International Journal of Analysis and Applications

Volume 17, Number 3 (2019), 361-368

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

DOI: 10.28924/2291-8639-17-2019-361

LINEAR FUNCTIONALS CONNECTED WITH STRONG DOUBLE CESARO

SUMMABILITY

FATIH NURAY1 AND NIMET PANCAROḠLU AKIN2,∗

1Department of Mathematics, Afyon Kocatepe University, Turkey

2Department of Mathematics and Science Education, Afyon Kocatepe University, Turkey

∗Corresponding author: npancaroglu@aku.edu.tr

Abstract. D. Borwein characterized linear functionals on the normed linear spaces wp and Wp. In this

paper we extend his results by presenting definitions for the double strong Cesaro mean. Using these new

notions of strongly p-Cesaro summable double sequence and strongly p-Cesaro summable bivariate function

we present extensions of D. Borwein’s results.

1. Introduction

The first definitions and investigations of the convergence of double sequences are usually atributted to

F. Pringsheim [12], who studied such sequences and series more than hundred years ago. Pringsheim defined

what we call the P limit and gave examples of convergence (P convergence) of double sequences with and

without the usual convergence of rows and columns. G. H. Hardy [4], considered in more details the case

of convergence of double sequences where, besides the existence of the P limit, rows and columns converge.

F. Moricz [6–8] discovered an alternative approach to the Hardy convergence, which significantly influenced

the whole theory.

The following notion of convergence for double sequences was presented by Pringsheim in [11]. A double

sequence x = {xnm} of real numbers is said to be convergent to L ∈ R in Pringsheim’s sense if for any ε > 0,

Received 2019-01-10; accepted 2019-02-12; published 2019-05-01.

2010 Mathematics Subject Classification. Primary 40A05; Secondary 40C05.

Key words and phrases. double sequence; measurable function; bivariate function; Cesaro summable function.

c©2019 Authors retain the copyrights

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

361

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-361


Int. J. Anal. Appl. 17 (3) (2019) 362

there exists Nε ∈ N such that |xnm − L| < ε, whenever n,m > Nε. In this case we denote such limit as

follow:

P − lim
n,m→∞

xnm = L.

A classical notion of sequence space is the following:

wp = {x = (xn) : lim
N→∞

1

N

N∑
n=1

|xn − `|p = 0}.

In [2], D. Borwein extended the sequence space wp to the function space Wp, the space of real valued functions

x, measurable (in the Lebesque sense) in the interval (1,∞) for which there is a number ` = `x such that

lim
T→∞

1

T

∫ T
1

|x(t) − `|p = 0.

By a linear functional we mean one that is real-valued, additive, homogeneous and continuous. It is to be

supposed throughout that 1 ≤ p < ∞ and that 1
p

+ 1
q

= 1.

2. Main Results

We begin to the main results with following definitions:

Definition 2.1. Let x = {xnm} be a real double sequence. Then the double sequence x is said to be strongly

p-Cesaro summable to ` if

P − lim
N,M→∞

1

NM

N∑
n=1

M∑
m=1

|xnm − `|p = 0.

The space of all strongly p-Cesaro summable double sequences will be denote by w2p. Observe that this

space is normed by

‖x‖2 = sup
N,M≥1

(
1

NM

N∑
n=1

M∑
m=1

|xnm − `|p
)1

p

.

Definition 2.2. Let x be a real valued bivariate function, measurable (in the Lebesque sense) in the (1,∞)×

(1,∞). Then the bivariate function x is said to be strongly p-Cesaro summable to ` if

lim
T,R→∞

1

TR

∫ T
1

∫ R
1

|x(t,r) − `|pdrdt = 0.

The space of all strongly p-Cesaro summable bivariate functions will be denote by W2p . Observe that this

space is normed by

‖x‖2 = sup
T≥1,R≥1

(
1

TR

∫ T
1

∫ R
1

|x(t,r) − `|pdrdt

)1
p

.

Given any real double sequence α = {αnm}. We define a double sequence {mnm(α,p)} by



Int. J. Anal. Appl. 17 (3) (2019) 363

mnm(α,p) =




sup

2n ≤ v < 2n+1; 2m ≤ u < 2m+1
{|vuαvu|}, if p = 1

(
1

2n+m

∑2n+1−1
v=2n

∑2m+1−1
u=2m |vuαvu|

q
)1

q

, if p > 1.

Given any real real valued bivariate function α(t,r) measurable in (1,∞) × (1,∞). We define a double

sequence {Mnm(α,p)} by

Mnm(α,p) =




ess.sup

2n ≤ t < 2n+1; 2m ≤ r < 2m+1
{|trα(t,r)|}, if p = 1

(
1

2n+m

∫ 2n+1
2n

∫ 2m+1
2m

|trα(t,r)|q
)1

q

, if p > 1.

Theorem 2.1. (i) If f is a linear functional on W2p , then there is a real number a and a real valued

bivariate function α, measurable in (1,∞) × (1,∞) such that

f(x) = a` +

∫ ∞
1

∫ ∞
1

α(t,r)x(t,r)drdt (2.1)

for every x ∈ W2p and
∞∑
n=0

∞∑
m=0

Mnm(α,p) < ∞. (2.2)

(ii) If a is a real number and α is a real valued bivariate function, measurable in (1,∞)×(1,∞), satisfying

(2.2), then (2.1) defines a linear function on W2p with

‖f‖2 ≤ |a| + 2
2
p

∞∑
n=0

∞∑
m=0

Mnm(α,p)

and the integral in (2.1) is absolutely convergent for every x ∈ W2p .

Proof. Let L2p be the linear space of real valued bivariate functions x measurable in (1,∞)×(1,∞) for which∫ ∞
1

∫ ∞
1

|x(t,r)|pdrdt < ∞,

with norm

‖x‖L2p =
(∫ ∞

1

∫ ∞
1

|x(t,r)|pdrdt
)1

p

.

Clearly, if x ∈ L2p, then x ∈ W2p , ` = 0 and ‖x‖2 = ‖x‖W2p ≤‖x‖L2p. Consequently the restriction to L
2
p of the

given linear functional f on W2p is linear on L
2
p. It follows from standard results that there is a real valued

bivariate function α, measurable in (1,∞) × (1,∞), such that

f(x) =

∫ ∞
1

∫ ∞
1

α(t,r)x(t,r)drdt (2.3)



Int. J. Anal. Appl. 17 (3) (2019) 364

for all x ∈ L2p and either

ess.sup {|α(t,r)|} < ∞ if p = 1

1 ≤ t < ∞

1 ≤ r < ∞

or ∫ ∞
1

∫ ∞
1

|α(t,r)|qdrdt < ∞ if p > 1.

To show that α must necessarily satisfy (2.2) we consider the cases p = 1 and p > 1 separately. If p = 1,

let Mnm = Mnm(α, 1). There is a measurable set enm of positive measure |enm| in the (2n, 2n+1)×(2m, 2m+1)

such that

|trα(t,r)| > Mnm −
1

2n+m

for all (t,r) ∈ enm.

Let

x(t,r) =




2n+m

enm
sign(α(t,r)), if (t,r) ∈ enm,n ≤ s,m ≤ u

0, otherwise.

Then x ∈ L21 and so, by (2.3),

‖f‖2‖x‖2 ≥ f(x) =
∫ ∞
1

∫ ∞
1

x(t,r)α(t,r)drdt

=

s∑
n=0

u∑
m=0

∫ ∫
enm

2n+m

|enm|
|α(t,r)|drdt

≥
1

4

s∑
n=0

u∑
m=0

1

|enm|

∫ ∫
enm

|trα(t,r)|drdt (2.4)

≥
1

4

s∑
n=0

u∑
m=0

(Mnm −
1

2n+m
).

Furtermore, for 2z ≤ T < 2z+1 ≤ 2s+1, 2h ≤ R < 2h+1 ≤ 2u+1,

1

TR

∫ T
1

∫ R
1

|x(t,r)|drdt ≤
1

2z+h

∫ 2z+1
1

∫ 2h+1
1

|x(t,r)|drdt

=
1

2z+h

z∑
n=0

h∑
m=0

∫ ∫
enm

x(t,r)|drdt

≤
1

2z+h

z∑
n=0

h∑
m=0

2n+m < 4,

and for T > 2s+1, R > 2u+1

1

TR

∫ T
1

∫ R
1

|x(t,r)|drdt ≤
1

2s+12u+1

∫ 2s+1
1

∫ 2u+1
1

|x(t,r)|drdt < 1.



Int. J. Anal. Appl. 17 (3) (2019) 365

Hence ‖x‖2 < 4 and so, by (2.4),

4‖f‖2 +
1

4

∞∑
n=0

∞∑
m=0

1

2n+m
= 4‖f‖2 + 1 ≥

1

4

∞∑
n=0

∞∑
m=0

Mnm,

which establishes (2.2) in this case. If p > 1, let Mnm = Mnm(α,p) and let

x(t,r) =




(tr)q

2n+m
|α(t,r)
Mnm

|
q
p sign(α(t,r)), if

2n ≤ t < 2n+1 ≤ 2z+1;

2m ≤ r < 2m+1 ≤ 2u+1 and Mnm 6= 0

0, otherwise.

Then x ∈ L2p and so, by (2.3),

f(x) =

∫ 2z+1
1

∫ 2u+1
1

|α(t,r)x(t,r)|drdt =
z∑

n=0

u∑
m=0

∫ 2n+1
2n

∫ 2m+1
2m

|α(t,r)x(t,r)|drdt

=

z∑
n=0

u∑
m=0

Mnm. (2.5)

Furtermore, for 2z ≤ T < 2z+1 ≤ 2s+1, 2h ≤ R < 2h+1 ≤ 2u+1,

1

TR

∫ T
1

∫ R
1

|x(t,r)|pdrdt ≤
1

2z+h

∫ 2z+1
1

∫ 2h+1
1

|x(t,r)|pdrdt

=
1

2z+h

z∑
n=0

h∑
m=0

∫ ∫
enm

|x(t,r)|pdrdt

≤
22p

2z+h

z∑
n=0

h∑
m=0

2n+m < 22p+2,

and for T > 2z+1, R > 2h+1

1

TR

∫ T
1

∫ R
1

|x(t,r)|pdrdt ≤
1

2z+12h+1

∫ 2z+1
1

∫ 2h+1
1

|x(t,r)|pdrdt < 4p.

Hence ‖x‖2 < 22+
2
p and so, by (2.5),

∞∑
n=0

∞∑
m=0

Mnm ≤ 22+
2
p‖f‖2,

which established (2.2) in this case.

Suppose now p ≥ 1, Mnm = Mnm(α,p) and x ∈ W2p . Then by Hölder inequality∫ ∞
1

∫ ∞
1

|α(t,r)x(t,r)|drdt =
∞∑
n=0

∞∑
m=0

∫ 2n+1
2n

∫ 2m+1
2m

|α(t,r)x(t,r)|drdt

≤
∞∑
n=0

∞∑
m=0

Mnm

(
2p(1−

1
p
)(n+m)

∫ 2n+1
2n

∫ 2m+1
2m

∣∣∣∣x(t,r)tr
∣∣∣∣p drdt

)1
p

≤
∞∑
n=0

∞∑
m=0

Mnm

(
2−(n+m)

∫ 2n+1
2n

∫ 2m+1
2m

|x(t,r)|pdrdt

)1
p



Int. J. Anal. Appl. 17 (3) (2019) 366

≤ 2
2
p‖x‖2

∞∑
n=0

∞∑
m=0

Mnm. (2.6)

It follows that ∫ ∞
1

∫ ∞
1

|α(t,r)x(t,r)|drdt < ∞

whenever x ∈ W2p , and in particular since the characteristic function of (1,∞) × (1,∞) is in W2p , that∫ ∞
1

∫ ∞
1

|α(t,r)|drdt < ∞.

Suppose next that x ∈ W2p and ` = `x. Let

y(t,r) = x(t,r) − `

ynm(t,r) =


 y(t,r), if 1 ≤ t ≤ n, 1 ≤ r ≤ m;0, if t ≥ n and r ≥ m.

Then y ∈ W2p , ynm ∈ L2p and

‖ynm −y‖2 = sup
T≥n,R≥m

(
1

TR

∫ T
n

∫ R
m

|x(t,r) − `|p
)1

p

= o(1) as n,m →∞.

But

|f(ynm −y)| = |f(ynm) −f(y)| ≤ ‖ynm −y‖2‖f‖2,

and so, by (2.3),

f(y) = P − lim
n,m→∞

f(ynm) = P − lim
n,m→∞

∫ n
1

∫ m
1

y(t,r)α(t,r)drdt

=

∫ ∞
1

∫ ∞
1

x(t,r)α(t,r)drdt− `
∫ ∞
1

∫ ∞
1

α(t,r)drdt.

Since both integrals on the right hand side have been shown to be absolutely convergent. Taking δ to be

characteristic function of (1,∞) × (1,∞) we see that

f(x) = f(y + `δ)f(y) + `f(δ) =

∫ ∞
1

∫ ∞
1

x(t,r)α(t,r)drdt + a`

where a = f(δ) −
∫∞
1

∫∞
1
α(t,r). This completes the proof of part (i.

(ii) It follows from (2.6) that if x ∈ W2p , ` = `x and Mnm = Mnm(α,p), then

|f(x)| =
∣∣∣∣
∫ ∞
1

∫ ∞
1

x(t,r)α(t,r)drdt + a`

∣∣∣∣ ≤‖x‖22 2p ∞∑
n=0

∞∑
m=0

Mnm + |a`|. (2.7)

Further, by Minkowski’s inequality

(
1 −

1

TR

)1
p

|`| ≤

(
1

TR

∫ T
1

∫ R
1

|x(t,r) − `|pdrdt

)1
p

+

(
1

TR

∫ T
1

∫ R
1

|x(t,r)|pdrdt

)1
p

and the first term on the right hand side is o(1). Hence |`| ≤ ‖x‖2 and consequently, by (2.7),

|f(x)| ≤ ‖x‖2

(
|a| + 2

2
p

∞∑
n=0

∞∑
m=0

Mnm

)



Int. J. Anal. Appl. 17 (3) (2019) 367

for every x ∈ W2p . The additive and homogenous functional f defined by (2.1) is therefore also continuous

on W2p and

|f(x)| ≤ |a| + 2
2
p

∞∑
n=0

∞∑
m=0

Mnm.

Finally, by (2.6), the integral in (2.1) is absolutely convergent. Thus the proof is completed. �

Theorem 2.2. (i) If f is a linear functional on w2p, then there is a real number a and a real double

sequence α = {αnm} such that

f(x) = a` +

∞∑
n=1

∞∑
m=1

αnmxnm (2.8)

for every x = {xnm}∈ w2p and

∞∑
n=0

∞∑
m=0

mnm(α,p) < ∞. (2.9)

(ii) If a is a real number and α = {αnm} is a real double sequence satisfying (2.9), then (2.8) defines a

linear function on w2p with

‖f‖2 ≤ |a| + 2
2
p

∞∑
n=0

∞∑
m=0

mnm(α,p)

and the series in (2.8) is absolutely convergent for every x = {xnm}∈ w2p.

Proof. Given any real double sequence x = {xnm}, define a bivariate function x∗ by

x∗(t,r) = xnm for n < t ≤ n + 1; m < r ≤ m + 1,n = 1, 2, 3, ...,m = 1, 2, 3, ....

It is easily verified that this defines a one to one correspondence between w2p and a linear subspace (W
2
p )
∗ of

W2p such that

`x∗ = `x and ‖x∗‖2 ≤‖x‖2 ≤ 2
2
p‖x∗‖2.

Hence given a linear functional on W2p , the functional f
∗ defined by

f∗(x∗) = f(x)

is linear on (W2p )
∗. Consequently, by the Hahn-Banach theorem and Theorem2.1, there is a real number a

and a real valued bivariate function α∗, integrable over (1,∞) × 1,∞), such that
∞∑
n=0

∞∑
m=0

Mnm(α
∗,p) < ∞

and, for every x ∈ w2p,

f(x) = f∗(x∗) = a`x∗ +

∫ ∞
1

∫ ∞
1

α∗(t,r)x∗(t,r)drdt = a`x +

∞∑
n=1

∞∑
m=1

αnmxnm



Int. J. Anal. Appl. 17 (3) (2019) 368

where αnm =
∫n+1
n

∫m+1
m

α∗(t,r)drdt. Furthermore, for α = {αnm},
∞∑
n=0

∞∑
m=0

mnm(α,p) ≤
∞∑
n=0

∞∑
m=0

Mnm(α
∗,p);

and this completes the proof of (i).

(ii) If x = {xnm} ∈ w2p mnm = mnm(α,p) and ` = `x then by Hölder’s and Minkowski’s inequalities, as

in the proof of (ii) of Theorem2.1,

f(x) = a` +

∞∑
n=1

∞∑
m=1

αnmxnm ≤ |a`| +
∞∑
n=1

∞∑
m=1

|αnmxnm|

≤ |a`| + 2
2
p‖x‖2

∞∑
n=0

∞∑
m=0

mnm ≤‖x‖2

(
|a| + 2

2
p

∞∑
n=0

∞∑
m=0

mnm

)
.

The functional f defined by (2.8) is therefore linear on w2p,

‖f‖2 ≤ |a| + 2
2
p

∞∑
n=0

∞∑
m=0

mnm

and the series in (2.8) absolutely convergent. This completes the proof. �

References

[1] S. Banach, Theorie des operations lineaires, Warsaw, 1932.

[2] D. Borwein, Linear functionals connected with strong Cesaro Summability, J. London Math. Soc. 40 (1965), 628-634.

[3] T. J. I. A. Bromwich, An introduction to the theory of infinite series, second ed., Cambridge, 1926 (repr. Macmillan,

London, 1955).

[4] G. H. Hardy, On the convergence of certain multiplie series, Proc. Cambridge Philos. Soc. 19 (1916-1919), 86-95.

[5] F. Moricz, Tauberian theorems for Cesaro summable double sequences, Studia Math. 110 (1994), 83–96.

[6] F. Moricz, On the convergence in a restricted sense of multiple series, Anal. Math. 5(1979), 135–147.

[7] F. Moricz, Some remarks on the notion of regular convergence of multiple series, Acta Math. Hungar. 41 (1983), 161–168.

[8] F. Moricz, Extensions of the spaces c and c0 from single to double sequences, Acta Math. Hungar. 57 (1991), 129–136.

[9] Mursaleen and Osama H. H. Edely Statistical convergence of double sequences, J. Math. Anal. Appl. 288 (2003), 223–231.

[10] R. F. Patterson, Double Sequence Core Theorems, Internat. J. Math. and Math. Sci. 22 (1999), 785–793.

[11] A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53 (1900), 289-321.

[12] A. Pringsheim, Elementare Theorie der unendlichen Doppelreihen, Sitzung-Berichte. der math.-phys. Classe der Akad. der

Wissenschafften zu Munchen 27 (1897), 101–152.


	1. Introduction
	2. Main Results
	References