Absolute Summability Of Fourier Series And Orthogonal Series

t

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1067 Yasuo Okuyama

Absolute Summability of Fourier Series and Orthogonal Series

Springer-Verlag Berlin Heidelberg New York Tokyo 1984

Author Yasuo Okuyama Department of Mathematics, Faculty of Engineering Shinshu University Wakasato, Nagano 380, Japan

AMS Su bject Classification (1980): 42 A 28, 42 C 15 ISBN 3-540-13355-0 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-38?-13355-0 Springer-Verlag New York Heidelberg Berlin Tokyo Library of Congress Cataloging in Publication Data. Okuyama, Yaauo, 1937- Absolute summability of Fourier series and orthogonal series. (Lecture notes in mathematics; 1067) Bibliography: p. Includes index. 1. Fourier series. 2. Series, Orthogonal. 3. Summability theory. I. Title. II. Series: Lecture notes in mathematics (Springer-Verlag) ; 1067. CIA 3.L 28 no. 1067 [QA 404] 510 s [515'.2433] 84-10713 ISBN 0-38?-13355-0 (U.S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort', Munich. © by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210

Preface

The purpose of these lecture notes is to study the absolute bility of Fourier

series and o r t h o g o n a l

series.

The a b s o l u t e

summa-

summabil-

ity is a g e n e r a l i z a t i o n of the concept of the a b s o l u t e convergence just as the s u m m a b i l i t y is an e x t e n s i o n of the concept of the convergence. The absolute c o n v e r g e n c e the v a r i a t i o n and the m o d u l u s several classical criteria series.

of Fourier

series is closely related with

of c o n t i n u i t y of functions.

for

the absolute c o n v e r g e n c e

There are of F o u r i e r

We would like to show that these criteria can be s y s t e m a t i c a l l y

p r o v e d from the point of v i e w of the best a p p r o x i m a t i o n and that we can offer some applications. On the other hand,

we consider the a b s o l u t e summabilitiesIN,Pnl and

IR,Pn,I I for the n o n - a b s o l u t e series.

convergent F o u r i e r series and o r t h o g o n a l

Then we can extend the concepts of the absolute c o n v e r g e n c e of

F o u r i e r series and o r t h o g o n a l series by the absolute

summability Just

as we can do to the c o n v e r g e n c e of Fourier series by the summability. C o n s e q u e n t l y we can give several c r i t e r i a for the n o n - a b s o l u t e gent F o u r i e r series and orthogonal

series systematically.

The obJ'ect of Chapter I is to make clear the b a c k g r o u n d of absolute

convergence of

Fourier

conver-

series

for both the

the

trigonometrical

system and the W a l s h system with the aid of Stechkin's Theorem and an i n e q u a l i t y on the best a p p r o x i m a t i o n and give also the similar results for o r t h o g o n a l

systems

satisfying some conditions.

In Chapter 2 we deal with the absolute N~rlund summability almost everywhere

of F o u r i e r

to Lal's Theorem

series and,

from T h e o r e m 2.9 which is equivalent

[38,39], we deduce

several results by the same m e t h o d

as that used in Chapter i. It is known that an o r t h o g o n a l

series converges a b s o l u t e l y under

some rather weaker c o n d i t i o n of coefficients. gously showed that F o u r i e r series is summable

F.W. Wang [90,91] analoIC,~I under

some

coef-

ficient conditions. In Chapter 3 we deal with the a b s o l u t e N S r l u n d s ~ a b i l i t y e v e r y w h e r e of o r t h o g o n a l series under some coefficient Also,

u s i n g the

L. L e i n d l e r

conditions.

structure t h e o r e m for the t r i g o n o m e t i r c

[43], we give some sufficient

sufficient conditions

system due to

conditions for the absolute

NSrlund summability of Fourier series under some m o d u l u s of c o n t i n u i t y of a function.

almost

Furthermore,

conditions

on

the

we show that some

for the absolute NSrlund summability of orthog-

onal series are the best p o s s i b l e ones.

IV

Similarly,

in Chapter 4 we prove a t h e o r e m which is e q u i v a l e n t

the c o n d i t i o n on the a b s o l u t e Riesz due to

F. M$rciz

some sufficient be summable

s u m m a b i l i t y of o r t h o g o n a l

[56] and , a p p l y i n g the equivalent

c o n d i t i o n s for o r t h o g o n a l

to

series

theorem,

we give

series and F o u r i e r

series to

IR,Pn,I I almost everywhere.

In Chapters

5 and 6 we extend the r e s u l t s on the a b s o l u t e

s u m m a b i l i t y of F o u r i e r

series and

Varshney

[89],

A. K u m a r

[37], r e s p e c t i v e l y ,

M. !zumi

its c o n j u g a t e

and S. !zumi

[28, 29],

series

due

M. Mudiraj

NSrlund

to O. P.

[57]

and we also treat those results

and

systemat-

ically. It is well known that the a b s o l u t e of f(x) at a point

of f(x) in the whole i n t e r v a l summability

convergence

of F o u r i e r

is not a local p r o p e r t y but depends

IC, i I or

(0,2w).

series

on the b e h a v i o u r

Also we can e a s i l y show that the

IR, l o g n , i I of a Fourier

series is not a local

p r o p e r t y of the g e n e r a t i n g function. In Chapters

7

and

8

we e s t a b l i s h the general

theorems

on

the

local p r o p e r t y and the n o n - l o c a l p r o p e r t y of the a b s o l u t e Riesz summability and the absolute NSrlund point,

respectively,

s u m m a b i l i t y of Fourier

and we also treat

series

at a

several known and new results

systematically. I would like

to

express my hearty thanks to

Professor

Hiroshi

H i r o k a w a who gave me the chance to lecture at Chiba U n i v e r s i t y in the autumn of 1982.

Finally,

! would like to thank Mrs Azuma Hoshina for

typing the manuscript.

Nagano, March,

1983

Contents

Chapter

Chapter

Chapter

Chapter

Chapter

Chapter

i

Absolute

Convergence

I.I

Introduction

1.2

Comparison

1.3

Weight

of O r h t o g o n a l

Series

........... 1

........................................

Theorem

Absolute

..................................

Convergence

1 3

......................... 5

1.4

Convergence

2

A b s o l u t e N 6 r l u n d S u m m a b i l i t y Almost E v e r y w h e r e of F o u r i e r Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.1

Introduction

9

2.2

Analogous

of ~ICn IB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

........................................

Theorem

..................................

l0

2.3

Proof

...................................

12

2.4

A p p r o x i m a t i o n P r o p e r t y and A b s o l u t e N S r l u n d Summability ........................................

18

A b s o l u t e N ~ r l u n d S u m m a b i l i t y Almost E v e r y w h e r e of O r t h o g o n a l Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

3.1

Introduction

.......................................

24

3.2

A b s o l u t e N5rlund S u m m a b i l i t y of O r t h o g o n a l Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

3

of T h e o r e m

7

3.3

Approximation

3.4

Structure

3.5

Rademacher

and A b s o l u t e

Theorem

Summability

for T r i g o n o m e t r i c

Trigonometric

Series

............. 27

System

......... 30

.................... 32

3.6

Impossibility

4

A b s o l u t e Riesz S u m m a b i l i t y Almost E v e r y w h e r e of O r t h o g o n a l Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

4.1

Introduction

37

4.2

Equivalence

4.3

Sufficient

Conditions

4.4

Rademacher

Trigonometric

4.5

Impossibility

5

A b s o l u t e Norlund S u m m a b i l i t y Factors of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

5.1

Introduction

47

of I m p r o v e m e n t

of C o n d i t i o n s

......... 34

....................................... Theorem

................................ .............................. Series

of I m p r o v e m e n t

39

.................... 42

of Conditions

......... 44

....................................... Factors

38

5.2

Summability

5.3

Some A p p l i c a t i o n s

................................

48

..................................

52

5.4

Other

Conditions

...................................

54

5.5

Some

6

A b s o l u t e N 6 r l u n d S u m m a b i l i t y Factors of Conjugate Series of F o u r i e r Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Corollaries

...................................

59

6.1

Introduction

6.2

Conjugate

6.3

Some L e m m a s

6.4

Proofs

6.5

Equivalent

.......................................

Theorems

61

.................................

62

........................................

63

of T h e o r e m s Relation

.................................

66

................................

72

Vl

Chapter

Chapter

6.6

Some

7

Local Property Fourier Series

Applications

7.1

Introduction

7.2

Local

.................................

74

of A b s o l u t e R i e s z S u m m a b i l i t y of ....................................

77

......................................

Property

7.3

Some

7.4

Non-Local

7.5

Proofs

8

Local Property Fourier Series

....................................

Applications Property

of T h e o r e m s

8.1

Introduction

8.2

Local

8.3

Proofs

8.4

Izumi

8.5

Proofs

Notes

83

................................

85

......... .......................

86

...................................... ....................................

of Theorems Theorems

................................

...................................

of Izumi

Theorems

.........................

............................................

References Subject

.................................

o f A b s o l u t e N S r l u n d S u m m a b i l i t y of ....................................

Property

.......................................

Index

77 79

....................................

91 91 92 95 102 103

112 113 118

Chapter i

Absolute C o n v e r g e n c e of O r t h o g o n a l Series

I.I.

Introduction.

and integrable

Let

f(x)

be a periodic

function with period

(L) over (-w,w). We write f c L or

f e L(-w,w).

2w Then

the integrals an

~-

(x)

cos

nx

dx

(n bn exist,

= ~i ]~f(x)

=

0,1,2,...)

sin nx dx

which are called the

Fourier coefficients.

We construct the

series g a0 +

(a n cos nx + b n sin nx) = n=l

which may be convergent

or not.

~ A (x) n=O n

This ls called the Fourier series of

f(x) and we write f(x)~~

1

T

~ (a n cos nx + b n sin nx ) . n=l

a0 +

Let ~(6,f)

= ~(6)

=

sup

If(x+h)-f(x)i

and

=

sup ([2 O 0 we have ~(~) < C6 ~ , with C independent we shall say that

of

6 ,

f belongs to the class As; in symbols, f c A

Concerning absolute

convergence of F o u r i e r series, we know

the

following theorems. Theorem i.i.

(Bernstein

[96]).

If f(x) belongs to the class

A (a > 1/2), then its F o u r i e r series converges absolutely. Theorem 1.2.

(Bernstein [96]).

If the series

~ ~(I/n,f)/~ n--1 converges, then its Fourier series converges absolutely. Theorem 1.3.

(Zygmund [96]).

and is of bounded variation, ly.

If f(x) belongs to the class Aa(e > 0)

then its F o u r i e r series converges absolute-

Theorem 1.4. (Zygmund [94]).

If f(x) is of bounded variation and

m(6,f) = O{I/(log i/6) n} , as 6 + 0

for n >

2,

then its Fourier series converges absolutely. Theorem 1.5. (Salem [71]). the series

If f(x) is of bounded variation and

co

[ ~(i/n, f)i/2/n n=l converges, then its Fourier series converges absolutely. Theorem 1.6. (Sz~sz [79]).

If f(x) belongs to

LP(I < p ~ 2) and

the series n~l~p(i/n,f)/n /P converges, then its Fourier series converges absolutely, where 1/p+i/p'= i. We assume that a sequence # = {~n } forms a complete orthonormal system over a set of finite measure and ¢

is the linear space spanned n

by the first n elements of #. Let f(x) be a periodic function and integrable over (a,b).

We sup-

pose that the orthogonal series of f(x) is given by

f(x)~ where

an=

[ a ¢ (x)

n=l n n

f(X)¢n(X)dx. a

We put

llfllp = { ib If(x)IPdx} i/p a T h e n we d e f i n e

E(P)(f)n = inf {llf-Pllp : P ~ ~n }" It is well known that E(2)(f) = ( ~nlajl2)I/2. n

Stechkin

[77]

Theorem 1.7.

proved

j_

the

following

theorem.

Let f(x) ~n~iCnCn(X).

If the series converges.

~ n -I/2 E(2)(f) converges, then the series ~ ICnl n=l n n=l

Proof.

By Schwarz's

inequality,

k+llckl k=l ~

This theorem view point

a positive

constant

is an extension more

criteria

However,

of Bernstein

those

above

or less independently,

do not seem clear enough. of classical

be

One of the trials

was given by Mclaughlin

from the

classical

and their interto clarify

the

[51], but it still

some gaps to be filled up.

Our purpose

of this chapter theorem

and give also the similar conditions. Comparison

~0(x)

r0(x)

= r0(x+l) , rn(X)

2n(2)+

(A)

(B)

satisfying

are defined

some

by

= -i (1/2 ~ x < I)

= r0(2n, x)

(n = 1,2,...).

orthonormal

over a set of finite measure, n elements

to have the following property)

in different (de l a V a l l e e

properties

[21] and Paley

an (L I-) complete

operators

functions

For basic

to Fine

of Walsh functions,

[65].

system

and denote

of ¢.

n(i) are unique-

¢ = {~n } of bounded by @

The system under

the linear

n

space

consideration

We denote

by A a constant

we have which may be

contexts. P o u s s i n pr, o p e r t y )

Gn: L l ÷ ¢ 2 n

is

properties:

For I ~ p < q ~ ~, and P ~ Cn

< A n~IIPIIp, ~ = I/p-I/q. different

systems

= rn(1)(X)rn(2)(x)...rn(v)(x),

by n(i+l) < n(i).

(Nikolsky

on the best approximation

... + 2 n(v) ~ l, where the integers

is referred

by the first

postulated

r0(x)

of those

system with

are then given by

~ i, ~n(X)

Consider spanned

The Rademacher

= i (0 ~ x < 1/2),

ly determined the reader

and the Walsh

for orthogonal

r0(x)

for n = 2n(1)+

clear the background

system

and an inequality

results

Theorem.

The Walsh functions

functions

is to make

for both the t r i g o n o m e t r i c a l

the aid of Stechkin's

linear and p,

(f)
0, we have

f[-a]

=

k

[ k-(~CkqSk c L p

and

k--1

E(P)(f [-a])

< A n

n

-a~(p) mn (f)

=

Observe metric

•

that these properties

functions

Theorem

now reads,

1.8.

i ~ p < q < ~.

indicating

of trigono-

functions.

the conjugate

Let ¢ have the properties

exponent;

(A) and

(B) and let

Then we have n -I/q'E n(q)(f) =< A

n=l Proof.

are held by the system

as well as that of Walsh

Our theorem

By Cauchy's

co

n=0

~ CkCk ~ @n' write k--i

~ n-I/P'E(P)(f)n " n=l

condensation

principle,

what have to prove

is

co

2 n/q E(q)(f) 2

< A ~ 2 n/p E(P)(f) = n=0 2n "

This is reduced,

by property

(B)

to 2n/qll f - G n=0

n fll 2

Or, by the subadditivity

< A [ 2n/P IIf - G nfll q = n=0 2 P"

and the property

(A), we see that

IIf - a2nfllq ~ ~:n[l102k+lf- °2kfIIq co

co

Ak__[n2k°~I°2k+If-Gkfllp-- i,

8 < A [ = n=0

that

Rn = []f - G2nflIq"

by p r o p e r t y

(B).

q.e.d.

P

leading

to T h e o r e m

following.

1.7 a c t u a l l y

gives

the

oo

Theorem

For

i. ii.

0 < ~ < 2,

[ (n -I12 n=l

E(2)(f)) 8 < ~ n

implies

co

IenlS< n=l The indices

only change 2/ ~

and 2/(2-8)

Combining

Theorem

Corollary

1.4.

(n -I/p" The

in place

of the series

for example,

and C o r o l l a r y

1.4

inequqlity

we obtain

0 < B < 2.

Then

implies

with

inequality.

i.i0,

~ I c I~ < n= I n

~ n Y l e n IB may be t r e a t e d n=l

along the

for i < p < 2 and 0 < ~ < p',

(n6-1/P'E(P)(f)) ~ < ~ implies n

n= 1

is H S l d e r ' s

of C a u c h y - S c h w a r z

i ~ p ~ 2 and

E(P)(f)) ~ < ~ n

convergence

llne;

in the proof

I.ii with T h e o r e m

Let

n= I

same

needed

assures,

for

~ (n-I/P~E(P)(f[6]))~< n=l n

0 < 8 < 2, the c o n v e r g e n c e

of

[ (~ICnl)8. n=l

Thus we have Corollary

n= 1

1.5.

Let 1 ~ p ~ 2, 0 < ~ < 2

(n6-1/P~E(P)(f)) B < ~ n

and

implies

n=l

0 < B < p"

(n61Cnl) B
0 and B + e > 1/2. If f(x) belongs

series

of f(x)

is summable

IC,BI every-

[50] applying

the absolute

where. This theorem was extended NSrlund

summability,

to M. Izumi and S. Izumi Pn - Pn+I"

by McFadden

but the final result

In the same direction

[27] and S. N. Lal

[41].

We Write

is due

APn =

10 Theorem negative

2.2.

([27] and [41]).

and non-lncreaslng

sequences.

n__[iPn p n p-2 < ~ and

n[l

~(l/n)

=

hold,

Supplementing following

series

.

of f(x)

the result

theorem,

If the conditions

(i < p __< 2)


1/2. tween Theorem

Tsuchikura's

~ > 1/2, there exists a function

IC,~I ih (a,b) in (0,2~),

Thus this theorem

and applying

theorem.

of the Denjoy-

This theorem

is open for

seem that there is a difference

be-

2.4.

We shall require

the following

lemmas to prove

(see [38 and 39]). co

Lemma 2.1.

[38].

If

~ PP n -2 __< A, then n=l n PP n -I < A; n =

f I [50].

Proof. Now

(2.3.1)

h pp(t_l)d t < A; 0

(2.3.2)

i/n i n n -I/p 0 Wp(t)P(t-1)dt = 0[~p( / ) "].

For the proof of (2.3.1),

pP (t-l)dt

=

0

see Lemma

(5.44)

(2.3.3) of McFadden

pP(u)u -2 h- I

du

oo

= This establishes

h-l]

(2.3.2).

Again i/n ~0p(t) P(t-l) )dt ~0 co

1/2 + B -

~.

n

By Theorem 2.9, Theorem

2.10 is completed.

The case ~ = 0 of this theorem is an analogue Hys lop [24]. Theorem 2.11.

of the result

due to

Let B < ~.

If E(1)(f) = 0(I/n) and E(=)(f) = 0(n-~), n n is summable IC,6[ almost everywhere, where

then the series

~ nBAn(X) n=l > B- ~/2 and 6 > -1/2. Proof. By Lemma I.I', we have E(1)(f[ B]) = O(n -I+B) n

E(~)(f[B]) '

= O(n -~+B)

n

and f[~](x)

Since

~

~ nBAn(X). n=l

(E~)(f[~]))2 [3 = 0 (n=l n i+6-g+(~/2

-

o~/2.

2.11 is proved by Theorem 2.9.

Since f(x){ BV(0,2~) implies E(1)(f) = 0(i/n), the case ~ = 0 of n Theorem 2.11 is an analogue of results due to Chow [13]. Theorem 2.12. Let l < p < 2. If E(P)(f) = O(n -~) (~> 0) and ~ < ~, =

then the series 6 > I/p+~-~ Proof.

nBAn(X) n=l and 6 > -i/p'.

n

is summable

Ic,61 almost

everywhere,

where

By Lemma i.I', we have E(P)(f[B]) n

=

O(n-~+B)

and f[~](x)

~

~ nBAn(X). n=l

Thus we obtain

n=l

- o ,' [

nl/P'p n

l

\n=l nl- l/P+6 -13+a

) 1 / p

+B-

~.

21

Hence we establish Theorem Also, the case

2.12 by Theorem 2.9.

~ = 0 of this theorem

to Chow [13] and Mcfadden summability

of Fourier

is an analogue

[50] for the results

of results due

on the absolute Ces~ro

series of a function which belongs to the class

Lip (c~,p). We see from Lusin-Denjoy's ries i.i', 2.10,

Theorem

2.5 that the following

Corolla-

1.2" and 1.3" are the results which are deduced from Theorems

2.11 and 2.12, Corollary

respectively.

i.I'.

E(2)(f) n

co

n~-i/2(lanI+

I bn[)

= 0(n-~),

a > 0 and B< ~ imply

< ~.

n=l Corollary imply,

1.2".

E(1)(f) n

for (0 0, if ~ Pn = ~" Therefore we may suppose that I

E

n~N Pn n P = pnp~_l I~j N pn-j (P~

PnP lP i

Pn-j k Pn-j~ ~jajrj(t) I dt~ ~

~ ~

(3.5.2)

where N = N(E) is determined by the well known Khinchin inequality: n P J I ~NPn_j (pn E j

P n-j) x t)ik dt Pn-j jajrj ( (3.5.3)

n p => A{ ~ p2_j~ ( ~n- n j =N

Pn-j) Pn-j

2 ~2

jlajl2}k/2

33 From (3.5.2) and (3.5.3) we can conclude the convergence of the series (3.2.1), since repeating the similar argument as above, the integer N may be replaced by i. Lemma 3.3. Let 1 ~ k ~ 2 and {pn } be the same sequence as in Theorem 3.6. Put Aj(x) = pjcos(jx + ej). If the sezles n

Pn n=l

2

Pn

P

pnpL 1 {j

2

k/2

(3.5.4)

n-0

converges for every x in a set of positive measure, then the series { n

Pn

n=l

Pn 2 Pn-i( j =I ~ Pn

PnPk_l

P " 2 2 k/2 pn_~-J) pj} n-J

(3.5.5)

converges. Conversely, the convergence of (3.5.5) implies that of (3.5.4) for every x. Proof. We may suppose that the sum (3.5.4) is uniformly bounded by a constant A in a set E, m(E) > 0 and denote, for the simplicity k1 Pn Pn-___~j [ . ~n = PnPnlPn - -- ' 6n,j = Pn-j [ pn Pn-j Then we have I -- • ~ { n:l n E j

cos2(jx+ej)

dx < Am(E). =

(3.5.6)

Using the Minkowski inequality, we get j ] I >_ [ ~n{ n_[_ ([E6n, n=l j 1

pj[ cos(jx +ej)Idx) 2} k/2

n 2 [ k/2 -- n:l ~ ~n{j~16n'j p~ (]Elc°s(jx+ej)Idx)2}

(3.5.7)

By the Riemann-Lebesgue theorem, we have I Ic°s(jx+e )Idx > I c°s2(jx+eJ )dx E J = E I I (l+cos 2(jx+Sj)) dx (3.5.8) = I re(E) + 21- I cos 2(Jx+ %) dx 2 E _> ~ m(E) for sufficiently large J, say j ~ N.

34 Therefore,

by (3.5.7)

and

(3.5.8)

n I > [ ~n{ ~ 6~ P~ = n=l j=N 'J n 2 ~An~l~n{j~N6n,j

k/2 ( ~ m(E)) 2}

k/2 P~ }

By the same reason as in Theorem the convergence Theorem 3.6.

of (3.5.5).

3.7.

3.6, we replace

The converse

N by 1 and we conclude

is obvious.

Let i ~ k ~ 2 and let {pn } be the same as in Theorem

If the series

(3.5.5)

converges,

then almost

all series

of

J (anCOS nx + bnsin nx),

(3.5.9)

where An(X) = PnCOS(nX+0n ) = anCOS nx + bnsin nx, are summable IN,Pnl k for almost every x, and if the series (3.5.5) diverges, then almost all series

of the series

Proof.

(3.5.9)

Considering

easy consequence

of Theorem

is also a consequence

are non-summable IN,Pnl k for almost

the series [ rn(t)An(X) , the first part 3.3 putting

of Theorem

= Aj(x).

3.6 and Lemma

known Paley-Zygmund

argument

3.6.

of Improvement

Impossibility

ljaj

is an

The latter part

3.3 following

the well-

(cf.[96,p.214]). of Conditions.

We shall

show that

the positive

number s in L(~)(t) is indispensable in Corollaries P 3.2 and 3.3 for the case of trigonometric series. 3.1,

every x.

3.1,

Theorem 3.8. Let I ~ k < 2. In the assumptions of Corollaries ( 3.2 and 3.3 the positive number ~ in Lp s) or Ls+q ~(~) is indispensable. Proof.

Corollary Proof

We treat

the case

(v) of Corollary

3.3, because the other cases of Corollary

3.1(v).

of a Rademacher-trigonometric

3.1 and the case

can be shown

It is sufficient

s e r i e s [ anrn(t)cos

to show the existence nx which

is non-summable

IN,Pnlk for almost every (t,x) in (0,1)x(0,2w) and the series is convergent for E = 0. For this purpose we put a

Ls+l(n) I/k

= n

n L(0)(n) 1-17k~(°)

s

, ,l/k_

Ls+q+l~n)

, )i/k

.

Ls+q+2~n

Then we have [ I an 12 n Ls+l(

n -2/kL(0)(,2-2/k~(0) (n)2/k-I ) s n) Ls+q+ I

= [ n T(0) ~n ~IjLs+q+2(n)2/k ~s+q+l'

< ~

(v) of

similarly.

foI' 1 ~ k < 2.

(3.3.5)

35

On the other hand, since Pn = (n+2)-l~(0)(n+2)-l--~Ls, we see that Pn Z Ls+l(n) and Pn

Pn-j - 0(j L(0)(j)) s+l Pn-j

Pn

for j =

{ ~ 2 k=l Pk-llakl

X Pn n=l PnPn-I

of the series

proved

(4.2.1)

(4.1.4)

Let m0(n)

Then we have

2} 1/2

m0(n)-i

{ ~ m=O

>

Vm+ I k=v +i

=

n~= 1 P n P n - 1

m0(n)-i

{ I

m=O

is equivalent

o

2 2} 1/2 Pk_zlakl

m

Pn

theorem.

(4.1.4)

implies

by the same method

implication.

Vm+l

P~2 m

I k=v

lakl2} 1/2 +1

m

If

Thus we see that

the following

of the series

of the series Pn {~ 2 2} 1/2 n=l PnPn_l k=iPk-iIak I

That the convergence

(see

that

of

as that used be the integer

39

mo(n)-i Pn { ~ n=l PnPn-i m=O

> =

>

Vi+l

I

2 1/2 p2 Cm } Vm

Pn

m0(n)-i 2

Pc}

{[ j=l n=vj+l PnPn-i m=0

Vj+l

I

Pn

I

j-i

{I

2 1/2

vm m

p2 Cm 2}

1/2

= j=l n=vj+l PnPn-I m=O Vm Vi+l

Ic..P

--> j=l J-I vj_ I n=v.+l J

=

c

1 Pn-i

i ) Pn

( P(vj-1)P-~ ~)P(v'-')

j~ICj_I

j=

(

J

by virtue of the fact that P(Vj_l)/P(vj)-P(Vj_l)/P(vj+ I) ~ 2J-i/2 j- 2J-i/2 j+l = i/2-i/4 = 1/4.

4.3. Sufficient Conditions. Next, applying Theorems 4.2 and 4.3, we shall give some sufficient conditions for the absolute Riesz summability of orthogonal series and Fourier series. Corollary 4.1. Let p be a non-negative integer and s a positive integer. (i) If a>0 and the series

7

(

~lanl2Ls(n)aLs O)(n) -IL(~)~n~ p+s" "

(4.3 Z)

converges for some e > 0, then the series ~ anOn(X) is summable IR,exp Ls(n)a,l 1 almost everywhere. (ii) If ~>0 and the series Ilanl 2 Lp(e)(n)

(4.3.2)

converges for some c > 0, then the series ~anCn(X) is summable IR,n~,II almost everywhere. (iii)

If ~ >0 and the series (~) ~lanl 2n~ Lp (n)

(4.3.3)

40 converges for some e > 0, then the series [ anCn(X) is summable IR,exp n~,iI almost everywhere. (iv) If ~ > 0 and the series an

2L(0)(n)-iL(a)(n) s p+s"

(4.3.4)

converges for some ¢ > 0, then the series [ an#n(X) is summable IR,Ls(n)e, ll almost everywhere. (v) If ~ > 0 and the series [lanI2n Ls(n)

-~

(¢).

(4.3.5)

Lp+s(n)

converges for some ¢ > 0, then the series [ anOn(X) is summable IR,exp n/Ls(n)~,iI almost everywhere. Proof. As these results are analogously proved, we shall prove here only the case (i). By Theorems 3.2 and 3.3 and the Schwarz inequality, we have

Pn nil PnPn_l

:

nX I =

{ ~

2

2} 1/2 k=iPk-llak I

I

L(C)(n)p~ ~ 2} 1/2 p+s 2 p2 p2 Pk-llak I n n-1 k=l

n

{

nl/2L(a)(n]l/2 p+s" "

(4.3.6) 1/2

< { I =

1

~(¢)(n) n=l n bp+ s

oo < A { ~ [ak12

:

k=l

2

Pk-1

}

~

n L!i~(n)p~

{ [

~

[

n

n=k

X Pk-1 2

p2

n=l

n

p2 p2 n

n-i

2 1/2 lakl }

k=l

n-i

L(e) (n) 2 p+s Pn

n

1/2

}

On the other hand, if we put Pn = exp Ls(n)~ , we see that Pn ~ n-iLs (n) ~L~0) (n) -lexp Ls(n)~ . Therefore, we obtain " 2 ~o n L(E)( p+s n)Pn n=k p2 p2 n n-i

n L(S)(n)

< A [ p+s = n=k (exp Ls(n)~)4

{n

-i~ . , ~ ( 0 ) a}2 Ls 0,

t~'(t)/~2(t)

is equivalent

Let is a

{pn } be non-negative positive

is non-increasing, is non-increasing,

and non-increasing.

non-decreasing

t2~'(t)/~2(t) where

~(t),

function

Suppose

such that

is non-decreasing t > 0, is a positive

and bounded

If the conditions T

~ k=l

(k)~(k) ~(k)

< ~

(5 4.9) '

56

k= n

l(k)lJ(k) k Pk

_ 0 (l(n____~) ) , n = l , n

2 ....

(5.4.10)

and k(C/t)¢(t) ~ BV(0,w) hold,

for a constant

(5.4.11)

C(> ~)

then the series k(n)~(n)An+l(t) n=l

is summable

IN,Pnl , at t =x.

If we put in our theorem (~ ~ 0), then tk'(t)/k2(t) t2k'(t)/k2(t)

=at/(log

k(t)

=~/(log

= (log t) e and u(k) t) ~+I

= i/(log k) ~

is non-increasing

t) ~+I is non-decreasing.

Further,

and we can easily

see that k=2

k'(k)~(k) k(k)

~ = k=2 k(log k) I+~

< ~

and k=n Thus Theorem

k(k)~(k) k Pk

_

5.8 contains

Similarly,

Theorem

We need some lemmas

"

5.4.

if we put ~(t) = t ~ and Pn =

then Theorem 5.5 is deduced Lemma 5.2. If tk'(t)/k2(t)

[ i _ k n))_ (log n)~) k=n k Pk 0 (--n - 0 ( pn

from Theorem

F(n+~)/F(~)F(n+I)(0

~ a ~ i),

5.8.

for the proof of Theorem

5.8.

Let k(t), t > 0, be a positive non-decreasing function. is non-increasing and t2~'(t)/k2(t) is non-decreasing,

then we have 1 Proof.

cos kt dt I < A k'(k)

By an integration

J =

k'(C/t)/tk2(C/t)

increasing,

by parts,

C I ~ ~'(C/t) k 0 t2k--~C/t)

is non-decreasing

_ C [ w/k k'(C/t) - k ~0 tk2(C/t)

{ ~ }

sin kt dt

sin kt t dt

sin kt dt

sin kt dt.

and ~'(C/t)/t2k2(C/t)

we obtain C [w/k k'(C/t) I J I ~ ~ ~O t2q2(C/t)

C(> ~) .

we have

I~ cos kt Fsin kt ] w i lw d 0 ~ dt = L kk-~-~--/~)J 0 - ~ 0

= Since

for a constant

in non-

57

C kl'(~k)

[n/k sin kt dt

< A l'(k) k2(k)

=

Hence we complete the proof of Lemma 5.2. Proof of Theorem 5.8. Let t be the n th NSrlund mean of the n series .-[ZnlnAn+l(t)" Then we have n

tn - pnl k-~0Pn-kk(k)Z(k)Ak+l(x)

(5.4.12)

where Ak+l(X) = Y

0 ¢(t)cos(k+l)t

(5.4.13)

dt.

Hence we have by (5.4.12) and (5.4.13) n

t n - tn_l =

[ (Pn-k k=l Pn

Pn-k-l]l(k)~(k)Ak+l(X )

Pn-1 j (5.4.14)

= ~ o¢(t){

( Pn-k k=l

Now, we put g(t) = l(C/t)¢(t)

Pn

Ppn - k - 1 .J ) X ( k ) ~ ( k ) e o s ( k + l ) t } d t n-i

.

for 0 < t ~ n.

Then, by (5.4.13) and an integration by parts, we have Ak+l(X) = ~g(~)

0

cos(k+l)t 2 l(C/t) dt - ~

Putting T = [C/2t], we have by (5.4.14 ~ Itn-tn_ I' = 0, ~ + ~ < I,

and

(log C/t)~@(t) (BV(0,w), then the series An(t ) n=O {log(n+2)} I-B

is summable

I N,i/(n+2){log(n+2)}~l ,

at t = x . For ~ = 6 = 0, this corollary CorollarY

5.8.

is due to Varshney

I f a >0 and (log C/t)¢(t) c BV(0,w),

then the series An(t) n=0

is summable

IN,{log(n+2)}~/(n+2)l,

[87].

of

60

at t = x. Corollary

5.8 is due to Izumi and Izumi

not hold for ~ = 0 by Pati's theorem rollary

is due to Varshney

Corollary

5.9.

[29].

This corollary

[68], and the case ~ =

does

I in the co-

[89].

If (log log C/t)B@(t) ( B V ( 0 , w )

for 0 ~ B< i,

then the series An(t ) is summable n=0 log(n+2){log

IN,i/(n+2)log(n+2)l,

log(n+2)} 1-g

at t = x. Corollary

5.10.

and

If ~ > 0

(log log C/t)@(t) ( B V ( 0 , ~ ) , then the series An(t ) [ log(n+2) n=O

IN,{log log(n+2))~/(n+2)log(n+2)I,

is summable

at t = x . Corollary

5.11.

If ~_>_0 and t~(t)

~ BV(0,~),

then the series

~

n

n--I {log(n+l) } I+~

is summable

An(t)

IC,~I,

at t = x, where s>0. For Corollary for the case 8 = 0.

5.11,

the reader is also referred

As these corollaries Corollary

sre similarly proved,

to Matsumoto

[47]

we shall prove here only

5.7.

Proof of Corollary

5.7.

In Theorem

5.8, we put

Pk = i/(k+2){l°g(k+2)}~, l(t) = {log(t+2)} B and ~(k) = I/log(k+2). Then we have k Pk : [ I ~ {log(k+2)}l-~, ~=0 (~+2){log(~+2)} ~ = I' (k)D(k)

~ =

i

k= I (k+2) {log(k+2) } 2

and

X

~(k)~(k)

k=n

k Pk

= O(

1

1-~]

~(n) : o( -y--]

{log(n+2) }

Hence we see that all assumptions proof is complete.

of Theorem 5.8 hold.

.

n Therefore,

the

Chapter Absolute

NSrlund

of Conjugate 6.1.

Introduction.

6

Summability

Series

Factors

of Fourier

Let {pn] be a sequence

Series

of positive

constants.

We write {pn } E M: Pn+i/Pn

~ Pn+2/Pn+l

APn = Pn - Pn+l T

i.e.,

:

;

[nlt],

the greatest

integer

in ~/t.

Let f(x) be a periodic over

(-~,~).

series

We assume

of f(t)

function

without

with period

2~ and integrable

any loss of generality

(L)

that the Fourier

is given by

(anCOS nt + b sin nt) = ~iAn(t) n n

n= I and

~ i (n : 0,1,2,...);

(6.1.1)

f(t)dt = 0. The series ~conjugate ~ to (6.1.1)

n= I

-(bnCOS

is

nt - a sin nt) : ~iBn(t). n n=

We write ~x(t)

= @(t)

Concerning series

increasing function

½ {f(x+t)

the absolute

of a Fourier

Theorem

:

6.1.

series, Let

sequences.

such that

- f(x-t)].

N~rlund Kumar

{pn] and {APn} Let l(t),

{l(t)/t}

summability

[37] proved

factor

of the conjugate

the following

be both non-negative

t > 0, be a positive

is non-increasing

two theorems. and

non-

non-decreasing

for t > 0.

If the conditions

k:n

kl(k)pk = 0 [l~n___~)), n = l , 2 n

I

~X(C/t)Id,(t)l 0

and

I~ 0

X(C/t)l~(t)l t

hold for some constant

....

(6.1.3)


~ hold,

(6 1.7

then the series

Bn(t) [ log(n+i) n=l is summabie

IN,Pnl, at t = x.

Theorem

6.2 includes

as special

[70], which is the conjugate

case the theorem

analogue

of the result

of Ram and Lal

due to Varshney

[87]. 6.2.

Conjugate

Theorems.

6.1 and 6.2, we may expect

Comparing a result

Theorem for the

5.3 with both Theorems IN,Pnl

summability

of the

series

~ l(n)~(n)Bn(X ). n=l Our theorems are as follows:

Theorem

6.3.

increasing.

function and u(t), {n ~(n)}

Let {pn } and {Ap n} be both non-negative

Assume

that l(t),

t > 0, is a positive

t > 0, is a positive

and {u(n)/Pn}

bounded function

are n o n - d e c r e a s i n g

and non-

non-decreasing such that

and {l(t)u(t)/t}

is non-

increasing. If the conditions

k~ n

l(k)u(k)__ __ k Pk

IT 0

X(C/t)u(C/t) l~(t) I dt < ~ t

-

X(n)] -rn

- 0 (~j,

n

=

1,2,...,

(6.2

I)

(6.2 2)

and IT l(C/t)Id~(t)l 0 for some constant

C > ~ hold,

[ l(n)~(n)Bn(t) n=l is summable

IN,Pnl , at t = x.


0, is a p o s i t i v e

bounded

to T h e o r e m

then we see that

because

and n o n - i n c r e a s i n g

Theorem

reduces

function

{l(n)~(n)/n}

(6.1.3)

and ~(n)

Assume is a

is n o n - i n c r e a s i n g .

and

= 0

(6.2.3)

the series 1 (n)~(n)Bn(t) n=l

is summable

IN,Pnl,

Theorem which

was g i v e n

6.3.

Some

6.1.

{ ( P n - Pn-k )/k] Lemma

[76].

6.3.

6.4.

I~ 0 hold,

sequence {Apn}

and

and

the

and n o n - i n c r e a s i n g ,

are both n o n - n e g a t i v e

Let ~(t),

- Pn )/k)

and

{n~(n)]

function

Proof.

bounded such that

is n o n - d e c r e a s i n g .

constant C > ~

(6 2.2)

series

~/(n+l)

Since

I~(t)l E BV(0,~),

then

fixed

and

n

n=2 where

is non-

t > 0, be a p o s i t i v e

t(n)~(n) Ig(On) I converges

then

and non-

and n o n - i n c r e a s i n g ,

non-decreasing

l ( C / t ) ~ ( C / t ) l~(t) I dt < ~ for some t

then

of our theorems.

{Pn_k/Pn] ÷ I as n ÷ ~ for each

[92]).

[18],

5.2.

in k for I =< k =< n.

n ~ 2, {(Pn-k

t > 0, be a p o s i t i v e

is n o n - i n c r e a s i n g

of Dikshit

for the proofs

If {pn ) is n o n - n e g a t i v e

(cf.[37],

If ~(t) ~ BV(0,~)

of T h e o r e m

is n o n - n e g a t i v e

for i ~ k ~ n

and l(t),

{l(t)~(t)/t}

lemmas

If {pn } and

[29].

of the t h e o r e m

analogue

some

If {pn]

is n o n - d e c r e a s i n g

Lemma function

We need [76].

then

Lemma

conjugate

is a n o n - d e c r e a s i n g

6.2.

increasing, decreasing.

{Pn_k/Pn] k > 0.

as the

Lemmas.

Lemma

at t = x.

6.4 is a g e n e r a l i z a t i o n

~

8n< q/n, n = 2,3,...

the h y p o t h e s i s

~(t) ~ BV(0,~)

we can write l~(t)l = ~l(t)

-

~2(t)

implies

that

where @l(t) and92(t) are positive, bounded and non-decreasing, functions. Using the condition that {k(t)~(t)/t} is positive and non-increaslng, we have w/n i ~/(n-l) C

91(t) {I(C/t)z(C/t) } dt t2 i/t l(~n)H(~n) in

(6.3.1) C l(~n)~(~n) dt ~ n

t~/(n-l)~l(t) ]~/n ~

~i(-~ )

by virtue of the fact that ~i(t) is positive and non-decreasing. Similarly, we have

i

~/(n-l) * 2 ( t )

t 2 { k(C/t)H(C/t) i/t } it

z/n

I(C (n-l))~(~C (n-l)) < C =

C

(n-i)

j

~2(t)

~/(n-1)

-

~/n

-

t2

6.3.2)

dt

X( C (n-1))~(C (n-i))
0n, then we have Hn(t) = 0

for n __> • + I.

Therefore, since {k(k)} is non-decreasing, and Isin ktl 0, be a non-decreasing

to an infinity as t ÷ ~ condition

function.

If the

holds for a suitable constant y, then the set of the conditions

(6.2.2)

d d-Y ~ ( C / t )

and (6.1.3) (6.2.3).

and ~(t), t > 0, be a positive

function tending

~ V

is equivalent

l(C/t)~(C/t) t

to the set of the conditions

To prove this theorem,

we require the following

Lemma 6.7. Let l(t) be a non-decreasing positive function. If the condition dt

l(C/t) = y ~(C/t)~(C/t) t

(6.1.3)

and

two lemmas.

function and ~(t) be a

?3

holds for a suitable constant y, then the c o n d i t i o n l(C/t)~(t) c BV(0,~) is equivalent

to the c o n d i t i o n

(6.5.1) (6.1.3) w h e n e v e r the c o n d i t i o n

(6.2.2)

holds. Proof.

I

This lemma easily follows from the fact that

t(C/t)id@(t)[

=

Id{X(C/t)¢(t)}I

< =

0

I I~(t)l

+

0

dX(C/t)l

0

Id{X(C/t),(t)}l

X(C/t)~i(C/t l,(t)I

+ IYI

0

(6.5.2)

0

dt

< ~

t

and

I

Id{l(C/t)@(t)}[

~

I

(C/t)Id@(t)l +

0
w,

then the following theorem is known. Theorem in (0,w),

6.6.

(ii)

is a convex

[42].

sequence

X log(n+l)UnBn(t) By Lemma

If (i) ¢(t)(log

I~(t)I/t is integrable such that ~ n - I u n

is absolutely

is convergent,

harmonic

6.8, we see that Theorem

C/t) is of bounded variation

over the same interval and {~n } summable,

6.4 contains

then the series

at the point t = x. Theorem

6.6.

Chapter Local Property

of Absolute

of Fourier 7.1.

Introduction.

integrable

Riesz Summability

Series

Let f(t) be a periodic

(L) over

that the Fourier

7

(-~,~).

series of f(x)

S[f] = n=l

function with period

We assume without

27 and

any loss of generality

is given by

[ , say, (a n cos nt + b sin nt) ~ ~iAn(t) n n=

and

i

z

f(t)dt

= O.

We use the following Cx(t) @(t)

Dn(t)

notations:

1 = ¢(t) = ~ { f(x+t) = I t I ¢(u)Idu 0 sin(n+½)t 2 sin t/2

=

+ f(x-t)}

;

;

;

L0(t) = i, Ll(t) = log t, L p ( t ) = L l ( L p _ l ( t ) ) = log...log L~e)(t) where,

= Ll(t)...Lp_l(t)(Lp(t))l+e

Also we define a function

i(t)

that l(n) = In for n = 1,2,... Similarly

p(t)

numbers,

A denotes

a positive

sequence.

continuous

by the sequence

interval

(0,2~).

vergence

of [ An(t)/n

absolute

[i0] have

at a given point

but depends

Further

constant

such

{pn] and we put

that

is

convergence

on the behaviour

we can easily

log(n+l)

(0,~)

•

It is well known that the absolute is not a local property

We put Al n = I n - l n + l .

in the interval

and l(t) is linear for every non-integral

is defined

= I t p(u)du 0

P(t)

Kestelman

as positive

them by ins.

Let {In } be a monotone decreasing

t.

1,2,...),

(e ~ 0, p

if the right hand sides are not determined

we replace

t(p times)

not

the

same.

of f(x) in the whole

show that even the absolute

is not a local property.

shown that the summability

is not a local property

always

of S[f] at a point

Bosanquet

IC,I I of a Fourier

of the generating

con-

and series

function.

78 Mohanty series Mohanty

[53]

has

further

[ An(t)/log(n+l)

remarked

[53] have independently

of a Fourier

series

that the summabillty

is also not a local property. proved

at a given point

The main purpose

that the summability

of this chapter

exp n

1

is to treat

exp(log

n

I n~(log n)l+c

6

0 < ~ < i

0 < ~

X(i/t)

t log l~t

t log~i/t

More precisely,

Further-

JR,log n,l I of [ An(t)/log Iog(n+l)

Theorem 7.1. The IR,Pn,l I summability at t = x is a local property where

n

IR,logn,l I

of the function. the local property

the absolute Riesz summability ]R,Pn,I I of Fourier [46] proved the following two theorems.

P

[25] and

is not a local property.

more we can see that the summability is not a local property

]C,I I of the Izumi

n

series.

of the series

exp(log

i (log(n+l))6+c

of

Matsumoto [ I nAn(t) n=l

log n) 6

I (log log(n+l)) 6+c 0 i

(8.2.1)

Pn-j - Pn-j-i n=j +I

imply the condition

of Theorem

and

(8.2.2)

give

2j

Pn-i

n=j +i

n=2j +i

PJ - P0 ~ ___< pj + A ~ n=2j+l


0, =

(8.5.21)

we have, by (8.4.4), n-i V3 ~ n=l ~ iPnPn-i j=l ~ (pnPn -°-~~ I Pn_iPn_j )lJ+IIajl -- I lj+llajl ( Pn-j-i -Pn-j + Pn-j-i ( 1 j=± n=j+l Pn Pn-i ~jJa. J < A [ 0....< ~. = j :I Pj

(8.5.22) 1 )] Pn

We see from (8.3.4) that n=j +i

pnPn_j - PnPn_j PnPn-i