On the absolute summability factors of Fourier series at a given point

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C OMPOSITIO M ATHEMATICA

F U C HENG H SIANG

On the absolute summability factors of Fourier series at a given point

Compositio Mathematica, tome 17 (1965-1966), p. 156-160

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156

Fu

Cheng Hsiang

1.

A

series 03A3an

^{is said}^to^be

absolutely

^summable

(A )

^or^sum-

mable

J A |

if the function

is of bounded variation in the interval

(0,1)

^Let

an

^{denote the}

n-th Cesàro mean of order oc of the series

03A3an, ^i.e.,

If the series

converges,

then,

^we_saythat the series

03A3an

^is

absolutely

^summable

(C, oc)

^or^summable

JC, ocl.

It is known that

[2]

^if^a^{series is}^sum-

mable

1 CI,

then it is also

summable |A|.

2.

Suppose

^now^that

f(x)

^is^a^function

integrable

^{in the}^sense^of

Lebesgue

^and

periodic

^with

period

^203C0. ^Let

Whittaker

[5] proved

that the series

is summable

lAI |

^almost

everywhere.

^Prasad

[5] improved

^this

result

by showing

that the series

03A3An(x)

^when

multiplied by

one of the factors:

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157

where a is _any

positive quantity

^and

logl n

⁼

log

^n,

loe n

⁼

log (logk-1 n),

^is^summable

|A|

^at^the

point

^x.

Let

(03BBn)

^be^a^convexand bounded sequence

[3],

^Chow

[1]

^has

demonstrated that the series

is

summable 1 C, 11

^almost

everwhere,

if the series

i ^n-103BBn

^conver-

ges.

In the

present

note, ^we^areinterested

particularly

ⁱⁿ^the^case^of

|C, Il summability.

^For^a^fixed

point of x,

^we^write

We are

going

^to^establish^at^{first the}

following

THEOREM 1.

If

as t - +0, ^thenthe series

is

summable 1 C, 1| f or

every ^e> 0.

3.

In

the proof

^of^the

^theorem,

^we

require

^the

following

^lemmas.

LEMMA

1[4].

^Let^ce> -1 and let

in

^{be the}n-th Cesàro ^mean

of

order ^oc

o f

the sequence

(nan),

^then

where

03C303B1n

is the n-th Cesàro mean

o f

^order^03B1

o f

^the^series

1

^{an .}

LEMMA 2. Write

then

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if nt ~ ^1.This proves ^thefirst

part

^ofthe lemma. And the second

part

^{of the}^lemma^is^evident.

4.

Now,

^{we are}ⁱⁿ^a

position

^toprove the theorem. We ^use

cos ntdt.

Let

03C4n(x)

be the n-th Cesàro ^meanof first order of the sequence

{nAn(x)(log n)-(1+03B5)},

^then

Abel’s transformation

gives

say.

By

^{Lemma 2,}^we^have

Tb.us,

^on

writing

say, ^{we see}that

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159

Now,

It follows that

I4

⁼

O{(log n)-(1+03B5)}.

^As

^before,

^we^write

say.

Then,

And

Now,

It follows that

By

^Lemma^1,^we^need

only

^to^prove

that 03A3|03C4n(x)|/n

^converges.

And ^{from the}above

analysis,

it concludes that

This

completes

^the

proof

of Theorem ^1.

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For the

conjugate

^series

we can derive an

analogous

theorem. Write

Then,

^we

get

^the

following

THEOREM 2.

If

as t ^-+0, then the series

is

summable |C, 1|

^at^x.

REFERENCES H. C. CHOW,

[1] ^{On the}summability factors of Fourier series, Journ. London Math. Soc.,

16 (1941), ^215-220.

M. FEKETE,

[2] On the absolute summability (A) of infinite series, ^Proc.Edinburgh ^Math.

Soc. (2), ³(1982), ^132-134.

S. IZUMI and T. KAWATA,

[3] ^NotesônFourier series III, âbsolutesummability, ^Proc.Imperial Academy ôf Japan, 14(1988), ^32-35.

E. KOGBETLIANTZ,

[4] Sur les séries absolument sommables par la methode des moyennes arith-

métiques, Bull. Sciences Math. (2), ⁴⁹(1925), ^284-256.

B. N. PRASAD,

[5] ^{On the}summability ^ofFourier series and the bounded variation of power

series, Proc. London Math. Soc., ¹⁴(1939), ^162-168.

T. PATI,

[6] On the absolute summability factors of Fourier series, Indian Journ. Math.,

1 (1958), ^41-54.

National Taiwan University (Oblatum 18-1-64) Taipei, Formosa, ^China

On the absolute summability factors of Fourier series at a given point

C OMPOSITIO M ATHEMATICA

F U C HENG H SIANG