Linear Functionals Connected with Strong Double Cesaro Summability

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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 PANCARO ¯}_{GLU AKIN}2,∗

1_{Department of Mathematics, Afyon Kocatepe University, Turkey}

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

∗_{Corresponding author:} _{[email protected]}

Abstract. D. Borwein characterized linear functionals on the normed linear spaceswpand Wp. In this

paper we extend his results by presenting definitions for the double strong Cesaro mean. Using these new notions of stronglyp-Cesaro summable double sequence and stronglyp-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

sequencex={xnm}of real numbers is said to be convergent toL∈Rin Pringsheim’s sense if for anyε >0,

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

2010Mathematics 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.

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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 X

n=1

|xn−`|p= 0}.

In [2], D. Borwein extended the sequence spacewpto the function spaceWp, 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

Z 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. Letx={xnm} be a real double sequence. Then the double sequencexis said to be strongly

p-Cesaro summable to`if

P− lim

N,M→∞ 1 N M

N X

n=1 M X

m=1

|xnm−`|p = 0.

The space of all strongly p-Cesaro summable double sequences will be denote by w2

p. Observe that this

space is normed by

kxk2= sup N,M≥1

1 N M

N X

n=1 M X

m=1

|xnm−`|p !p1

.

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

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

lim

T ,R→∞ 1 T R

Z T

1 Z R

1

|x(t, r)−`|p_drdt_{= 0.}

The space of all stronglyp-Cesaro summable bivariate functions will be denote byW2

p.Observe that this

space is normed by

kxk2= sup T≥1,R≥1

1 T R

Z T

1 Z R

1

|x(t, r)−`|p_drdt !p1

.

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mnm(α, p) =         

       

sup

2n _≤_{v <}₂n+1_{; 2}m_≤_{u <}₂m+1

{|vuαvu|}, if p= 1

1 2n+m

P2n+1−1 v=2n

P2m+1−1

u=2m |vuαvu|q 1q

, if p >1.

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

sequence{Mnm(α, p)}by

Mnm(α, p) =         

       

ess.sup

2n _≤_{t <}₂n+1_{; 2}m_≤_{r <}₂m+1

{|trα(t, r)|}, if p= 1

1 2n+m

R2n+1 2n

R2m+1

2m |trα(t, r)| q

1

q

, if p >1.

Theorem 2.1. (i) If f is a linear functional on W2

p, then there is a real number aand a real valued

bivariate functionα, measurable in (1,∞)×(1,∞)such that

f(x) =a`+

Z ∞

1 Z ∞

1

α(t, r)x(t, r)drdt (2.1)

for everyx∈Wp2 and

∞

X

n=0

∞

X

m=0

Mnm(α, p)<∞. (2.2)

(ii) Ifais a real number andαis a real valued bivariate function, measurable in(1,∞)×(1,∞), satisfying

(2.2), then (2.1) defines a linear function on W2 p with

kfk2≤ |a|+ 2

2

p

∞

X

n=0

∞

X

m=0

Mnm(α, p)

and the integral in (2.1) is absolutely convergent for every x∈W2 p.

Proof. LetL2

pbe the linear space of real valued bivariate functionsxmeasurable in (1,∞)×(1,∞) for which Z ∞

1 Z ∞

1

|x(t, r)|p_{drdt <}_∞_,

with norm

kxkL2

p=

Z ∞

1 Z ∞

1

|x(t, r)|pdrdt

p1

.

Clearly, ifx∈L2_p, thenx∈W_p2,`= 0 andkxk2=kxkW2

p ≤ kxkL2p.Consequently the restriction toL 2 pof the

given linear functionalf onW2

p is linear onL2p. It follows from standard results that there is a real valued

bivariate functionα, measurable in (1,∞)×(1,∞), such that

f(x) =

Z ∞

1 Z ∞

1

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for allx∈L2

pand either

ess.sup {|α(t, r)|}<∞ if p= 1

1≤t <∞

1≤r <∞

or

Z ∞

1 Z ∞

1

|α(t, r)|q_{drdt <}_∞ _if_{p >}_1.

To show thatαmust necessarily satisfy (2.2) we consider the casesp= 1 and p >1 separately. If p= 1,

letMnm=Mnm(α,1).There is a measurable setenmof 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.

Thenx∈L2

1 and so, by (2.3),

kfk2kxk2≥f(x) = Z ∞

1 Z ∞

1

x(t, r)α(t, r)drdt

=

s X

n=0 u X

m=0 Z Z

enm

2n+m

|enm|

|α(t, r)|drdt

≥1

4

s X

n=0 u X

m=0

1

|enm| Z Z

enm

|trα(t, r)|drdt (2.4)

≥1

4

s X

n=0 u X

m=0

(Mnm−

1 2n+m).

Furtermore, for 2z_≤_{T <}₂z+1_≤₂s+1_{, 2}h_≤_{R <}₂h+1_≤₂u+1_,

1 T R

Z T

1 Z R

1

|x(t, r)|drdt≤ 1

2z+h Z 2z+1

1

Z 2h+1

1

|x(t, r)|drdt

= 1 2z+h

z X

n=0 h X

m=0 Z Z

enm

x(t, r)|drdt

≤ 1

2z+h z X

n=0 h X

m=0

2n+m<4,

and forT >2s+1_,_{R >}₂u+1

1 T R

Z T

1 Z R

1

|x(t, r)|drdt≤ 1

2s+1₂u+1 Z 2s+1

1

Z 2u+1

1

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Hencekxk2<4 and so, by (2.4),

4kfk2+

1 4 ∞ X n=0 ∞ X m=0 1

2n+m = 4kfk2+ 1≥

1 4 ∞ X n=0 ∞ X m=0

Mnm,

which establishes (2.2) in this case. Ifp >1, letMnm=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_M nm6= 0

0, otherwise.

Thenx∈L2

p and so, by (2.3),

f(x) =

Z 2z+1

1

Z 2u+1

1

|α(t, r)x(t, r)|drdt=

z X n=0 u X m=0 Z 2n+1

2n

Z 2m+1

2m

|α(t, r)x(t, r)|drdt

= z X n=0 u X m=0

Mnm. (2.5)

Furtermore, for 2z_≤_{T <}₂z+1_≤₂s+1_{, 2}h_≤_{R <}₂h+1_≤₂u+1_,

1 T R Z T 1 Z R 1

|x(t, r)|p_drdt_≤ 1

2z+h Z 2z+1

1

Z 2h+1

1

|x(t, r)|p_drdt

= 1 2z+h z X n=0 h X m=0 Z Z enm

|x(t, r)|p_drdt

≤ 2 2p 2z+h z X n=0 h X m=0

2n+m<22p+2,

and forT >2z+1_,_{R >}₂h+1

1 T R Z T 1 Z R 1

|x(t, r)|pdrdt≤ 1

2z+1₂h+1 Z 2z+1

1

Z 2h+1

1

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

Hencekxk2<22+

2

p _{and so, by (}_2.5_),

∞ X n=0 ∞ X m=0

Mnm≤22+

2

pk_fk₂_,

which established (2.2) in this case.

Suppose nowp≥1,Mnm=Mnm(α, p) andx∈Wp2.Then by H¨older inequality

Z ∞

1 Z ∞

1

|α(t, r)x(t, r)|drdt= ∞ X n=0 ∞ X m=0 Z 2n+1

2n

Z 2m+1

2m

|α(t, r)x(t, r)|drdt

≤ ∞ X n=0 ∞ X m=0

Mnm 2p(1−

1

p)(n+m)

Z 2n+1

2n

Z 2m+1

2m x(t, r) tr p drdt

!1p

≤ ∞ X n=0 ∞ X m=0

Mnm 2−(n+m) Z 2n+1

2n

Z 2m+1

2m

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≤22pk_xk₂

∞

X

n=0

∞

X

m=0

Mnm. (2.6)

It follows that

Z ∞

1 Z ∞

1

|α(t, r)x(t, r)|drdt <∞

wheneverx∈Wp2, and in particular since the characteristic function of (1,∞)×(1,∞) is inWp2, that

Z ∞

1 Z ∞

1

|α(t, r)|drdt <∞.

Suppose next thatx∈W2

p 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≥nandr≥m.

Theny∈W2

p,ynm∈L2p and

kynm−yk2= sup T≥n,R≥m

1 T R

Z T

n Z R

m

|x(t, r)−`|p !1_p

=o(1) as n, m→ ∞.

But

|f(ynm−y)|=|f(ynm)−f(y)| ≤ kynm−yk2kfk2,

and so, by (2.3),

f(y) =P− lim

n,m→∞f(ynm) =P−n,mlim→∞

Z n

1 Z m

1

y(t, r)α(t, r)drdt

=

Z ∞

1 Z ∞

1

x(t, r)α(t, r)drdt−`

Z ∞

1 Z ∞

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) =

Z ∞

1 Z ∞

1

x(t, r)α(t, r)drdt+a`

wherea=f(δ)−R∞ 1

R∞

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

(ii) It follows from (2.6) that ifx∈W2

p,`=`x andMnm=Mnm(α, p), then

|f(x)|=

Z ∞

1 Z ∞

1

x(t, r)α(t, r)drdt+a`

≤ kxk22

2

p

∞

X

n=0

∞

X

m=0

Mnm+|a`|. (2.7)

Further, by Minkowski’s inequality

1− 1

T R

1p

|`| ≤ 1

T R

Z T

1 Z R

1

|x(t, r)−`|p_drdt !1_p

+ 1 T R

Z T

1 Z R

1

|x(t, r)|p_drdt !1_p

and the first term on the right hand side iso(1). Hence|`| ≤ kxk2 and consequently, by (2.7),

|f(x)| ≤ kxk2 |a|+ 2

2

p

∞

X

n=0

∞

X

m=0

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for everyx∈W2

p. The additive and homogenous functional f defined by (2.1) is therefore also continuous

onW_p2 and

|f(x)| ≤ |a|+ 22p

∞

X

n=0

∞

X

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 w2

p, then there is a real number a and a real double

sequence α={αnm} such that

f(x) =a`+ ∞

X

n=1

∞

X

m=1

αnmxnm (2.8)

for everyx={xnm} ∈w2p and

∞

X

n=0

∞

X

m=0

mnm(α, p)<∞. (2.9)

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

linear function onw2 p with

kfk2≤ |a|+ 2

2

p

∞

X

n=0

∞

X

m=0

mnm(α, p)

and the series in (2.8) is absolutely convergent for everyx={xnm} ∈wp2.

Proof. Given any real double sequencex={xnm}, define a bivariate functionx∗ by

x∗(t, r) =xnm forn < 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 betweenw_p2and a linear subspace (W_p2)∗of

W2

p such that

`x∗=`_x andkx∗k₂≤ kxk₂≤2 2

pk_x∗k₂_.

Hence given a linear functional onW_p2, the functionalf∗ defined by

f∗(x∗) =f(x)

is linear on (W2

p)∗. Consequently, by the Hahn-Banach theorem and Theorem2.1, there is a real numbera

and a real valued bivariate functionα∗, integrable over (1,∞)×1,∞), such that ∞

X

n=0

∞

X

m=0

Mnm(α∗, p)<∞

and, for everyx∈w2 p,

f(x) =f∗(x∗) =a`x∗+

Z ∞

1 Z ∞

1

α∗(t, r)x∗(t, r)drdt=a`x+

∞

X

n=1

∞

X

m=1

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whereαnm= Rn+1

n

Rm+1

m α

∗_{(t, r)drdt. Furthermore, for}_α₌_{_α

nm},

∞

X

n=0

∞

X

m=0

mnm(α, p)≤

∞

X

n=0

∞

X

m=0

Mnm(α∗, p);

and this completes the proof of (i).

(ii) If x={xnm} ∈ wp2 mnm =mnm(α, p) and`=`x then by H¨older’s and Minkowski’s inequalities, as

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

f(x) =a`+ ∞

X

n=1

∞

X

m=1

αnmxnm≤ |a`|+

∞

X

n=1

∞

X

m=1

|αnmxnm|

≤ |a`|+ 22pkxk₂

∞

X

n=0

∞

X

m=0

mnm≤ kxk2 |a|+ 2

2

p

∞

X

n=0

∞

X

m=0

mnm !

.

The functionalf defined by (2.8) is therefore linear onw2 p,

kfk2≤ |a|+ 2

2

p

∞

X

n=0

∞

X

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 andc0 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.