International Journal of Analysis and Applications

Volume 16, Number 2 (2018), 209-221

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-16-2018-209

ON GENERALIZED LOCAL PROPERTY OF |A; δ|k-SUMMABILITY OF FACTORED
FOURIER SERIES

B. B. JENA1, VANDANA2,∗, S. K. PAIKRAY1 AND U. K. MISRA3

1Department of Mathematics, Veer Surendra Sai University of Technology, Burla 768018, Odisha, India

2Department of Management Studies, Indian Institute of Technology, Madras, Tamil Nadu-600 036, India

3Mathematics, National Institute of Science and Technology, Pallur Hills, Golanthara 761008, Odisha,

India

∗Corresponding author: vdrai1988@gmail.com

Abstract. The convergence of Fourier series of a function at a point depends upon the behaviour of the

function in the neighborhood of that point and it leads to the local property of Fourier series. In the proposed

paper a new result on local property of |A; δ|k-summability of factored Fourier series has been established

that generalizes a theorem of Sarigöl [13] (see [M. A. Sariögol, On local property of |A|k-summability of

factored Fourier series, J. Math. Anal. Appl. 188 (1994), 118-127]) on local property of |A|k-summability

of factored Fourier series.

1. Introduction and Motivation

Suppose
∑
an be a given infinite series with sequence of partial sum (sn) and let A = (anv) be a lower

triangular matrix with nonzero diagonal entries. Then A defines the sequence-to-sequence transformation

Received 2017-09-21; accepted 2017-12-07; published 2018-03-07.

2010 Mathematics Subject Classification. 40F05; 40D25.

Key words and phrases. Fourier series; lower triangular matrix; |A; δ|k-summability; local property.
c©2018 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

209

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-16-2018-209


Int. J. Anal. Appl. 16 (2) (2018) 210

from the sequence s = (sn) to A(s) = (An(s)), with

An(s) =
n∑
v=0

anvsv. (1.1)

A series
∑
an is summable |A|k (k ≥ 1) if, (see [13])

∞∑
n=1

|ann|1−k|An(s) −An−1(s)|k < ∞, (1.2)

and the series
∑
an is summable |A; δ|k (k ≥ 1) if, (see [6])

∞∑
n=1

|ann|1−k−δk|An(s) −An−1(s)|k < ∞. (1.3)

Let us consider two lower triangular matrices Ā and  associated with A as follows:

ānv =

n∑
r=v

anr, (n,v = 0, 1, 2, ..., )

and

ânv = ānv − ān−1,v. (n = 1, 2, 3, ..., ).

In special case, when A = (N̄ ,pn) then |A,δ|k-summability reduces to |N̄ ,pn; δ|k-summability and for

k = 1, (|N̄ ,pn; δ|) is equivalent to |R,pn; δ|-summability (see [2]). Also, if we take A = (C,α) with (α > −1),

then |A,δ|k-summability becomes |C,α, (α− 1)(1 − 1/k)δ|k in Flett’s notation. Furthermore, for double ab-

solute factorable summability matrix (see [11]).

We use the notations

∆cn = cn − cn+1 and ∆̄cn,v = cnv − cn−1,v, c−1,0 = 0, (n,v = 0, 1, 2, ..., ).

A sequence (λn) is called a convex sequence if,

∆2(λn) ≥ 0 for every n ∈ Z+,

where



Int. J. Anal. Appl. 16 (2) (2018) 211

∆2(λn) = ∆(λn) − ∆(λn+1) and ∆(λn) = λn −λn+1.

Let f(t) ∈ L(−π,π) be a 2π periodic function. Without loss of generality let us consider that a0 = 0

in the Fourier series expansion of f(t) that is,

∫ π
−π

f(t)dt = 0. (1.4)

Thus the Fourier series expansion of f(t) becomes:

f(t) =

∞∑
n=1

(an cos nt + bn sin nt) =

∞∑
n=1

An(t). (1.5)

It is well known that the convergence of the Fourier series at t = x is a local property of f [16] (i.e., it

depends only on the behavior of f in an arbitrarily small neighborhood of x) and hence the summability of

the Fourier series at t = x by any regular linear summability method is also a local property of f. Moreover,

as regards to the approximation of Fourier series of functions see the recent results [9], [10] and [5].

2. Preliminaries

Dealing with Riesz summability and local property of Fourier series, Mohanty [12] has established that

|R, log(n), 1|-summability of a factored Fourier series

∑ An
log(n + 1)

(2.1)

of a function f(t) at any point t = x is a local property of the generating function of f(t) but the summability

|C, 1| of this series is not. Subsequently, replacing the series (2.1) by

∑ An(t)
(log log(n + 1))δ

(δ > 1). (2.2)

Matsumoto [7] as obtained a new result on local property of |R,pn, 1|-summability.

Generalizing the above result Bhatt [1] proved the following theorem:

Theorem 2.1. Suppose (λn) is a convex sequence such that
∑

λn
n

is convergent, then the |R, log(n), 1|-

summability of a factored Fourier series
∑
An(t)λn log(n) at any point t = x is a local property of f(t).



Int. J. Anal. Appl. 16 (2) (2018) 212

By replacing the factor λnlog(n) in a most general form, Mishra [8] has proved the following theorem.

Theorem 2.2. Suppose (pn) be a sequence satisfying following conditions:

Pn = O(npn),

Pn∆pn = O(pnpn+1).

Then the |N̄ ,pn|-summability of a factored Fourier series

∞∑
n=1

An(t)λnPn(npn)−1 (2.3)

at any point t = x is a local property of f(t), where (λn) is a convex sequence.

Replacing |N̄ ,pn|-summability in Mishra’s result, Bor [3] proved a more general form on |N̄ ,pn|k-

summability method. Quite recently, Bor [4] introduced the following result on |N̄ ,pn|k-summability of

a factored Fourier series at any point t = x as a local property of f(t) under more appropriate conditions

then those given in the theorem.

Theorem 2.3. Let the positive sequence (pn) and a sequence (λn) be such that

∆Xn = O(n
−1);

∞∑
n=1

1

n
{|λn|k + |λn+1|k}Xk−1n 5 ∞;

∞∑
n=1

(Xkn + 1)|∆λn| 5 ∞,

where Xn = (npn)
−1Pn. Then the |N ,pn|k-summability of a factored Fourier series

∞∑
n=1

λnXnAn(t) at any

point t = x is a local property of f(t).

Later Sarigöl (see [13]) has proved the following



Int. J. Anal. Appl. 16 (2) (2018) 213

Theorem 2.4. Suppose that A = (anv) is a positive normal matrix satisfying

an−1,v = anv, (n 5 v + 1)

ān,0 = 1 (n = 0, 1, 2, ..., )

n−1∑
v=1

avvân,v−1 = O(ann),

∆xn = O(n
−1),

where Xn =
1

(nann)
. If a sequence (λn) satisfying following conditions

∞∑
n=1

n−1{|λkn| + |λn+1|
k}Xkn−1 5 ∞,

∞∑
n=1

(Xkn + 1)|∆λn| 5 ∞.

Then the |A|k-summability of a factored Fourier series
∞∑
n=1

λnXnAn(t) at any point t = x is a local property

of f(t).

Again to improve upon and generalize Theorem 2.4, Sulaiman [14] has proved the following theorem for

a normal matrix.

Theorem 2.5. Let A = (anv) is a normal matrix satisfying

|ân,v+1| ≤ |ann|,

∞∑
n=v+1

|ân,v+1| ≤∞,

n−1∑
v=1

|avv||ân,v+1| = O(|ann|),

∆Xn = O(
1

n
),



Int. J. Anal. Appl. 16 (2) (2018) 214

where Xn =
1

(nann)
. If a sequence (λn) satisfying the following conditions

∞∑
n=1

n−1{|λkn| + |λn+1|
k}Xkn−1 5 ∞,

∞∑
n=1

Xn|∆λn| 5 ∞.

Then the |A|k-summability of a factored Fourier series
∞∑
n=1

λnXnAn(t) at any point t = x is a local property

of f(t).

3. Main result

In the present paper, we have established a new result on local property of |A,δ|k-summability of factored

Fourier series

∞∑
n=1

λnXnAn(t) in the form of a theorem as follows.

Theorem 3.1. Suppose A = (anv) is a positive normal matrix such that

an−1,v ≥ an,v (n 5 v + 1); (3.1)

ān,0 = 1 (n = 0, 1, ..., ); (3.2)

n−1∑
v=1

avvân,v−1 = O(ann); (3.3)

m+1∑
n=v+1

ân,v+1a
−δk
nn = O(v

δk); (3.4)

m+1∑
n=v+1

a−δknn |∆̄anv| = O(v
δk), ; (3.5)

∆Xn = O(n
−1), (3.6)

where Xn =
1

(nann)
. If a sequence (λn) satisfying the following conditions

∞∑
n=1

n−1{|λ|k + |λn+1|k}Xknn
δk 5 ∞; (3.7)

∞∑
n=1

(xkn + 1)|∆λn|n
δk 5 ∞. (3.8)



Int. J. Anal. Appl. 16 (2) (2018) 215

Then the |A,δ|k-summability of a factored Fourier series
∞∑
n=1

λnXnAn(t) at any point t = x is a local prop-

erty of f(t).

Remark 3.1. The element ânv = 0 for each n,v. In fact, it is easily seen from the positiveness of the

matrix, (3.1) and (3.2), that â00 = 1,

ânv = ān0 − āv−1,0 +
v−1∑
j=0

(an−1,j −anj)

=

v−1∑
j=0

(an−1,j −anj) = 0 (1 5 v 5 n)

and equal to zero otherwise.

In order to prove the above theorem we need the a lemma as follows.

Lemma 3.1. Suppose that the matrix A and the sequence (λn) satisfy the conditions of the theorem, and

that (sn) is bounded. Then factored Fourier series

∞∑
n=1

λnXnAn(t) is summable to |A,δ|k (k = 1, δ = 0).

Proof. Let (Tn) be an A− transformation of
n∑
i=1

λiXiAn(t), then

Tn =

n∑
i=0

anisi =

n∑
i=1

ani

i∑
v=1

λvXv =

n∑
v=1

λvXv

n∑
i=v

ani =

n∑
v=1

ānvλnXv

∆̄Tn = Tn −Tn−1 =
n∑
v=1

(ānv − ān−1,v)λvXv =
n∑
v=1

ânvλvXv

∆̄Tn =

n−1∑
v=1

(ânvλvXv)sv + annλnXnsn

but, ∆(ânvλvXv) = λvXv∆ânv + ∆(λvXv)ân,v+1

= λvXv∆̄anv + (Xv∆λv + ∆Xvλv+1)ân,v+1.



Int. J. Anal. Appl. 16 (2) (2018) 216

∆̄Tn =

n−1∑
v=1

ân,v+1Xv∆λvsv +

n−1∑
v=1

ân,v+1λv+1∆Xvsv +

n−1∑
v=1

∆̄anvλvXvsv + annλnXnsn

= Tn,1 + Tn,2 + Tn,3 + Tn,4, (say).

To complete the proof, it is sufficient to show that by using Minkowski’s inequality

∞∑
n=1

a1−k−δknn |Tn,m|
k < ∞ (m = 1, 2, 3, 4).

Using Hölder inequality and (3.1), (3.2), (3.8),

Let

I1 =

m+1∑
n=2

a1−k−δknn |Tn,1|
k

5
m+1∑
n=2

a1−k−δknn

{
n−1∑
v=1

ân,v+1Xv|∆λv||sv|

}k

= O(1)

m+1∑
n=2

a1−k−δknn

{
n−1∑
v=1

ân,v+1Xv|∆λv|

}k

= O(1)

m+1∑
n=2

a−δknn

n−1∑
v=1

ân,v+1X
k
v |∆λv|

{
(ann)

−1
n−1∑
v=1

ân,v+1|∆λv|

}k−1
.

Since,

ân,v+1 =

n∑
r=v+1

(anr −an−1,r) =
n∑
r=0

(an−1,r −an,r)

5
n−1∑
r=0

(an−1,r −anr) = ān−1,0 − ān0 + ann = ann.

⇒
n−1∑
v=1

ân,v+1|∆λv| 5 ann
n−1∑
v=1

|∆λv| = O(ann).



Int. J. Anal. Appl. 16 (2) (2018) 217

I1 = O(1)

m+1∑
n=2

a−δknn

n−1∑
v=1

ân,v+1X
k
v |∆λv|

= O(1)

m∑
v=1

Xkv |∆λv|
m+1∑
n=v+1

ân,v+1a
−δk
nn

= O(1)

m∑
v=1

Xkv |∆λv|v
δk

= O(1).

Using Hölder inequality, and (3.3), (3.4), (3.6), (3.7),

I2 =

m+1∑
n=2

a1−k−δknn |Tn,2|
k

5
m+1∑
n=2

a1−k−δknn

{
n−1∑
v=1

ân,v+1|λv+1||∆xv||sv|

}k

= O(1)

m+1∑
n=2

a1−k−δknn

{
n−1∑
v=1

ân,v+1|λv+1|avvXv

}k

= O(1)

m+1∑
n=2

(ann)
−δk

n−1∑
v=1

ân,v+1|λv+1|kavvXkv

{
(ann)

−1
n−1∑
v=1

avvân,v+1

}k−1

= O(1)

m+1∑
n=2

(ann)
−δk

n−1∑
v=1

ân,v+1|λv+1|kavvXkv

= O(1)

m∑
v=1

avvX
k
v |λv+1|

k
m+1∑
n=v+1

a−δknn ân,v+1

= O(1)

m∑
v=1

avvX
k
v |λv+1|

kvδk

= O(1)

m∑
v=1

1

v
Xkv |λv+1|

kvδk

= O(1).



Int. J. Anal. Appl. 16 (2) (2018) 218

Using Hölder inequality, and (3.1), (3.2),

I3 =

m+1∑
n=2

a1−k−δknn |Tn,3|
k

5
m+1∑
n=2

a1−k−δknn

{
n−1∑
v=1

|∆̄anv||λv|Xv|sv|

}k

= O(1)

m+1∑
n=2

a1−k−δknn

{
n−1∑
v=1

|∆̄anv||λv|Xv

}k

= O(1)

m+1∑
n=2

a−δknn

n−1∑
v=1

|∆̄anv||λv|kXkv

{
(ann)

−1
n−1∑
v=1

|∆̄anv|

}k−1
.

We know

n−1∑
v=1

|∆̄anv| =
n−1∑
v=1

(an−1,v −anv)

= ān−1,0 − ān,0 + an0 −an−1,0 + ann

= an0 −an−1,0 + ann ≤ ann.

I3 = O(1)

m+1∑
n=2

a−δknn

n−1∑
v=1

|∆̄anv||λv|kXkv

= O(1)

m∑
v=1

|λv|kXkv
m+1∑
n=v+1

a−δknn |∆̄anv|

= O(1)

m∑
v=1

|λv|kXkvv
δkavv

= O(1).



Int. J. Anal. Appl. 16 (2) (2018) 219

Finally, using (3.7),

I4 =

∞∑
n=1

a1−k−δknn |Tn,4|

5
∞∑
n=1

a1−k−δknn {ann|λn|Xn|sn|}
k

= O(1)

∞∑
n=1

a1−k−δknn {ann|λn|Xn}
k

= O(1)

∞∑
n=1

(ann)
−δkann|λ|kXkn

= O(1)

∞∑
n=1

(ann)
−δk|λ|kXkn

1

n

= O(1).

Thus the proof of the above Lemma is established.

Proof of the Theorem 3.1. Since the convergence of the Fourier series at a point is a local property of

its generating function f(t), the theorem follows by formula from chapter II of the book (see details [17])

and from the above Lemma 3.1.

Applications. Now we apply the theorem to the weighted mean in which A = (anv) is defined as

anv = pvP
−1
n , when (0 5 v 5 n) where Pn = p0 + p1 + ... + pn; therefore, it is well known that

ānv = P
−1
n (Pn −Pv−1)

and

ân,v+1 = (PnPn−1)
−1pnPv.

One can now easily verify that taking δ = 0 the conditions of the theorem reduce to those of Theorem 2.3.

We may now ask weather there are some examples (other then weighted mean methods) of matrices

A that satisfy the hypotheses of the theorem. For this, apply the theorem to the Cesàro method of order α

with (0 5 α 5 1) in which A is given by [15]

anv =
Aα−1n−v
Aαn

.



Int. J. Anal. Appl. 16 (2) (2018) 220

It is well known that

ānv =
Aαn−v
Aαn

and

ânv =
vAα−1n−v
nAαn

.

It is now seen that by taking account of Aαn ≈
nα

Γ(α+1)
conditions (3.1)-(3.8) are satisfied. Therefore the

above theorem is same as the following result.

Corollary 3.1. Let k ≥ 1 and 0 ≤ α ≤ 1. If (λn) a convex sequence satisfying following condition-

s:
∞∑
n=1

nαk−α−k{|λ|k + |λn+1|k}nδk 5 ∞,

∞∑
n=1

|∆λn|nδk 5 ∞.

Then the |C,α, (α− 1)(1 − 1
k

)δ|k summability of a factored Fourier series
∞∑
n=1

λnXnAn(t) with Xn = A
α

n
at

any point t = x is a local property of the generating function f(t).

4. Conclusion

The result obtained here is more general in the sense that, by substituting δ = 0, the |A; δ|k-summability

reduces to |A|k-summability.

Acknowledgment

The authors would like to express their heartfelt thanks to the editors and anonymous referees for their

most valuable comments and constructive suggestions which leads to the significant improvement of the

earlier version of the manuscript.

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	1. Introduction and Motivation
	2. Preliminaries
	3. Main result
	4. Conclusion
	Acknowledgment
	References