ON STRONG SUMMABILITY OF THE FOURIER SERIES VIA DEFERRED RIESZ MEAN Текст научной статьи по специальности «Математика»

DOI: 10.15393/j3.art.2024.15090

UDC 517.521

J. Sahqq, B. B. Jena, S. K. Paikray

ON STRONG SUMMABILITY OF THE FOURIER SERIES VIA DEFERRED RIESZ MEAN

Abstract. The strong summability technique has attracted a remarkably large number of researchers for better convergence analysis of infinite series as well as Fourier series in the study of summability theory. The of strong summability method was introduced by Fekete (Math. Es Termesz Ertesito, 34 (1916), 759-786). In this paper, we introduce the notions of strong deferred Cesaro, deferred Norlund, and deferred Riesz summability methods. We then consider our proposed strong deferred Riesz summability mean to establish and prove a new theorem for the summability of the Fourier series of an arbitrary periodic function. Moreover, for the effectiveness of our study, we present some concluding remarks demonstrating that some earlier published results are recovered from our main non-trivial Theorem.

Key words: strong summability, deferred Cesaro summability,

[DN,Pn \ 2]-summability, arbitrary periodic function,

Fourier series

2020 Mathematical Subject Classification: 30D40

1. Introduction. Let (an) and (bn) be sequences of non-negative integers with

and let un be an infinite series with the sequence of partial sums (sn).

We define the deferred Cesaro mean of order a of (sn) as

an < bn (n e N) and lim bn = 8,

DQ

ia

'n

E

2 Sbn-rEl

Г

ia—2

© Petrozavodsk State University, 2024

where

E\

a—1

bn

г pbn +1)

Г(Ъп + 1)Г(а)

(a > 0),

and it is denoted by (DC, a).

The series un is said to be (DC, a)- summable to l if

lim DC* = l.

Moreover, it is said to be strongly (DC, a)- summable with index q (or (DC, a, q)- summable (a > 0,q > 0)) to l if

2 \DC?—1 - l\q = o(bn) (n ^8).

r=a„ + 1

Suppose that (pn) is a sequence of non-negative numbers with

Pn = Yj Pr (P—i = p—i = 0) (Po ^ 0). (1)

r=a„+1

We now consider the deferred Norlund (DN,pn) mean of the sequence (sn) generated by the sequence of coefficients (pn) as

t

П

1

p

1 n

У1 Pbn—r sr

r=a„+1

(2)

The given series un is said to be strongly (DN,pn)- summable with

index q (or [DN,pn, q] (q > 0)-summable) to l if

-1 Vn

^ \Pr \-\tr - l\q = 0(1), (3)

\ n\ r-an+1

where

/

i r

1

~p

1 n

^ Sr VPbn—r

r"«„+1

1

~p

± n

У1 urPbn

r=a„+1

r

(VPbn—r " Pb„—r

Pb„—r—1).

Note that if we take

Pn = У Ц—rPr

r=a„+1

instead of pn, then the (^^,pra)-summability with index q can further be extended to [DN.p^.q] (a ^ 1,q ^ 1)-summability.

Also, note that if we take

Рьп-г — Ebn_r

in (2), then (tn) mean reduces to the Cesaro mean of order a of the sequence (sn).

Next, we define the deferred weighted (DN,pn) mean of the sequence (sn) generated by the sequence of coefficients (pn) as

T

n

1

p

П

^ I SrPr.

r=an+1

(4)

The given series XI un is said to be strongly (DN,pra)-summable with index q (or [DN,pn,o], (q > 0)-summable) to l if

Vn

тг \Pr\-\P - t\< — 0(1). (5)

I n\ r-an +1

where

1

p

n

^ Sr Vpr

r=a„ + 1

1 Vn

UrPr. (Vpr " Pr - Pr-l).

P* r"Z+1

Again if, we take

Pt

Et: 1pr

r=a„+1

in place of pn, then the (DN,pra)-summability with index q can also be extended to [DN,p% ,q] (a ^ 1,q ^ 1)-summability. Subsequently, if we take

Pb

n

a—1 ЬЬп

in (2), then (Tn) mean reduces to the Cesaro mean of order a of the sequence (sn).

2. Preliminaries and Known Results. Let sn(g) be the partial sum of the Fourier series of an arbitrary periodic function g of period 2L, such

that

П П

sn(g) = ao + ^ (ak cos -^-x + bk sin -?-x^ = ^ (ж)

fc-i

k" 0

where

1

= 2L

g(t)dt

L

>

L

ak = — Г g(t) cos 2^2dt (k e N) L j L

—L

and

1

L

knt

bk = — g(t) sin —— dt (k e N).

L

Again, under the Dirichlet kernel, the partial sum takes the form

sn(g) = \ g(\ + x)Vn(X)d\,

where

Vn(X) =

sin

pn+ 1/2)\ж L

sin

(Ш

(' 7 X = t — x)

is the Dirichlet kernel.

We use the following notations throughout the paper:

ф = фх(х) = 2 [g(x + x) + g(X — x) — 2sn]; = rpr/p^ = rpr/Pr;

— К sin

Ф(ж) = | \ф(и)\ди; Nn(x) = — Yi

pr-

{т+1/2)хж

L

r—a„ +1

sin (g)

a(x) = Pr cos(rx); fj(x) = ^ pr sin(rx);

r=a„ +1

r=a„ +1 (т+1/2)хж

r 1J sin \ (т+1/2)хж

h = V ^ (I ф(х) — ‘ W

1 „to рл i n ’ sin (g) ) •

h2 = V

m=0

Pm

~Pr

x

0

<5

1 '

Sin

- ф(х)-

1{r

(т+1/2)хж

L

sin (!)

dx;

h = V 7? (gj <«*)—

m-0 ^

(т,+ 1/2)от L

sin (S)

dx;

lo(t) = t + 1; h(t) = log (t + 1);

l2(t) = loglog(i + 1),... and so on, for t > 0;

and r = [1/t], the largest integer less than or equal to 1/t.

Based on a finding of Hardy and Littlewood [8] about the summa-bility of Fourier series, Fekete [7] investigated and introduced the notion of strong summability method. Subsequently, a few researchers have imposed the idea of strong summability techniques in their research works. Additionally, matrix summability (also known as matrix transformation) is crucial for understanding summability theory because it generalizes many summability techniques, including Cesaro summability, Norlund summability, Riesz summability, and so on. Moreover, the notion of statistical convergence via various summability means has recently attracted the wide-spread attention of many researchers due mainly to the fact that it is more powerful than the classical versions of the convergence. In this context, attention of the interested researchers are drawn towards the recent published works [1]- [6], [12]- [16], [20]- [24], [26], and [27].

Besides, in the year 1996 Mittal and Kumar [19] established certain results based on the strong Norlund summability means and, later on, Mittal (see details below, [18]) demonstrated a result on the strong Norlund summability of the Fourier series. Recently, Jena et al. (see details below, [10]) established a result on the strong Riesz summability of the Fourier series of a 2ж periodic function.

Theorem 1. [18] If

Ф(ж) = о

X

bred uia)

(x ^ 0),

(6)

8

then the series Uk(x) is [N,pn^, 2]-summable to the sum l, for x = t,

к" 0

provided that

Pn

j

m=0

1

m = 0,1, 2,...,

(7)

holds.

Theorem 2. [10] If

Ф(t) = о

(

t

VShul/h

)

(t ^ 0),

(8)

then the series

8

un(t) is strongly [IV,pn^, 2]-summable to the sum s (the

n=0

same sum), for t = x, provided that

Pn

к

^m(^)

m=0

1

к

0,1, 2,...,

(9)

holds.

From the literatures cited above, it is clear that a few works are carried out based on the strong summability of the Fourier series of 2ж periodic functions. However, no such result has been developed for the strong summability of the Fourier series of arbitrary periodic functions. Motivated essentially by the above mentioned investigations and developments, we have first introduced the notion of strong deferred Cesaro, deferred Norlund, and deferred Riesz summability methods. We then considered our proposed strong Riesz summability mean to establish and prove a new theorem for summability of the Fourier series of an arbitrary periodic function. Moreover, in the last section we present some concluding

remarks in which some earlier published results are recovered from our main non-trivial Theorem.

3. Axillary Lemmas. We need to prove the following lemmas for the proof of our main Theorem 3.

Lemma 1. Let (an) and (bn) be sequences of non-negative integers and if

Ф(ж) = o(x), x ^ 0,

(10)

then

Z1 = o(1) and Z3 = o(1) as ^ 0.

r

Proof. By nn-Lebesgue lemma and the regularity condition [10], Z3 = o(1) as > 0.

r

Since (an) and (bn) are sequences of non-negative integers and

, so, by (10), we have

DNr (x) = O(r) is uniform in

0,-

r

Zi

1 /r

О

$(x)\DNr (x)\dx

0

0(г)Ф(1/г) =

= 0(r)o(1/r) = o(1) as 1 ^ 0.

r

The proof is completed. □

Lemma 2. Let {pn} be a non-negative and non-increasing sequence, and let (an) and (bn) be sequences of non-negative integers. If condition (10) of Lemma 1 holds, then

s s

11

Z2 = ф(х)а(х)4х + Ф(х)Р(x) cot('Kx/2L)dx + o(1). (11)

Ъгг Ъгг

1/r 1/r

Proof. Since pr Z pr+1, фп = 0(1), we have

2

LPr

ф(х)

1/r

£ ' £

m=0 т=т+1

sin

Pm

{т+1/2)хж

L

sin (S)

dx =

1

+

s 1 г ф'х)

LPr srn (g )

1/r

S

1 ф'х)

LPr srn (g )

a(x) sin + p(x) cos(^£j

dx+

- (m + 1/2)хж

E i л»sin(—i—

m=r+ 1

dx =

— 22,1 + 22, 2. (12)

Thus, for proving the validity of (11), it suffices to show I2,2 — o(l) as r ^8.

For

m=r ' 1

and pm ^ pm+1, we have s

2 Pm Sin

T 1

(m + 1 / 2)хж L

)— °(7)

22, 2 —

l

ф(х)

LPr sin (g)

1/r

v (m + 1/2)хж

PmSin ^

m=r+1

dx —

(7 1

V 1/r

— 01 W1 dx

ж

Now, using (7) and (10) and applying integration by parts, we have

s

To 2 = О

1

LPr

(

Ф(х)р(1/х)

X2

1/r Ф(Х) dxx n.‘i=1 lm(l/x)

1/r

d

1

= О

LPr

p(1/x)

x

л 5

)C •!

1/r

Ф(х)

Х2П L1 lm(1/x)

)

О

1

Ф(ж) d

1

lp,. x dx nt=1 urn

'1/r

Furthermore, к being fixed, we can write

1

Pr — lk+1(r) — l(r).

Clearly, we have

2-2, 2 — 0(Jr) + 0(1) +

+ О

1

Ll(r)

/ 5

J

V1/r

-dx + - т------- dx —

ldx nLi ui/x)

ХП m=1 lm(1/x) dx km-1 lm(1/x)

— 0(1) + i УЩ 7

1{r

1/r X Пш-1 1ш(1/х) l(1/x)

-dx+

+ 0

l(r)

&

,„.1 lrn{1/x), 1/W

— 0(1) + о(УЩ/УЩ) + o(1/i(r )) — 0(1) as r ^8. (13)

1

1

1

This completes the proof of Lemma 2. □

4. Main Theorem. In this section, we wish to consider our strong deferred Riesz summability mean to prove a new theorem for summability of the Fourier series of an arbitrary periodic function, as follows:

Theorem 3.

if

Let (an) and (bn) be a sequences of non-negative integers;

Ф(ж) — о

(

x

v^bur/h

:)

(x ^ 0 ),

14)

8 D

then the series ^ un(x) is strongly [DN,рП’, 2]-summable to 1 for x — t,

n— 0

provided that

-1

(k e{0}u N) (15)

holds.

Pn —

m=0

5. Proof of the Main Theorem. Following Zygmund (see details, [28], p. 50), we have:

Tr (x) - l — 2 ‘Jr {*. M - l} —

m-0 Гг

, L

JL n i i sin

.£ S i i "

(т+1/2)рж

L

sin (Ш

)

dp =

1 /r s L

{! / О Jj

\ *н

0 1/r S

= X\ + X2 + X3 (say).

(m+1/2)p-w

L

sin (2P

dp =

In order to establish the theorem, it is enough to prove

n

^ |Tr(x) — l|2 = o(n) (n ^ 8).

r=0

Using the results of Lemma 1 and Lemma 2, equation (16) leads to |Tr (ж)— l| =

LP,

s s

ft(p)a(p)dp +— ft(p)ft(P)2cot(^£) dP + °(1)

LPr

1/r

1/r

<

1

ft(p)a(p)dp + ft{p)ft(p)2 cot (2x)dp ‘ °(1) (r - n)

1/n 1/n

Moreover, a(p) = 0(PT) = ft(p); so, using monotonicity of [/(1/p)]-1 have:

1

LPr

ft(p)a(p)dp

1/n

= о jt dP

Г 1/n '

&

°(^Щ) J |^p)|l(1/p)dp =

0{u(r))

(*mi/p)n,n —

1 / n

pl(1/p)

F(p)dp

=ofe)Kb

pH m=1 1ш(1/р) 6

+

пи. ui/p)) 1//j

(16)

(17)

(18)

, we

' 0 (

/(r)

(1)

P

= О - + O

)

dp

Vnlf-1. Ui/р)> »П"- Ul/p)

1

1

l(r){ Yit-1 1ш(г)} / 1/п

dp = o(1) as-------> 0

n

Similarly,

1

LPr

np

2L

1/n

= o(1) as-----> 0.

n

Now, using Minkowski’s inequality, we get:

r—0

<

D \Tr (x) - l\‘

21Tp( j + J (p)2cot (^L) dp\ I

Г—0 ^ r '1/n 1/n ' '

{lP( J Hv)a(v)dv + j 0(^)/3(u)2cot (2L^ ^ j

^ ' 1/n 1/n ' '

ff1/2

+ o(n)1/2 =

= [j1 + 32 + 33 + Ju]1/2 + 0(n)1/2. (19)

Thus, to complete the proof, we need to verify that j1,j2,j3, and j4 individually tend to 0 as n ^8.

Let

31 =

1

L2

6 6 ( n Л

J J ф(Р)а(Р)ф(^)о.(у)\ 2 p£ > dpdv

1/n 1/n Vr 0 J

As we have &

1ф(х)\1(1/фх = [ф(р)1(1/р)]1/п + ф(р)

РП m= 1 lm(1/P)

1/n

1/n

dp =

1

1

= о

(

p/(l/p)

) „..'Д Vn*;"1! «2 lm{1/p)^

l

= о

л/п^'-1, ШрУ 1/

PK1{P) + (__________

'in

dp =

Vn™'-1, «1 m •

dp =

-1 Im(1/P)TI т-1 1ш(1/р) 1%

= О

(1) G - 0

So, for a(p) = 0(PT) and using (6), we get:

<5 <5

•?1 - 2 J №(P)\1(1/P)dp ^ ^v^^dv = o(1) (Д — o).

1/ra

1/n

Next, as s

\ф(у)\а(у) 2cot (2^)

dv =

\ф(у)\а(п)

dv =

1/n

Ф(п)

l(v)

1/n

&

1/n

1/n

i / ЮМ \

dv v

dv =

Ф(ь)

l(v)

ф(у)1 (

1/n p V V^Om-1 lm(1/v)

= О

' ^(V)~

1/n

1(1/v)

)

dv =

(

+ о

(

ьш-ьььь

■)‘

1

+

/п

6

iti U1/v)) bnf-1 U1/e)

dv = o(1)--------> o')

n

1 / n

this implies

ь ь , n л

T2 \ J Ф(р)а(р)Ф(у)Ф(у)2с^ (2| 2 ~p2 dPdv

1/n 1/n ^Г-° Г '

d

о

1

n f ° ° \

- J2 2 Jv \Ф{р)\а{р)Лр №(v)\a(v) cot(^j dv =

= о

0

).

41/n 1/n

S

>(lq J \Ф(р)Нр№ j \ф(у)\а(у) co^(^£) dv = o(l) (

1/n 1/n

In the similar way, we can obtain:

j3 = o(l) and j4 = o(l) as--------> 0.

n

Hence, the proof of the theorem is completed.

6. Concluding Remarks and Discussion. In this final section of our investigation, we offer a number of additional remarks and observations regarding our findings that we have proved here.

Remark 1. Let (an) and (bn) be sequences of non-negative integers and

8

satisfy the condition (6) of Theorem 1; then the series un(x) is strongly

[DN,pn \ 2]-summable to 1, for x = t, provided that

^ -1

П" 0

Pn =

lm(ft)

m=0

(k p {0} y N)

(20)

holds.

Remark 2. If (an) = 0 and (bn) = n in Theorem 3, then, under the

8

conditions (14) and (15), ^ un(t) is strongly [TV,pn \ 2]-summable to 1.

П" 0

Remark 3. If the condition (14) of Theorem 3 and the condition (1.8)

8

of [17] are satisfied, then the series ^ un(x) is [DC, l, 2]-summable to the

П" 0

sum 1 at x = t.

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Received November 21, 2023.

In revised form, May 10 , 2024.

Accepted May 14, 2024.

Published online June 01, 2024.

Jyotiranjan Sahoo

Department of Mathematics, Veer Surendra Sai University of Technology

Burla 768018, Odisha, India

E-mail: jsahoo520@gmail.com

Bidu Bhusan Jena

Faculty of Science (Mathematics), Sri Sri University

Cuttack 754006, Odisha, India

E-mail: bidumath.05@gmail.com

Susanta Kumar Paikray

Department of Mathematics, Veer Surendra Sai University of Technology

Burla 768018, Odisha, India

E-mail: skpaikray_math@vssut.ac.in