An inclusion theorem for double Cesaro matrices over the space of absolutely k-convergent double series

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Applied Mathematics Letters

journal homepage:www.elsevier.com/locate/aml

An inclusion theorem for double Cesàro matrices over the space of absolutely k-convergent double series

Ekrem Savaş

^a,∗

, B.E. Rhoades

^b

aDepartment of Mathematics, Istanbul Ticaret University, Üsküdar, Istanbul, Turkey

bDepartment of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA

a r t i c l e i n f o

Article history:

Received 15 July 2008

Received in revised form 5 February 2009 Accepted 30 March 2009

Keywords:

Absolute summability factors Double triangular summability

a b s t r a c t

In this work it is shown that if C(α, β)^{and C}(γ , δ)are double Cesàro matrices with α ≥ γ > −¹, β ≥ δ > −1, then any double series that is| C(γ , δ) |k-summable is also|C(α, β) |k-summable, k≥1.

© 2009 Published by Elsevier Ltd

A doubly infinite Cesàro matrix C

(α, β)

is a doubly infinite Hausdorff matrix with entries h_mnij

=



m+α−i−1 m−i

 

n+β−j−1 n−j



m+α α



n+β β

 , α, β ≥

⁰

.

A series

P P

a_mnwith partial sums s_mnis said to be absolutely C

(α, β)

-summable, of order k

≥

1, if

∞

X

m=1

∞

X

n=1

(

^mn

)

^k−1

|

_∆11

σ

m^(α,β)−1,n−1

|

^k

< ∞,

⁽¹⁾

where

σ

mn^(α,β)

:=

¹

m+α α



n+β β



m

X

i=0 n

X

j=0



m

+ α −

ⁱ

−

1 m

−

i

 

n

+ β −

^j

−

1 n

−

j



s_ij

and where, for any double sequence

{

u_mn

}

, and for any fourfold sequence

{

a_mnij

}

, we define

∆11u_mn

=

_umn

−

_um+1,n

−

_um,n+1

+

_um+1,n+1

,

∆11a_mnij

=

a_mnij

−

a_m+1,n,i,j

−

a_m,n+1,i,j

+

a_m+1,n+1,i,j

,

∆ija_mnij

=

a_mnij

−

a_m,n,i+1,j

−

a_m,n,i,j+1

+

a_m,n,i+1,j+1

,

∆i 0a_mnij

=

a_mnij

−

a_m,n,i+1,j

,

^and

∆0ja_mnij

=

a_mnij

−

a_m,n,i,j+1

.

The one-dimensional version of(1)appears in [1].

∗Corresponding author.

E-mail addresses:ekremsavas@yahoo.com,esavas@iticu.edu(E. Savaş),rhoades@indiana.edu(B.E. Rhoades).

0893-9659/$ – see front matter©2009 Published by Elsevier Ltd doi:10.1016/j.aml.2009.03.015

For brevity, let H

=

C

(α, β)

. Associated with H are two matrices H andH defined by

ˆ

h

¯

_mnij

=

m

X

µ=i n

X

ν=j

h_mnµν

,

⁰

≤

i

≤

m

,

⁰

≤

j

≤

n

,

^m

,

ⁿ

=

0

,

¹

, . . . ,

and

h

ˆ

_mnij

=

_∆11h

¯

m−1,n−1,i,j

,

⁰

≤

i

≤

m

,

⁰

≤

j

≤

n

,

^m

,

ⁿ

=

1

,

²

, . . . .

It is easily verified that_h

ˆ

0000

= ¯

h₀₀₀₀

=

h₀₀₀₀. In [2] it was shown that, if A is a double triangular matrix with the row sums in each direction constant (i.e.,

P

m

µ=0a_mnµν

= P

m−1

µ=0a_m−1,nµν

=

a

(

ⁿ

, ν)

^{, and}

P

n

ν=0a_mnµν

= P

n−1

ν=0a_m,n−1,µ,ν

=

b

(

^m

, µ)

^), then

ˆ

a_mnij

=

i−1

X

µ=0 j−1

X

ν=0

∆11a_m−1,n−1,µ,ν

.

⁽²⁾

Using the standard interpretation that the right hand side of(2)is zero whenever i or j is zero, formula(2)remains valid for these choices. (Under the hypotheses that the row sums are constant in each direction, a direct calculation verifies that a

ˆ

_mn0j

= ˆ

a_mni0

= ˆ

a_mn00

=

0.) Eq.(2)is certainly true for double Cesàro matrices, since the row sums in each direction are 1.

Therefore_h

ˆ

mni0

= ˆ

h_mn0j

=

0.

σ

mn^(α,β)

=

m

X

i=0 n

X

j=0

h_mnijs_ij

=

m

X

i=0 n

X

j=0 i

X

µ=0 j

X

ν=0

h_mnija_µν

=

m

X

µ=0 n

X

ν=0 m

X

i=µ n

X

j=ν h_mnija_µν

=

m

X

µ=0 n

X

ν=0

h

¯

_mnµνa_µν

,

and a direct calculation verifies that

X_mn

:=

_∆11

σ

m^(α,β)−1,n−1

=

m

X

i=1 n

X

j=1

h

ˆ

_mnija_ij

,

⁽³⁾

since

h

¯

_m−1,n−1,m,ν

=

h_m−1,n−1,µ,n

= ˆ

h_m,n−1,µ,n

= ˆ

h_m−1,n,m,n

=

0

.

Theorem 1. Let

α ≥ γ > −

¹

, β ≥ δ > −

^{1, and}

P

m

P

na_mnbe a double series with partial sums s_mn. If

P

m

P

na_mnis

|

C

(γ , δ)|

k-summable, then it is also

|

C

(α, β)|

k-summable.

We will require the following lemmas in the proof of the theorem.

Lemma 1.

τ

mn^(α,β)

=

mn∆11

σ

m^(α,β)−1,n−1, where

τ

mn^(α,β)

:=

¹

m+α α



n+β β



m

X

i=0 n

X

j=0



m

+ α −

ⁱ

−

1 m

−

i

 

n

+ β −

^j

−

1 n

−

j



ija_ij

Proof. By analogy to formula (2.1.3) of [1] we may write 1



n+β β



n

X

j=0



n

+ β −

^j

−

1 n

−

j



ja_ij

=

n

(σ

in

− σ

i,n−1

),

where

σ

in

=

¹



n+β β



n

X

j=0



n

+ β −

^j

−

1 n

−

j



j

X

ν=0

a_iν

.

Define b_ij

= P

j

ν=⁰a_νj. Then we may write 1

m+α α

m

X

j=0



m

+ α −

^j

−

1 m

−

j



jb_ij

=

m

( ¯σ

mj

− ¯ σ

m−1,^j

),

where

σ ¯

mj

=

¹

m+α α

m

X

i=0



m

+ α −

ⁱ

−

1 m

−

i



i

X

µ=0

b_µj

=

¹

m+α α

m

X

i=0



m

+ α −

ⁱ

−

1 m

−

i



s_ij

.

Thus

τ

mn^(α,β)

=

ⁿ

m+α α

m

X

i=0



m

+ α −

ⁱ

−

1 m

−

i



i

(σ

in

− σ

i,ⁿ−1

)

=

n





1



n+β β



n

X

j=0



n

+ β −

^j

−

1 n

−

j



1

m+α α

m

X

i=0



m

+ α −

ⁱ

−

1 m

−

i



ib_ij

− 

¹

n+β−1 β



n−1

X

j=0



n

+ β −

^j

−

2 n

−

j

−

1



1

m+α α

m

X

i=0



m

+ α −

ⁱ

−

1 m

−

i



ib_ij





=

mn





1



n+β β



n

X

j=0



n

+ β −

^j

−

1 n

−

j

 (

1

m+α α

m

X

i=0



m

+ α −

ⁱ

−

1 m

−

i



s_ij

− 

¹

m+α−¹ α



m−1

X

i=0



m

+ α −

ⁱ

−

2 m

−

i

−

1



s_ij







− 

¹

n+β−1 β



n−1

X

j=0



n

+ β −

^j

−

2 n

−

j

−

1

 (

1

m+α α

m

X

i=0



m

+ α −

ⁱ

−

1 m

−

i



s_ij

−

¹



m+α−1 α



m−1

X

i=0



m

+ α −

ⁱ

−

2 m

−

i

−

1



s_ij









 =

mn∆11

σ

m^(α,β)−1,ⁿ−1

.



Lemma 2. For any

, η ≥

⁰

, α, β > −

^1,

τ

mn^(α+,β+η)

=

¹

m+α+

α+



n+β+η β+η



m

X

µ=1 n

X

ν=1



m

+  − µ −

¹ m

− µ

 

n

+ η − ν −

¹ n

− ν

  µ + α α

  ν + β β

 τ

_µν^(α,β)

.

Proof. The summations on the right hand side can be written in the form L₁

:=

m

X

µ=1



m

+  − µ −

¹ m

− µ



n

X

ν=1



n

+ η − ν −

¹ n

− ν



µ

X

i=1

X

ν

j=1

 µ + α −

ⁱ

−

1

µ −

ⁱ

  ν + β −

^j

−

1

ν −

^j



ija_ij

=

m

X

i=1 n

X

j=1

ija_ij

m

X

µ=i



m

+  − µ −

¹ m

− µ

  µ + α −

ⁱ

−

1

µ −

ⁱ



n

X

ν=j



n

+ η − ν −

¹ n

− ν

  ν + β −

^j

−

1

ν −

^j

 .

Using identity (2.1.1) from [1],

m

X

µ=i



m

+  − µ −

¹ m

− µ

  µ + α −

ⁱ

−

1

µ −

ⁱ



=

m−i

X

t=0



m

+  −

^t

−

i

−

1 m

−

t

−

i

 

t

+ α −

¹ t



=



m

+ α +  −

ⁱ

−

1 m

−

i

 .

Similarly,

n

X

ν=j



n

+ η − ν −

¹ n

− ν

  ν + β −

^j

−

₁

ν −

^j



=



n

+ β + η −

^j

−

₁ n

−

j



.

Thus L₁

m+α+

α+



n+β+η β+η

 =

¹

m+α+

α+



n+β+η β+η



m

X

i=₁ n

X

j=₁



m

+ α +  −

ⁱ

−

1 m

−

i

 

n

+ β + η −

^j

−

1 n

−

j



ija_ij

= τ

mn^(α+,β+η)

.



To prove the theorem, we shall first prove that

(

_∞

X

m=1

∞

X

n=1

(

^mn

)

⁻¹

| τ

mn^(α+,β+η)

|

^k

)

1/k

≤

A

∞

X

m=1

∞

X

n=1

(

^mn

)

⁻¹

| τ

mn^(α,β)

|

^k

!

1/k

,

⁽⁴⁾

where A is a positive constant depending only on k

, α, β, , η

, and where

, η ≥

⁰

, α, β > −

^1.

Let us have

λ >

1 to be chosen later, and let

λ

⁰denote the conjugate index of

λ

. Using the identity inLemma 2and Hölder’s inequality we have

| τ

mn^(α+,β+η)

| ≤

¹

m+α+

α+



n+β+η β+η



m

X

µ=1 n

X

ν=1



m

+  − µ −

¹ m

− µ

 

n

+ η − ν −

¹ n

− ν



1/λ

×

 µ + α α

  ν + β β



1/k

| τ

_µν^(α,β)

|

×



m

+  − µ −

¹ m

− µ

 

n

+ η − ν −

¹ n

− ν



1/λ⁰

 µ + α α

  ν + β β



1/k0

≤

¹

m+α+

α+



n+β+η β+η





m

X

µ=¹ n

X

ν=¹



m

+  − µ −

¹ m

− µ



×



n

+ η − ν −

¹ n

− ν



k/λ

 µ + α α

  ν + β β



| τ

_µν^(α,β)

|

^k



1/k

×



m

X

µ=1 n

X

ν=1



m

+  − µ −

¹ m

− µ

 

n

+ η − ν −

¹ n

− ν



k0/λ⁰

 µ + α α

  ν + β β



1/k0

.

Now choose

λ

^{so that k0}

( −

¹

)/λ

^{0and k0}

(η −

¹

)/λ

⁰

> −

^{1. Then}

m

X

µ=1 n

X

ν=1



m

+  − µ −

¹ m

− µ

 

n

+ η − ν −

¹ n

− ν



k0/λ⁰

 µ + α α

  ν + β β



≤

Bm^{k0(−1)/λ0+α+1}n^{k0(η−1)/λ0+β+1}

.

Hence

| τ

mn^(α+,β+η)

|

^k

≤

Bm^−σn^−φ

m

X

µ=1 n

X

ν=1



m

+  − µ −

¹ m

− µ

 

n

+ η − ν −

¹ n

− ν



k/λ

 µ + α α

  ν + β β



| τ

_µν^(α,β)

|

^k

,

where

σ = −

^k

( −

¹

)

λ

⁰

−

^k

(α +

¹

)

k⁰

+

k

(α + ) = α +

¹

+

^k

( −

¹

) λ ,

and

φ = β +

¹

+

^k

(η −

¹

) λ .

Hence

1

mn

| τ

mn^(α+,β+η)

|

^k

≤

Bm^−1−σn^−1−φ

m

X

µ=¹ n

X

ν=¹



m

+  − µ −

¹ m

− µ

 

n

+ η − ν −

¹ n

− ν



k(−1)/λ

× µ

^α+1

µ

⁻¹

ν

^β+1

ν

⁻¹

| τ

_µν^(α,β)

|

^k

.

Therefore

M

X

m=1 N

X

n=1

(

^mn

)

⁻¹

| τ

mn^(α+,β+η)

|

^k

≤

B

M

X

µ=1 N

X

ν=1

λ

µν

(µν)

⁻¹

| τ

_µν^(α,β)

|

^k

,

where

λ

µν

= µ

^α+1

ν

^β+1

X

^M

m=µ N

X

n=ν

m^−1−σn^−1−φ



m

+

k

( −

¹

)/λ − µ

m

− µ

 

n

+

k

(η −

¹

)/λ − ν

n

− ν



≤ µ

^α+1

ν

^β+1

∞

X

m=µ

∞

X

n=ν

m^−1−σn^−1−φ



m

+

k

( −

¹

)/λ − µ

m

− µ

 

n

+

k

(η −

¹

)/λ − ν

n

− ν



≤ µ

^α+1

ν

^β+1

∞

X

m=µ

∞

X

n=ν



m

+

k

( −

¹

)/λ − µ

m

− µ

 

n

+

k

(η −

¹

)/λ − ν

n

− ν

 Z

1 0

(

¹

−

x

)

^σxmdx

Z

1 0

(

¹

−

y

)

^φyndy

= µ

^α+1

ν

^β+1

Z

1

0

(

¹

−

x

)

^σxµ−1

(

_∞

X

m=µ



m

+

k

( −

¹

)/λ − µ

m

− µ



x^m−µ

)

dx

× Z

1

0

(

¹

−

y

)

^φyν−1

(

_∞

X

n=ν



n

+

k

(η −

¹

)/λ − ν

n

− ν



y^n−ν

)

dy

≤ µ

^α+1

ν

^β+1

Z

1

0

(

¹

−

x

)

^{σ −k(−1)/λ−1xµ−1dx}

Z

1

0

(

¹

−

y

)

^{φ−k(η−1)/λ−1yν−1dy}

≤

B

µ

^α+1

µ

^{k(−1)/λ−σ}

ν

^β+1

ν

^{k(η−1)/λ−φ}

=

B

,

since

α +

¹

+

k

( −

¹

)/λ − σ =

^{0 and}

β +

¹

+

k

(η −

¹

)/λ − φ =

^0.

To finish the proof, in(4)set

 = γ − α, η = δ − β

^. Define

A

= (

{

s_mn

} :

∞

X

m=1

∞

X

n=1

(

^mn

)

^k−1

|

a_mn

|

^k

< ∞ )

.

If a doubly infinite matrix A sums every double sequence inA, then A is said to be absolutely kth-power conservative.

Corollary 1. For

α, β ≥

^{0, C}

(α, β)

is absolutely kth-power conservative.

Proof. InTheorem 1set

γ = δ =

^0. 

The one-dimensional version ofCorollary 1appears in [1].

Acknowledgements

We wish to thank the referees for their careful reading of the manuscript and for their helpful suggestions. The second author acknowledges support from the Scientific and Technical Research Council of Turkey for the preparation of this work.

References

[1] T.M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London Math. Soc. 7 (1957) 113–141.

[2] Ekrem Savaş, B.E. Rhoades, Double absolute summability factor theorems and applications, Nonlinear Anal. 69 (2008) 189–200.