives 3. Approximating Functions 4. General Estimates 5. On the Functions φ(·) Specifying the Sets L^ 6. Estimates of the Quantities \\r%{t, /?)||i for c = σ - h and h > 0 7. Estimates of the Quantities ||f£(i,/3)||i for c = θσ, 0 < θ < 1, and φ G 2lc 8. Estimates of the Quantities | | / 3 ) | | i for c = 2σ - η(σ) and V G 2loo 9. Estimates of the Quantities (i,0)||i for c = θσ, 0 < θ < 1, and φ G 2t0 10. Estimates of the Quantities ||5σ)0(ί, /3)||i 11. Basic Results 12. Upper Bounds of the Deviations pa(J\ •) in the Classes CYJ'^ 13.

and Some Remarks on the Approximation of Functions of High Smoothness

505 510 522 525 539 545 553 553 557 560 572 586

597 598 601 607 615 624 626 632 634 635 636 639 648 666

Contents 14.

ix

Strong Means of Deviations of the Operators Fa(f-,x)

668

10. APPROXIMATION OF CAUCHY-TYPE INTEGRALS

679

1. 2. 3. 4. 5. 6. 7.

680 696 702 713 716 723 727

Definitions and Auxiliary Statements Sets of ^-Integrals Approximation of Functions from the Classes Landau Constants Asymptotic Equalities Lebesgue-Landau Inequalities Approximation of Cauchy-Type Integrals

T)+

11. APPROXIMATIONS IN THE SPACES Sp

741

1. 2. 3. 4. 5. 6. 7. 8. 9.

741 745 747 750 755 771 777 808

10. 11.

Spaces ^-Integrals and Characteristic Sequences Best Approximations and Widths of p-Ellipsoids Approximations of Individual Elements from the Sets ψΞφ Best re-Term Approximations Best re-Term Approximations (q > p) Proof of Lemma 6.1 Best Approximations by q-Ellipsoids in the Spaces Application of Obtained Results to Problems of Approximation of Periodic Functions of Many Variables Remarks Theorems of Jackson and Bernstein in the Spaces Sp

811 817 822

12. APPROXIMATIONS BY ZYGMUND AND DE LA VALLEE POUSSIN SUMS

847

1. 2. 3. 4. 5.

Fejer Sums: Survey of Known Results Riesz Sums: A Survey of Available Results Zygmund Sums: A Survey of Available Results Zygmund Sums on the Classes C g ^ De la Vallee Poussin Sums on the Classes and W^H^

848 860 863 866 871

6.

De la Vallee Poussin Sums on the Classes

877

and

BIBLIOGRAPHICAL NOTES (Part Π)

881

REFERENCES (Part Π)

885

Index

917

PREFACE

What is new in this monograph? First of all, it is worth noting methods that enable one to solve, within the framework of a common approach, traditional problems of approximation theory for large collections of functions, including the well-known Weyl-Nagy and Sobolev classes as particular cases as well as classes of functions defined by convolutions with arbitrary summable kernels. The results obtained with the use of these methods are also new. Systematic investigations in this direction were originated in the 1980s under the influence of works by B. Nagy, S. M. Nikol'skii, S. B. Stechkin, V. K. Dzyadyk, Ν. I. Akhiezer, N. P. Korneichuk, Α. V. Efimov, S. A. Telyakovskii, etc. In the same years, the concept of (φ, β)-derivative defined for a given function / by a given sequence of numbers ψ = ψ (k), k = 1 , 2 , . . . , and numbers β was formed. The ordinary rth derivative, r = 1 , 2 , . . . , of a periodic function is a particular case of the (φ, /3)-derivative for y(k) = k~r and β = r. Derivatives in the Weyl and Weyl-Nagy sense are also particular cases of (ψ, /?)-derivatives. Generally speaking, the sequences φ(k) in the definition of derivatives may be arbitrary. However, it turned out that, in many cases, it suffices to consider only sequences convex downwards (in the present monograph, the set of such sequences is denoted by 9Jt). This simplifies investigations without considerable loss of generality. It turns out that any summable (or continuous) 27r-periodic function / necessarily has the (φ, /3)-derivative, which remains summable (continuous), and, furthermore, φ £ 9Jt. Moreover, if the set of periodic functions for which (φ, /3)-derivatives exist for given φ and β is denoted by L | , then the following equality is true: U L,t = L, where L denotes the set of 27r-periodic functions integrable over the period. This equality means, in particular, that, using the notion of (φ, ;3)-derivatives, one can classify all functions from L. It was established that the common part of all sets LZ for φ £ 9JI consists solely of trigonometric polynomials. This xi

xii

Preface

implies that, under this classification, every function f Ε L falls only into its "own" set L'ß, and only polynomials remain indistinguishable (they are contained in each of these sets). Clearly, any classification of functions has the right to exist only if its efficiency is confirmed. Apparently, the main problem of approximation theory is to establish the properties of approximation characteristics of a function on the basis of the postulated properties of this function. In the case of approximation of 27r-periodic functions, it is customary to use the convergence rates of Fourier series, best approximations by trigonometric polynomials, approximations by polynomials generated by linear methods of summation of Fourier series, approximations by interpolation polynomials, etc., as such characteristics. Functions with the same a priori properties are combined into classes, so that the facts established for a given class are valid for every representative of it. In this case, there arises the possibility of formulating new problems for entire classes of functions. Among them, there are problems of the properties of upper bounds for a given class of deviations of Fourier sums, best approximations, approximations by linear methods, etc., problems of various widths of given classes, etc. Therefore, the success of any classification of functions depends, first of all, on whether the properties used as its basis are adequate to the objective chosen. In this respect, the classification considered can withstand even strong criticism because all approximation characteristics investigated are completely expressed (most often, in explicit form) in terms of parameters defining classes of functions. For example, the orders of upper bounds of the best uniform approximations by trigonometric polynomials of degree η — 1 on classes of functions for which (ψ, β)-derivatives are bounded are equal to ip(n). An analogous quantity in the Lebesgue spaces Lp has the same value for any ρ G [1, oo]. If a function / is the (^,/?)-derivative of a function F, then it is quite natural to call F the (ψ, /?)-integral of the function / . In this case, the set L^ is the collection of (ψ, β)-integrals of all functions / € L. Obviously, it makes no difference which concept (derivative or integral) is regarded as primary. For a long time, the concept of derivative has been primary in the definition of classes of functions. However, in recent years, a preference has been given to the concept of integral (indeed, the approximated objects are, as a rule, integrals). This concept dominates in the present monograph. As a rule, the main results are formulated for entire families of classes whose parameters are sequences running through a given set (e.g., the set 9JI or a certain part of it). Since they are quite general, their proof is not simple. All cumbersome

Preface

xiii

proofs are divided into several steps, the first of which, as a rule, is the derivation of integral representations for the quantities investigated, and the second step is devoted to the simplification of the obtained expressions according to the objective of the problem considered. Problems of approximation of 27r-periodic functions of one variable occupy the main place in the monograph. The most complete and final results are obtained in the case of approximation of these functions by their Fourier sums in the uniform metric and in the metric of the space L = L\. These results are presented in Chapter 5. Chapters 1-4 can be regarded as preparatory, though, undoubtedly, they are of independent interest. For, example, Chapter 1 is devoted to general problems of the theory of summation of Fourier series, in particular, problems of their regularity. In this chapter, we present the well-known Nagy and Nikol'skii statements that give sufficient regularity conditions and (apparently, for the first time in monographs) the known Telyakovskii theorem that enables one to determine the principal parts of the integrals of summable functions in terms of their Fourier coefficients in fairly general cases. General problems of the theory of summation of Fourier series (problems of saturation of linear methods) are considered in Chapter 2. An important place in the monograph is occupied by Chapter 3. One may begin reading this monograph with exactly this chapter, where problems of classification of periodic functions (from the notion of modulus of continuity to properties of ^-integrals) are systematically presented. Formulas for the integral representation of deviations of polynomials generated by linear processes of summation of Fourier series on sets of ^-integrals are deduced in Chapter 4. Parallel with traditional representations of these deviations in terms of convolutions with integrals over the period, we also present here representations with the use of convolutions with integrals taken over the entire real axis. Exactly these representations are most often used in what follows. In the derivation of these representations, an important role is played by functions that are integrable and continuous on the entire axis and orthogonal to every function from L. Approximations by Fourier sums and the best approximations in the spaces Lp for ρ > 1 are considered in Chapter 6. While, in the case of the spaces C and Li, asymptotic equalities for the upper bounds of deviations of Fourier sums are obtained in Chapter 5 for large collections of functions, in the spaces Lp, with rare exceptions, one has to be content with exact order relations. It is clear that one of these positive exceptions corresponds to the case ρ = 2. Time will show whether this is a consequence of inadequate analysis or of the disharmony

Preface

xiv

between the original data and the objectives formulated. Arguments in favor of the latter are given, in particular, by the constructions presented in Chapter 11, where it is proposed to consider functions (including functions of many variables) as elements of linear spaces with a norm defined in a special way. Chapter 7 is devoted to the determination of orders and exact values of the best polynomial approximations in the spaces C and L\ on the sets of ^-integrals. Orders are readily determined for large collections of functions. However, as for exact values, only a few new results are added to the well-known FavardAkhiezer-Krein, Nagy, Dzyadyk-Stechkin-Xiung Yungshen, and Korneichuk results, and, probably, the main achievements in this direction are yet to come. In Chapter 8, interpolation problems are considered. The Nikol'skii results on the approximation of differentiable functions by interpolation polynomials and new results for sets of i/j-integrals are presented here. It is shown that, in fact, approximations by interpolation polynomials are not inferior in quality to approximations by Fourier sums. The methods for the investigation of approximations of ^-integrals developed in Chapters 5-7 are fairly universal. In particular, these methods and new facts (which are actually at the level of definitions) enable one to investigate approximation characteristics of classes of functions locally integrable on the entire axis and defined by Cauchy-type integrals in Jordan domains with rectifiable boundary. The corresponding results are presented in Chapters 9 and 10. In Chapter 9, we investigate problems of approximation of locally integrable functions defined on the entire axis (not necessarily periodic) with the use of entire functions of the exponential type. Here, in fact, we construct a theory anas j^ous to that for the periodic case presented in Chapters 5-7: the notion of the (ψ, β)integral of a function / locally integrable on the numerical axis ( / € L) is introduced, sets analogous to the sets Vg are defined, and approximations of functions / 6 by entire functions of the exponential type are studied. Main attention is given to approximations by so-called Fourier operators (analogs of Fourier sums in the periodic case). Here, analogs of the spaces Lp are the sets Lp of functions / with the finite norm ||/||p defined as follows: α+2π _

P=

sup I [ \f(t)\pdt \ J

a€R1

\ VP j

, 1 < ρ < oo,

a

ess sup | / ( ΐ ) | , teH1

ρ = oo.

xv

Preface Therefore, the statements obtained for the sets

involve the corresponding

results for the sets The results presented in Chapter 11 are completely new (they were published in journals in 2001-2002), and, apparently, their analysis will be performed in the future. This material is a result of a search for new approaches to problems of the theory of approximation of functions of many variables and, in particular, functions periodic in each variable. There are many problems here, but the following may be regarded as the most important ones: the choice of approximating aggregates, the choice of classes of functions, and the choice of approximation characteristics. The extension of one-dimensional results to the multidimensional case immediately encounters the problem of the choice of approximating aggregates. In the one-dimensional case, the form of the simplest aggregate is determined by the natural order of a natural series, whereas in the multidimensional case, i.e., in the case where there is a set X [a Banach space of functions f(t) t e

Rm,

= f(ti,...,

tm),

of m variables], the choice of the simplest aggregates becomes prob-

lematic; it is well known that the first difficulties here begin with the problem of the choice of an analog of a partial sum for the multiple series

cfc> k = (ki,...,km), Σ kezm

where Zm

is the integer lattice in

(1)

Rm.

It seems quite natural to introduce "rectangular" sums and the corresponding approximating aggregates in the periodic case, namely trigonometric polynomials of the form Πι

nm

Σ •·• Σ h\— Tl\ km=

(2) 71m

However, partial sums of a multiple series can be introduced in many ways, e.g., as follows: Let {Goo n m However, this is obviously true only if (1.14) holds. Furthermore, according to the known Banach theorem, in order that a sequence of linear operators U n ( f ) that map a complete normed vector space F into its part possess the property lim 11/ — Un(f)\\f n—> oo

= 0,

it is necessary that ||C/ n || F = s u p { | | t / n ( / ) | | F : f e F , | | / | | F < 1} = 0(1),

η —> oo.

(1.18)

The space C is complete, and, by virtue of (1.4), it is obvious that ||C/ n (A)|| c =

sup \\Un(f-,x;A)\\c= ll/llc(f;x)\\c oo. Since η

n_jfc

Σ — i—n—k ι

i \(k + l)/2 ^ ( « - f' c )η — > \{n-k)/2 fc

+

if 0 < k < v, if

v

is called the Fejer kernel of order η - 1. Using equality (3.1), we get Fn(t)

= ( 2 n s i n i / 2 ) _ 1 ( s i n i / 2 + sin(3i/2) + . . . + sin((2n — l)t/2)) = (4n sin 2 1/2) —1 ((1 - cos t) + (cos t - cos 21) + . . . + (cos(n — l ) i — c o s n i ) ) = (1 - cosrai)/4nsin 2 i/2 =

sin 2 nt II 2 η sin

(3.13)

t/2

Therefore, Fn(t) > 0 for any t. Thus, for the Lebesgue constant of the Fejer method, we have π Fn

= l J

π

\Fn(t)\dt

= £ J Fn(t)dt

= 1.

(3.14)

18

Regularity of Linear Methods of Summation of Fourier Series

Chapter 1

3.3. According to equalities (1.5) and (1.8), the kernel of the de la Vallee Poussin method is the polynomial V£(t) = — — Δ

V

η-τη -~/ k=m

VkW =

+

( l - ~ ) cos kt, (3.15)

Σ

n—m ' \ k=m+1

η)

so that

V^+1(t) = Vn(t),

V0n = Fn(t).

(3.16)

It follows from (3.15), (3.12), and (3.13) that

V£(t) =

- (nFn(t) η—m

mFm(t))

sin 2 nt/ 2 — sin 2 mt / 2

cos mt — cos nt

(n — m)2 sin 2 i/2

4(n - m) sin 2 i/2

(3 17)

This yields the following estimate for the Lebesgue constants of the de la Vallee Poussin method: 7Γ

K

= l [\VZ(t)\dtcn, where c is an arbitrary positive number, then the quantities are uniformly bounded by the number 2/c. The values of can be computed for some combinations of the numbers η and m. In particular, it is easy to find the values V%n and V£ n - According to (3.17), we have T „«_ . , Vnn(t)

Since

=

cos nt — cos 2nt • 2i/o ' A 4n sin i/2

τ Vn

o„, . (*) =

[1, sign(cosi — cos2i) = < [-1,

cos nt — cos 3nt · 2./. /o ' g 8rasm i/2

ί€(0,2π/3), t € (2π/3,π),

and, for £ G (0, π), sign(cosi — cos3i) = sign cos t, the following Fourier expansions are true: „r . , ,.. 1 4^sin2/c7r/3 ο |sign(cos t — cos 2i)J = - - ( — > : cos kt, Λ ΤΓ ' J b 3 7Γ

k= 1

. ^ 4 ^ cos(2/c + l ) i S[signcosi] = - ^ ( - i ) * - ^ — ^ . fc=0

(3.19)

Section 3

Lebesgue Constants of Classical Linear Methods

19

Hence, S[signVn2n(i)]

1 2 γ/3 = - + — Vs cos nt - — cos 2nt + 3

π

...,

π

(3.19')

ο 4 4 S[signV„ (t)} = - cos nt - — cos 3 nt + ... . 7Γ 37Γ Taking into account these expansions and equality (3.15), we obtain 7Γ

V U

n

= 1 J

\Vnn(t)\dt

— 7Γ

n 1 J fl = I - + Σ cos _π V k—1

2n 1

kt

+

2

~ ( k \ \ Σ ( 1 - — J cos /ci 1 signV*n(t)dt U k=n+1 ^ ' /

1 2V3 = ö + —> 3 π

(3-20)

Vn n = - j ( l + i t c o B f c i + | £ _„. V k=1 fc=n+l 4 π

( ΐ - | Λ cos V

signed* / (3.21)

3.4. The kernel of the Rogosinski method, in view of (1.11), has the form 1 η—1 ^π 1 Rn(t) = - + 2 ^ c o s —cosAri = - ( Ί ) η ( ί + π / 2 η ) + Φ „ ( ί - π / 2 η ) ) , (3.22) fc=l or, with regard for (3.1), „ .. sin7r/2n cos nt Rn(t) = i 7 77Γ· 2 cosc — cos π/In Hence, signÄ„( - s i n — coskt. η ττ ^ k 2η fc=l

Therefore, taking into account equalities (3.19) and (3.19'), we get π

π

Rn = \Rn(t)\dt π J - π

/

η —1

\

= Rait) ( - + - V - sin TT J \ η TT k k 1 -π ^ ~

cos kt I dt 2n I '

1 . b - - + - y -sin—. Η IT k Η fc=ι

(3.24)

Now, using estimate (2.6), we obtain Rn


5

> 0'

(3 26)

·

and the kernel of the Zygmund method is the polynomial

=

- ( £ ) ' ) « * * * ,

s

>

0

·

Representations of these kernels in a form similar to (3.1), (3.13), (3.17), and (3.23) are unknown, except for the case where s = 1 in the Zygmund method. This complicates the direct determination of upper bounds for their Lebesgue constants. At the same time, the boundedness of these constants follows from Theorem 2.1. Indeed, setting

Section 4

Lower Bounds for Lebesgue Constants

21

we conclude that the system of numbers , k = 0,ra, A ^ = 0, satisfies conditions (2.1), (2.2), and (3.3) because, in this case, the expression

^

(1 - {k/nfY

(1 - {k/nf)5

_ ^

is the integral sum for the integral (i — u 2 y

1

,

}{i-(i-t)2)s dt

i^-S 0

0 1

1

0

0

Thus, the constants Rn of the Riesz method are uniformly bounded. By analogy, we establish the boundedness of the Lebesgue constants Ζ% of the Zygmund method. If = 1 - (k/n)s, k = Q~n, A ^ = 0, then conditions (2.1) and (2.2) are obvious. The expression n—1

ι -\K/U)

Σ n-k k=1

+ 1 ^ fc=l

1-yK/ri)

ι

l - ( vk - l ) / n n "

is the integral sum for the integral

f du J 1- u ο of a bounded function. Thus, condition (2.3) is satisfied, and, hence, the constants Z^ are uniformly bounded.

4. Lower Bounds for Lebesgue Constants 4.1. Inequality (2.6) means that, for an arbitrary trigonometric polynomial η Kn(t)

= — + Z

ak cos kt, k=l

22

Regularity of Linear Methods of Summation of Fourier Series

J

we have Έ f-'n-k fc=0

oo 7Γ J

—π

However, in the space C, every linear functional has the form π

^ J f(x +

t)dK(t),

(5.14)

where K(t) is a certain periodic function of bounded variation on the period. Therefore, taking into account the Helly theorem on the limit transition under the sign of Stieltjes integral, we get π

4 where Kn(t)

n

\ f ) = I J f(x + t)dKn(t) —π

= Un{f-x- A),

is a function of bounded variation on the period.

(5.15)

Section 5

Linear Methods Determined by Rectangular Matrices

27

Thus, the uniform convergence of series (5.1) yields relations (5.11), (5.13), and (5.15), which immediately implies the necessity of conditions (5.3) and (5.4). Further, setting π

Fn(f-x)

= Um(f-,x-,A) = ^ J f(x +

t)dKn(t)

—π in Theorem 5.2, we conclude that the condition π

(5.16)

\\Fm(f-x)\\ = ^-J\dKn(t)\oo κ

(5.17)

and, for a fixed η, Aj;n) = 0(l/lnÄ;),

k —> oo;

(5.18)

(ii) for every n, one can indicate a number πι (fixed or tending to infinity with n) such that Τ k=o k^m

m

1\m

— 1k\

K

< C,

(5.19)

28

Regularity of Linear Methods of Summation of Fourier Series

Chapter 1

where Δ2λoο n—> oo oo oo £ | A a f c | = V ( a ) < o o , Σ |A6fc| = V(b) < oo (b0 = 0), fc=0 fc=0 00

i/2

Σ I i=2 fc=l i=2

Λ Δα*-/-Λ £

(7.5)

(7.6)

1 = Β (a) < oo,

fc=l

oo ΣΜ/kKoo. (7.8) k=l Then, by virtue of Theorems 6.1, 6.5, and 6.6, series (7.4) converges for any χ € [—π,π], except, possibly, the point χ = 0. Its sum is continuous for all χ φ 0 and is integrable on the period. It follows from conditions (7.6) and (7.7) that, for any natural m, the following series are convergent: I ^ m+i-k Σ I i=2 k=1

— &am+i+k ι D / \ 1 1 = ^m(o),

a

Σι

=

k= 1 Thus, for any m G iV, the following quantities are finite:

(7 9)

i=2

m—2