On a sequence oî Fourier coefficients

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RO C ZN IK I POLSKIEGO TOW ARZYSTW A MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE X V (1971)

A N N A L ES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X V (1971)

G. K. D

w iv e d i

(Varanasi, India)

O n a sequence oî Fourier coefficients

I .

Let

£ a n

be a given infinite series with the sequence of partial sums (sn). Let {pn} be a sequence of constants, real or complex and let ns write

P n = Po + Pi + P2+--- + Pn { P - 1 = P - 1 = 0 ) . The sequence-to-sequence transformation,

1 П

( 1 - 1 ) К = P v 8 n - v ( P n ^ Q )

n u = 0

defines the sequence {tn} of the ISTorlund means of the sequence {sn}T generated by the sequence of constants {p nj . The series У,ап or the sequence

is said to be summable by Norland means [4], or summable (N, p n) to the sum s, if lim tn = s.

71 -+CG

The conditions of regularity of the method of summability (V , p n) defined by (1.1) are

lim - -

№ **-*00 P n**

**J > * l** = 0 ( \ p n\), k=0

(1.2) lim —^ = 0,

and (1.3) as n -> oo.

If p n is real and non-negative, (1.3) is automatically satisfied and then (1.2) is the necessary and sufficient condition for the regularity of the method of summability {N , p n) [1].

In the special case in which 1

P r

and, therefore

n -\-1

logw,

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as n -> oo, tn reduces to the familiar harmonic mean of {snj and if it is denoted by t'n, then the series ]?an or the sequence {$„,} is said to be summable by harmonic means or summable (H) to the sum s, if lim t'n — s [5].

n->

^OO

If the method of summability (Y, p n) is superimposed on the Cesâro means of order one, another method of summability (Y, p n)G11 is ob

tained.

2. Let f(x) be a periodic function with period 2 tt and integrable in the sense of Lebesgue over an interval ( — tt , n). Let the Fourier series of f(x) be

O O O O

(2.1) |« o + (апсошх+Ьп&тпх) = Jj?An(x),

n = l n= 0

and then the conjugate series of (2.1) is

oo oo

(2.2) J^(bnc,osnx—an&innx) == ^ B n(x).

n = 1 n = l

We write

y)(t) — /(a?-f t)— f(x — t) — L (where L is a constant), t

W(t) =

J \ i p ( u ) \ d u

, о

P m = P r> and Р (1Д)= Р Т, where r = [1 /t] is the integral part of Ijt.

3. In 1959, Yarshney [7] proved the following theorem:

T

h e o r e m

A. I f t

J \tp(u)\du = о о

t

log7

as t -> 0, then the sequence {nBn(x)} is summable ( Y , --- 1'(71 to the

value L/ tz . I l l ^ + 1 /

In this paper the particular sequence j—— is replaced by a more general sequence {pn}, and then we give a more general condition for the (Y , р п)Сх, summability of the sequence {nJBn(x}}. In what follows {p n}

is real non-negative and non-increasing sequence such that P n oo with n.

4. We establish the following theorem:

T

h e o r e m

. I f

(4.1)

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S e q u e n c e o f F o u r i e r c o e f f i c i e n t s

6S

as t -> 0, where X{t) and a(t) are functions of t such that X(t),a(t) and X(t) -t/a (t) increase monotonically with t, and

(4.2) X(n)Pn = 0 [ a ( P J ]

as n-> oo, then sequence {nBn(x)} is summable {N , p n)'Cx to the value L/it..

We require following Lemmas to prove our theorem:

L

e m m a

l([ 2 ])(i) I f {pn} is non-negative and non-increasing, then for 0 < a < 6 < oo, 0 < i < n- and any n

ь

k = a

where A is an absolute constant.

(Ü) ( 1 1 ) Р т < р т -*

L

em m a

2 . I f 0 < t < l{n , then

\Qn(t)\

1 vh / sin let cos kt

F K

2

j Pn~k

k — \

k t t = 0{n),

Proof.

I QJf)

П П

\Qn(t)\ = ° \ j r y ]i> n -k № t)\ = = 0 { n ) .

L

e m m a

. 3. I f 0 < t < тс, then

Proof. By Lemma 1 (i) and Abel’s transformation, we have sin kt cos kt '

1 fc=1 s

P n —k

{:

k t

sin kt t

01 P „ \ ^ J Pn~k Ы2

f c = l

P k

sin (n —k)t

(n— k)t и - ^{tPn A} П 1 =T+l ^j 71 — 1

V mi { n —k)t I

P k....t ~ ..., 7 -. 11 +

(n — k)t

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by Lemma 1 (ii) and since [7]

n I .

^ I sin Ш

I к — 7Г + 1 • 2

P ro o f of th e Theorem . If we denote the (c, 1) transform of the sequence {nBn{x)} by tn, we have after Mohanty and Nanda [3],

t n—Ljiz = i V rBr(x)— Lj7i n j

r — 1

ТГ TZ

r f 1 smnt 1 cosnt ) 1 r

/ V{t)\ ^ dnn/2 “ 2 ' +

ТГ ["sinw# coswil

nt2 t J I ^еЙ+о(1),

by Eiemann-Lebesgue Theorem.

On account of the regularity of the method of summability we have to show that under our assumptions

(4.4)

7t П

r wit) v i Г sm M cos fan

— r = e(1)-

Л ft' 7, _ ? *-

0 " fr=I ns 7i -> oo. We write

Qnit) ~ ,zPn n fc=1 L

sin M cos M

M2 t }

Therefore,

TZ

I — f vWQnWdt 1 In

= f y { t) Qn( t) dt+ f y>(t)Qn(t)dt+ f ip{t)Qn{t)dt

0 1 In <5

-^1 + ^2 +-^35 say, where 0 < ô < и.

Now by lemma 2 and hypothesis (4.1),

Vя r r” 1 \пХ(п)рпЛ

h = j v(t)Qn(t)dt = 0 [n f W(t)\dt\ = о I ( jp\~ I = 0(1)

о о l- J

as n -> oo, since 7i,pn < P n.

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Again by Lemma 3,

S e q u e n c e o f F o u r i e r c o e f f i c i e n t s

65

<5 <5

I 2 = f v( t ) Qn( . t ) dt = 0 ^ J

- <>Ш ' ™ т ) ¹ **] + » [ / i * " > £ * ] + » [ , / i**

— * 2 1 + * 2 2 + 1 2 3 , S a y .

Now

but

i *V

K f ,

= 0 ( 1 ), as n

<5

1 r P T

1™ ^~Fn / n t ) - f - n 1 In

t

k+1 /1 \

*/ \

W f

к

' '

1 In.

dt

] ~ ° { k ) + 0

1 Цп)рп P ri

\Pn a(-Pn)

1

ln

n —1 k+ 1

o W + i £ f у ( т ) р<«»йм’

fc=l к

Wtypjt

*(pk) ] = o{pk}, as fc -> o o ,

so

71— 1

I 22 = o ( l ) + ~ - V o { ^ } = o(l) n Ù ï

nd

n 1 I n n 1IÔ ' ' n k= 1 ' '

»-l

= o i D + o f F ^ p ^ i m ) = o(i) as w -> oo, from the hypothesis, so

(4.6) * 2 = 0 ( 1 ).

Since the method of snmmability is regular, we have

(4.7) Z3 = o(l)

as n -> ^oo, by Biemann-Lebesgue Theorem.

Hence collecting (4.5), (4.6) and (4.7) we get

* = o ( l ) which completes the proof.

R oczniki PTM — P r a c e M a tem a ty czn e XV 5

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I would like to express my warmest thanks to Dr. L.M. Tripathir for his kind help and generous encouragement. I am also thankful to the referee for his kind suggestions.

References [1] G.H. H a rd y , Divergent series, Oxford 1949.

[2] L. Me F a d d e n , Absolute Nôrlund summability, Duke Math J. 9 (1942), p. 168-207, [3] R. M o h a n ty and M. N a n d a , On the behaviour of Fourier coefficients, Proc,