On absolute matrix summability of a factored Fourier series

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXI (1979)

A. Ga n g u l y and B. K. Ma d a n (Sagar, India)

On absolute matrix summability of a factored Fourier series

1. Introduction. Let be an infinite series with sequence of partial sums {£„}, and let T = (an<k) be a triangular infinite matrix of real numbers (Hardy [2]). The series Z un is said to be absolutely summable T or summable

00

m , if Elf» — t n- iI < oo, where l

(id ) tn Z &п,к$к‘

k = 0

Let / be a function integrable (L) over < 0 ,2 n } and periodic with period 2k. We denote its Fourier series by

GO oo

/( Z

Z

n= 1 n= 0

The sequence {pn} is called the absolute matrix summability factor, or

\T\-summability factor of the Fourier series o f f at the point л: if Z /A.d„(x)

• - - __ . _ *.= П

is summable \T\, or simply Z P n (x)e T.

!f "=0

P n - k

(k ^ n), (k > n),

n

where P„ = Z Pv Ф 0, then tn defined by (1.1) is the same as (N , p n) mean

v = 0

generated by the sequence of coefficients {p„}. Similarly, if

^ n,k —

n — k + OL— 1 a — 1 n + Ct a 0

a > 0 for к ^ n,

for к > n,

tn mean is the same as (C, a) mean. We set

n

and An k = Yj a„'V with An<0 = 1 for every n ^ 0, where (an>k) are the ele-

v — k

ments of matrix T

2. Statement of result. In this paper we prove the following

Th e o r e m. I f

| f ( T W ( t ) l < CO,

then

Ï

n = 0

provided that

( 2 - 1 ) { a n,k }k = 0 a n d { a n - l , k ~ ~ a n , k + l } k = 0

are non-negative and non-decreasing sequences with respect to k,

( 2 - 2 ) { a n , k + l ~ a n ,k }k = 0

is a non-decreasing sequence with respect to k, and

(2.3) £ = 0 (f (*))

n=k+l n

for every non-negative integer k, where F (r) is a positive, non-decreasing function of t ^ 0 such that

(2.4)

00

I

Taking F{n) = 1, this reduces to the theorem of Nand Kishore and Hota [4] which is the generalization of the results obtained by Bosanquet [1] and T. Singh [5] for absolute Cesàro and absolute Norlund summa- bility, respectively. It may also be remarked that for F(n) = ri* (0 < a < 1), our theorem includes the result of Mohanty [3] for absolute Cesàro summa- bility as a special case.

3. Preliminary lemmas. We require the following eight lemmas (1):

L^{em m a} 1. Uniformly in 0 < t ^ n,

к

max I У F(n — v) sin (n — v)f| < — F(n),

o^k^n-i !vtb t

where F(n) is a function as defined in the theorem.

(*) К will denote a positive constant which is not necessarily the same at each occurrence.

P roof. We have, by Abel’s transformation,

к к- 1

I Yj F (n — v) sin (n — v) t\ ^ Y (Л F (n ~ v))--- 1--- F(n — k)

v = 0 v = 0 t t

from which the lemma follows.

Lemma 2 [4 ]. / / {a„,k}k=o is a non-negative and npn-decreasing sequence with respect to k, then for 0 ^ a < b ^ oo, 0 ^ t ^ k, and for every n,

b

I £ ei(n~k)t I ^ KA„(„_T, where r is the integral part of 1/t.

k = a

Lemma 3. I f (a„>k) is a sequence and F{n) is a function as defined in the theorem, then for 0 ^ a < b ^ с о , 0 ^ t ^ n, and for every n,

I Y F (n — v)a„'„-v ei("~v)t| ^ KF( n) An^ z.

v = 0

P roof. By Abel’s transformation and Lemma 2, we have

\ Y F { n - v ) a n „_v ei{n~v)t|

^

{

Л

v = 0 v = 0

^ KF( n) An^ t .

Lemma 4 [4 ]. I f {a„,k} 2=0 is non-negative and non-decreasing sequence with respect to k, such that An0 = 1, then, as n -> co, an k = О

for all к ^ n.

Lemma 5 [4 ]. I f {а„)к}2=0 and {a„>k+1 — a„)k}k=o w e non-negative and non-decreasing sequences with respect to k, such that A„t0 = 1, then (Л„_1>к — -v4„(k+i) is non-negative and is equal to 0( l / n) , as n -> oo uniformly for all к ^ n.

Lemma 6 [4 ]. If {a„,k}k=o and {a„_ 1>k —a„ik+1}2=o w e non-negative and non-decreasing sequences with respect to к such that An>0 = 1, then

is a non-negative and non-increasing sequence with respect to к for к < n.

Lemma 7 [4 ]. If {a„)k}k=0 is a non-negative and non-decreasing sequence with respect to k, such that {a„>k+1 — a„yk}k=b IS non-decreasing, then

is non-negative and non-increasing sequence with respect to к for к < n.

Lemma 8 [4 ]. //{ a „ ,k}k=o and {a„_ 1>k — ank+ i}£=o w e both non-negative and non-decreasing sequences with respect to k, then

is a non-negative and non-increasing sequence with respect to k.

Un,n — k Q-n, 0

n — k

■ ^ n , n —k+i A n - i , n - k + an>0 n — k

1 n — k + 1

4. Proof of the theorem. Using (1.1), we have, for our matrix,

П П

Zi @n,k ^k ~ Z* ^n,k Щ • Hence

fc = о k=0

t-n t-n— 1 Zj ( - ^n. k -Art—l , k ) ^ k Z (A„'k An — 1гк)Мк>

k—0 k = 1

as A„'о = A „ - ly0 = 1, where

2 :

uk = F (k)Ak(x) = — f 0 (t)F (k ) cos kt dt К 0

2 *J F(/c) sin ktd<P(t).

Then, formally, we have

°0 2 *

I | г „ - ( „ _ , к - | и > ( г ) | ( £ n = 1

kn о

" _ F(k) sin kt

l a 1л л,к ^ n - l . k l Г

fc = 1 K

Since by hypothesis j \d0{t)\ F(l /t ) < oo it suffices, for proving theorem, to show that

n = 1 к = 11

Let us write

Z (An,k~ An- l . k )

F (к) sin kt

= O F

n = 1 Z (An,k~ A n- l . k )

F (к) sin kt к — 1

on (0, 7t).

= 1 + I -

Since |sin/ct| ^ kt, we have

v « I t t |(A,-n - 4 . - ul)F(k)|

n = 1 к = 1

= t i

к = 1 л = к

^ t t F( k) Auk < t £ F(k) = 0 ( F (t».

к = 1 fc= 1

Further, let m = [и/2]. Then

v = Z I Z (An ,k - A „ - i,k ) ~ ^ - s i n kt\

n — X+ 1 fc = 1

oo m oo n

« I I I 1+ I l Z 1 = 1 (ИЧ-Л0.

say. Now,

W = ^£ ( A " ’k ^ n - l , k ) _ p ^ s i n k t

k= 1 ^

<

k = n — m

n-1 +

y 1 S A ”* - k An- ^ n-. . ^ F { n - k ) sin (in - k ) t

k=7-m n - k

V ' (Л"’в - fc ~ A"’n ~fc + 1 ~ F (n — k) sin (n ■- fc) t

» n - k

y (An,n - fc + 1 - Лп - 1, n - *+ 3»;0)_ (и _ fc) sin (n - fe) t

^ n — k

k = n — m

= П + П ,

where

-^n.n — к -^n.n — k+1 ^n,0 n — k

k = n — m

I

F {n — k) sin (n — k)t n -l

У > , n - f c S .?- i r ( n - f c ) s i n ( n - f c ) t

n — k

k = n — m

^ max I y JT (w _ v) sin (w _ v) Г I ,

Ш m-Kfc^n-1 v=0

= 0 ( T ) F ( n ) ^n,m ^n,0 m

= 0 ( i) F(n) n(n — 1)

by Lemma 7, by Lemma 1,

by Lemma 4,

and

= "y /4"’"~fc + 1 F (n -/c) sin (n-fc)t

k = n — m n — k

g Л"-,|’+1 Л -1.т + а.,о_ ^ ax _ I £ F ( n - v ) sin ( n - v ) ( I , by Lemma 6,

m m - l < f c ^ n - l 1v = 0

0 (t)F (n ) ^n,m+l An- i <m-\~ ü,n Q m

by Lemma 1, F jn)

n(n— 1)

= 0(1) by Lemma 5.

Hence

00 0 0 CO 00 F (n)

Z I Ï-.+ z y2-0( t) Z -^Tn

n = т + 1 n = t + 1 л = т + 1 n - z + 1 r l V 1 *■/

= 0 (F (i)) by hypothesis (2.4).

Next,

X = f ^ ^ " .-‘.■^-F (fc)s in fa k = m+ 1

£ f (k) sin kt

fc = m + 1

+

+ £ H.-Ot A..t+‘) F (kjsinkt

where Z i

k = m+ 1

= Z i + z 2,

£ "*'a + l) F (к) sink*

k = m+ 1

£ ^ F (к) sin kt

k = m+ 1 n — m — 1

£ an'n-~ - F (n — k) sin (n — k)t k=o n — k

n — m — 2

« z

^ n — m — 1

+ ---

г

= о № л и,п_

V

and, by Lemma 8, we observe that Z 2 = £ А" -'-к- - ^ кУ- F (k) sin kt

k = m+ 1 к

< An-1,

m + 1 ~n,m + 2 max | £ F (к) sin kt | m+l<A^n k = m + 1

n — m — 1

m + 1

>4 n - m - i

--- ^ ± ^ _ . щах У F (n — v) sin (n — v)t — m + 1 - ^ - - 1 ^m + 1 ^ A $ n v = 0

n — А — 1

—

v) sin

v)

1

v = o

^ 2 ■An— l , m + l A„m + 2

= 0 ( z ) F ( n ) m + 1

— 1 ,m + 1 ^ n . m + 2

max I Y F(n — v) sin (n — v)t I 0$k^n-m-l у = о

m + 1

= 0 (t)

F(n)

«(« — 1) / ’

, by Lemma 1,

by Lemma 5.

Therefore,

£ к I Z , + X z 2

n = z + 1 n = T + l я = г + 1

GO

o ( i) E

n — r + 1

F(n) . ---Л,

n ^{_ t}+ 0 (t) F(n)

n(n — 1)

= 0 ( F (t)) by hypothesis (2.3) and (2.4).

Collecting these, we have V = 0 ( F (t)).

This completes the proof of our theorem.

We are thankful to Dr. R. K. Jain for his valuable guidance and kind help during the preparation of this paper and also to the referee for his valuable suggestions.

References

[1] L. S. B o s a n q u e t, Note on the absolute summability (C) o f a Fourier series, J. London Math. Soc. 2 (1936), p. 11-15.

[2] G. H. H ard y , Divergent series, Clarendon Press, Oxford 1949.

[3] R. M o h a n ty , The absolute Cesàro summability of some series associated with a Fourier series, J. London Math. Soc. 25 (1950), p. 63-67.

[4] N and K is h o re and G. C. H o ta , On the absolute matrix summability of a Fourier series, Indian J. Math. 13, 2 (May 1971), p. 99-110.

[5] T. S ingh, Absolute NOrlund summability of a Fourier series, ibidem 6 (1964), p. 129-136.

DEPARTMENT OF MATHEMATICS UNIVERSITY OF SAGAR SAGAR, M. P., INDIA