International Journal of Analysis and Applications

Volume 18, Number 6 (2020), 957-964

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

DOI: 10.28924/2291-8639-18-2020-957

GENERALIZED ABSOLUTE RIESZ SUMMABILITY OF INFINITE SERIES AND

FOURIER SERIES

BAĞDAGÜL KARTAL∗

Department of Mathematics, Erciyes University, 38039 Kayseri, Turkey

∗Corresponding author: bagdagulkartal@erciyes.edu.tr

Abstract. In this paper, two known theorems dealing with |N̄,pn|k summability of infinite series and

Fourier series have been generalized to ϕ−|N̄,pn; β|k summability.

1. Introduction

A sequence (An) is said to be δ-quasi-monotone if An → 0, An > 0 ultimately and ∆An ≥ −δn, where

∆An=An −An+1 and δ = (δn) is a sequence of positive numbers (see [1]). A sequence (gn) is said to be of

bounded variation, denoted by (gn) ∈BV, if
∑∞
n=1 |∆gn| < ∞. Let

∑
an be a given infinite series with the

partial sums (sn). Let (ϕn) be a sequence of positive real numbers. The series
∑
an is said to be summable

ϕ−|N̄,pn; β|k, k ≥ 1 and β ≥ 0, if (see [22])

∞∑
n=1

ϕβk+k−1n |un −un−1|
k < ∞

where (pn) is a sequence of positive numbers such that

Pn =

n∑
v=0

pv →∞ as n →∞ (P−i = p−i = 0, i ≥ 1) ,

Received August 13th, 2020; accepted September 1st, 2020; published September 14th, 2020.

2010 Mathematics Subject Classification. 26D15, 40D15, 40F05, 40G99.

Key words and phrases. absolute summability; Fourier series; Hölder’s inequality; infinite series; Minkowski’s inequality;

Riesz mean; summability factor.

©2020 Authors retain the copyrights

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

957

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-18-2020-957


Int. J. Anal. Appl. 18 (6) (2020) 958

and

un =
1

Pn

n∑
v=0

pvsv.

For ϕn =
Pn
pn

and β = 0, ϕ −|N̄,pn; β|k summability reduces to |N̄,pn|k summability (see [2]). Taking

ϕn = n, β = 0 and pn = 1 for all values of n, ϕ −|N̄,pn; β|k summability reduces to |C, 1|k summability

(see [8]).

If we write Xn =

n∑
v=1

pv/Pv, then (Xn) is a positive increasing sequence tending to infinity with n.

In [3], the following theorem on δ-quasi-monotone sequences has been proved.

Theorem 1.1. Let (λn) → 0 as n →∞ and (pn) be a sequence of positive numbers such that Pn = O(npn)

as n →∞. Suppose that there exists a sequence of numbers (An) which is δ-quasi-monotone with∑
nXnδn < ∞,

∑
AnXn is convergent, and |∆λn| ≤ |An| for all n. If the condition

m∑
n=1

pn
Pn
|tn|k = O(Xm) as m →∞(1.1)

is satisfied, where (tn) is the n-th (C, 1) mean of the sequence (nan), then the series
∑
anλn is summable

|N̄,pn|k, k ≥ 1.

Lemma 1.1. [3] Under the conditions of Theorem 1.1, we have that

|λn|Xn = O (1) as n →∞,(1.2)

nXnAn = O(1) as n →∞,(1.3)

∞∑
n=1

nXn|∆An| < ∞.(1.4)

2. Main Result

There are some papers on absolute summability (see [4–6,9–12,16–18,23–25]). Now we generalize Theorem

1.1 as in the following form.

Theorem 2.1. Let (ϕn) be a sequence of positive real numbers such that

ϕnpn = O(Pn),(2.1)

m+1∑
n=v+1

ϕβk−1n
1

Pn−1
= O

(
ϕβkv

1

Pv

)
as m →∞.(2.2)



Int. J. Anal. Appl. 18 (6) (2020) 959

If all conditions of Theorem 1.1 are satisfied with the condition (1.1) replaced by

m∑
n=1

ϕβk−1n |tn|
k = O(Xm) as m →∞,(2.3)

then the series
∑
anλn is summable ϕ−|N̄,pn; β|k, k ≥ 1 and 0 ≤ β < 1/k.

3. Proof of Theorem 2.1

Let (In) indicates (N̄,pn) mean of the series
∑
anλn. Then, for n ≥ 1, we obtain

∆̄In = In − In−1 =
pn

PnPn−1

n∑
v=1

Pv−1avλv =
pn

PnPn−1

n∑
v=1

Pv−1λv
v

vav.

Applying Abel’s transformation, we get

∆̄In =
pn

PnPn−1

n−1∑
v=1

λv+1
v

Pvtv −
pn

PnPn−1

n−1∑
v=1

v + 1

v
pvλvtv

+
pn

PnPn−1

n−1∑
v=1

v + 1

v
Pvtv∆λv +

(n + 1)

nPn
pnλntn

= In,1 + In,2 + In,3 + In,4.

For the proof of Theorem 2.1, it is sufficient to show that

∞∑
n=1

ϕβk+k−1n | In,r |
k< ∞, for r = 1, 2, 3, 4.

First,

m+1∑
n=2

ϕβk+k−1n | In,1 |
k ≤

m+1∑
n=2

ϕβk+k−1n

(
pn

PnPn−1

)k (n−1∑
v=1

Pv |tv|
|λv+1|
v

)k

=

m+1∑
n=2

ϕβk−1n

(
ϕnpn
Pn

)k
1

Pkn−1

(
n−1∑
v=1

Pv |tv|
|λv+1|
v

)k
.

Here (2.1) gives
(
ϕnpn
Pn

)k
= O(1), also using Hölder’s inequality, we obtain

m+1∑
n=2

ϕβk+k−1n | In,1 |
k = O(1)

m+1∑
n=2

ϕβk−1n
1

Pn−1

(
n−1∑
v=1

Pv |tv|
k |λv+1|k

v

)(
1

Pn−1

n−1∑
v=1

Pv
v

)k−1
.

Now using the fact that Pv = O(vpv),

m+1∑
n=2

ϕβk+k−1n | In,1 |
k = O(1)

m+1∑
n=2

ϕβk−1n
1

Pn−1

(
n−1∑
v=1

pv |tv|
k |λv+1|k

)(
1

Pn−1

n−1∑
v=1

pv

)k−1
.



Int. J. Anal. Appl. 18 (6) (2020) 960

Then, we have

m+1∑
n=2

ϕβk+k−1n | In,1 |
k = O(1)

m+1∑
n=2

ϕβk−1n
1

Pn−1

n−1∑
v=1

pv |tv|
k |λv+1|k

= O(1)

m∑
v=1

pv|λv+1|k−1|λv+1| |tv|
k

m+1∑
n=v+1

ϕβk−1n
1

Pn−1
.

Here, by using (2.2) and (1.2),

m+1∑
n=2

ϕβk+k−1n | In,1 |
k = O(1)

m∑
v=1

ϕβkv
pv
Pv
|λv+1| |tv|

k

= O(1)

m∑
v=1

ϕβk−1v

(
ϕvpv
Pv

)
|λv+1| |tv|

k
.

Again, from (2.1), we obtain

m+1∑
n=2

ϕβk+k−1n | In,1 |
k = O(1)

m∑
v=1

ϕβk−1v |λv+1| |tv|
k
.

Hence, we get

m+1∑
n=2

ϕβk+k−1n | In,1 |
k = O(1)

m−1∑
v=1

∆|λv+1|
v∑
r=1

ϕβk−1r |tr|
k + O(1)|λm+1|

m∑
v=1

ϕβk−1v |tv|
k

= O(1)

m−1∑
v=1

|Av+1|Xv+1 + O(1)|λm+1|Xm+1 = O(1) as m →∞,

by using Abel’s transformation, hypotheses of Theorem 2.1, and Lemma 1.1.

Now, we have

m+1∑
n=2

ϕβk+k−1n | In,2 |
k = O(1)

m+1∑
n=2

ϕβk+k−1n

(
pn

PnPn−1

)k (n−1∑
v=1

pv |λv| |tv|

)k

= O(1)

m+1∑
n=2

ϕβk−1n

(
ϕnpn
Pn

)k
1

Pkn−1

(
n−1∑
v=1

pv |λv| |tv|

)k

= O(1)

m+1∑
n=2

ϕβk−1n
1

Pkn−1

(
n−1∑
v=1

pv |λv| |tv|

)k
.

Using Hölder’s inequality, we get

m+1∑
n=2

ϕβk+k−1n | In,2 |
k = O(1)

m+1∑
n=2

ϕβk−1n
1

Pn−1

(
n−1∑
v=1

pv |λv|
k |t

v
|k
)(

1

Pn−1

n−1∑
v=1

pv

)k−1

= O(1)

m∑
v=1

pv|λv|k|tv|k
m+1∑
n=v+1

ϕβk−1n
1

Pn−1
.



Int. J. Anal. Appl. 18 (6) (2020) 961

By (2.2), (2.1) and (1.2), we get

m+1∑
n=2

ϕβk+k−1n | In,2 |
k = O(1)

m∑
v=1

ϕβk−1v |λv||tv|
k.

Here, using Abel’s transformation as in In,1, we have

m+1∑
n=2

ϕβk+k−1n | In,2 |
k = O(1) as m →∞.

Again, using Hölder’s inequality, we have

m+1∑
n=2

ϕβk+k−1n | In,3 |
k = O(1)

m+1∑
n=2

ϕβk+k−1n

(
pn

PnPn−1

)k (n−1∑
v=1

Pv |tv| |∆λv|

)k

= O(1)

m+1∑
n=2

ϕβk−1n

(
ϕnpn
Pn

)k
1

Pkn−1

(
n−1∑
v=1

Pv |tv| |Av|

)k

= O(1)

m+1∑
n=2

ϕβk−1n
1

Pn−1

(
n−1∑
v=1

pv |tv|
k

(v |Av|)k
)(

1

Pn−1

n−1∑
v=1

pv

)k−1

= O(1)

m+1∑
n=2

ϕβk−1n
1

Pn−1

n−1∑
v=1

pv |tv|
k

(v |Av|)k−1(v |Av|).

Using (1.3), we get (v |Av|)k−1 = O(1), then

m+1∑
n=2

ϕβk+k−1n | In,3 |
k = O(1)

m∑
v=1

pv|tv|kv |Av|
m+1∑
n=v+1

ϕβk−1n
1

Pn−1
.

Now using the conditions (2.2) and (2.1), we get

m+1∑
n=2

ϕβk+k−1n | In,3 |
k = O(1)

m∑
v=1

ϕβk−1v |tv|
kv |Av| .

Then, we have

m+1∑
n=2

ϕβk+k−1n | In,3 |
k = O(1)

m−1∑
v=1

∆(v |Av|)
v∑
r=1

ϕβk−1r |tr|
k + O(1)m |Am|

m∑
v=1

ϕβk−1v |tv|
k

= O(1)

m−1∑
v=1

∆(v |Av|)Xv + O(1)m |Am|Xm

= O(1)

m−1∑
v=1

vXv|∆Av| + O(1)
m−1∑
v=1

|Av+1|Xv+1 + O(1)m |Am|Xm

= O(1) as m →∞,

by using Abel’s transformation, hypotheses of Theorem 2.1, and Lemma 1.1.



Int. J. Anal. Appl. 18 (6) (2020) 962

Finally, we get

m∑
n=1

ϕβk+k−1n | In,4 |
k = O(1)

m∑
n=1

ϕβk+k−1n

(
pn
Pn

)k
|λn|k−1|λn||tn|k

= O(1)

m∑
n=1

ϕβk−1n |λn||tn|
k.

Here, as in In,1, we get

m∑
n=1

ϕβk+k−1n | In,4 |
k = O(1) as m →∞.

Hence, the proof of Theorem 2.1 is completed.

4. Applications

There are some different papers dealing with applications of Fourier series (see [14, 15, 19–21]). Let f be

a periodic function with period 2π and Lebesgue integrable over (−π,π). The trigonometric Fourier series

of f is defined as

f(x) ∼
1

2
a0 +

∞∑
n=1

(an cos nx + bn sin nx) =

∞∑
n=0

Cn(x)

where

a0 =
1

π

∫ π
−π

f(x)dx, an =
1

π

∫ π
−π

f(x) cos(nx)dx and bn =
1

π

∫ π
−π

f(x) sin(nx)dx.

Write

φ(t) =
1

2
{f(x + t) + f(x− t)}

and

φ1(t) =
1

t

∫ t
0

φ(u)du.

If φ1(t) ∈ BV(0,π), then tn(x) = O(1), where tn(x) is the n-th (C, 1) mean of the sequence (nCn(x))

(see [7]). By using this, the following theorem has been obtained in [3].

Theorem 4.1. If φ1(t) ∈BV(0,π), and the sequences (pn), (λn) and (Xn) satisfy the conditions of Theorem

1.1, then the series
∑
Cn(x)λn is summable | N̄,pn |k, k ≥ 1.

The following theorem gives a generalization of Theorem 4.1 for ϕ−|N̄,pn; β|k summability.

Theorem 4.2. If φ1(t) ∈BV(0,π), and the sequences (pn), (λn), (An), (ϕn) and (Xn) satisfy the conditions

of Theorem 2.1, then the series
∑
Cn(x)λn is summable ϕ−|N̄,pn; β|k, k ≥ 1 and 0 ≤ β < 1/k.



Int. J. Anal. Appl. 18 (6) (2020) 963

5. Conclusions

If we take ϕn =
Pn
pn

and β = 0 in Theorem 2.1, then the condition (2.3) reduces to the condition (1.1),

and the conditions (2.1) and (2.2) are provided. Thus, Theorem 2.1 reduces to Theorem 1.1. If we take

ϕn = n, β = 0 and pn = 1 for all values of n, then we have a result for |C, 1|k summability of an infinite

series (see [13]). Also, if we take ϕn =
Pn
pn

and β = 0 in Theorem 4.2, then we get Theorem 4.1.

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the pub-

lication of this paper.

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	1. Introduction
	2. Main Result
	3. Proof of Theorem 2.1
	4. Applications
	5. Conclusions
	References