A generalized statistical convergence via ideals

Contents lists available atScienceDirect

Applied Mathematics Letters

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

A generalized statistical convergence via ideals

Ekrem Savas

^a

, Pratulananda Das

^b,∗,1

aIstanbul Ticaret University, Department of Mathematics, Üsküdar-Istanbul, Turkey

bDepartment of Mathematics, Jadavpur University, Kolkata-700032, West Bengal, India

a r t i c l e i n f o

Article history:

Received 10 June 2010 Accepted 21 December 2010

Keywords:

Ideal Filter

I-statistical convergence I−λ-statistical convergence I− [V, λ]-summability Closed subspace

a b s t r a c t

In this paper we make a new approach to the notions of [V, λ]-summability and λ^- statistical convergence by using ideals and introduce new notions, namely, I − [V, λ]^- summability and I−λ-statistical convergence. We mainly examine the relation between these two new methods as also the relation between I −λ-statistical convergence and I-statistical convergence introduced by the authors recently. We carry out the whole investigation in normed linear spaces.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The idea of convergence of a real sequence had been extended to statistical convergence by Fast [1] (see, also [2]) as follows: if N denotes the set of natural numbers and K

⊂

_{N then K}

(

^m

,

ⁿ

)

denotes the cardinality of K

∩ [

_m

,

ⁿ

]

. The upper and the lower natural density of the subset K are defined by

d

(

^K

) =

^limsupn→∞^K(1,n)

n and d

(

^K

) =

^liminfn→∞^K(1,n) n .

If d

(

^K

) =

^d

(

^K

)

then we say that the natural density of K exists and it is denoted simply by d

(

^K

)

. Clearly d

(

^K

) =

lim_n→∞^K(1,n)

n .

A sequence

(

^xn

)

of real numbers is said to be statistically convergent to L if for arbitrary

ε >

0, the set K

(ε) = {

ⁿ

∈

N

:

|

x_n

−

L

| ≥ ε}

has natural density zero. Statistical convergence turned out to be one of the most active areas of research in summability theory after the works of Fridy [3] and Salat [4].

The notion of ideal convergence was introduced first by Kostyrko et al. [5] as a generalization of statistical convergence [1,2] which was further studied in topological spaces [6]. More applications of ideals can be found in [7,6]. In another direction the idea of

λ

-statistical convergence was introduced and studied by Mursaleen [8] as an extension of the

[

V

, λ]

summability of Leindler [9].

λ

-statistical convergence is a special case of more general A-statistical convergence studied by Kolk in [10].

In this note we intend to unify these two approaches and use ideals to introduce the concept of I

− λ

-statistical convergence in line of our recent work [11], and investigate some of its consequences.

Throughout

(

^X

, ‖.‖)

will stand for a real normed linear space and by a sequence x

= (

^xn

)

we shall mean a sequence of elements of X . N will stand for the set of natural numbers.

∗Corresponding author.

E-mail addresses:ekremsavas@yahoo.com(E. Savas),pratulananda@yahoo.co.in(P. Das).

1 This research was done while the second author was a visiting scholar at Istanbul Commerce University, Istanbul, Turkey in 2010.

0893-9659/$ – see front matter©2010 Elsevier Ltd. All rights reserved.

doi:10.1016/j.aml.2010.12.022

2. Main results

A family I

⊂

2^Yof subsets a nonempty set Y is said to be an ideal in Y if

(

ⁱ

) ∅ ∈

^I

; (

ⁱⁱ

)

^A

,

^B

∈

I imply A

∪

B

∈

I

; (

ⁱⁱⁱ

)

^A

∈

I

,

^B

⊂

A imply B

∈

I, while an admissible ideal I of Y further satisfies

{

x

} ∈

I for each x

∈

Y . If I is an ideal in Y then the collection F

(

^I

) = {

^M

⊂

Y

;

M^c

∈

I

}

forms a filter in Y which is called the filter associated with I.

Let I

⊂

2^Nbe a nontrivial ideal in N. The sequence

(

^xn

)

n∈Nin X is said to be I-convergent to x

∈

X , if for each

ε >

^{0 the} set A

(ε) = {

ⁿ

∈

_N

: ‖

x_n

−

x

‖ ≥ ε}

belongs to I [5].

Definition 2.1. A sequence x

= (

^xk

)

is said to be I-statistically convergent to L

∈

X , if for every

ϵ >

0, and every

δ >

⁰

,



n

∈

_N

:

¹

n

|{

_k

≤

_n

: ‖

_xk

−

_L

‖ ≥ ϵ}| ≥ δ



∈

_I

.

For I

=

I_fin

,

I-statistical convergence coincides with statistical convergence.

Let

λ = (λ

n

)

be a non-decreasing sequence of positive numbers tending to

∞

such that

λ

n+1

≤ λ

n

+

1

, λ

1

=

1

.

The collection of such a sequence

λ

will be denoted by∆

.

The generalized de la Valée–Pousin mean is defined by

t_n

(

^x

) = λ

¹n

−

k∈In

x_k

,

where I_n

= [

n

− λ

n

+

1

,

ⁿ

] .

A sequence x

= (

^xk

)

is said to be I

− [

V

, λ]

- summable to L

∈

X , if I

−

lim_nt_n

(

^x

) →

^L

i.e. for any

δ >

⁰

,

{

n

∈

N

: |

t_n

(

^x

) −

^L

| ≥ δ} ∈

^I

.

If I

=

I_fin, I

− [

V

, λ]

-summability becomes

[

V

, λ]

summability [9].

We now introduce our main definition.

Definition 2.2. A sequence x

= (

^xk

)

is said to be I

− λ

-statistically convergent or I

−

S_λconvergent to L, if for every

ϵ >

⁰ and

δ >

⁰

,



n

∈

N

:

¹

λ

n

|{

k

∈

I_n

: ‖

x_k

−

L

‖ ≥ ϵ} ‖≥ δ



∈

I

.

In this case we write I

−

S_λ

−

lim x

=

L or x_k

→

L

(

^I

−

S_λ

)

. We also write I

−

lim

‖

x_k

‖ = ‖

L

‖

. For I

=

I_fin

,

^I

−

S_λ-convergence again coincides with

λ

-statistical convergence.

We shall denote by S

(

^I

),

^Sλ

(

^I

)

^and

[

V

, λ](

^I

)

the collections of all I-statistically convergent, I

−

S_λ-convergent and I

−[

V

, λ]

^- convergent sequences respectively.

Theorem 2.1. Let

λ = (λ

n

) ∈

∆^{. Then}

(i) x_n

→

L

[

V

, λ](

^I

) ⇒

^xk

→

L

(

^Sλ

(

^I

))

and the inclusion

[

V

, λ](

^I

) ⊂

^Sλ

(

^I

)

is proper for every ideal I.

(ii) If x

∈

m

(

^X

)

, the space of all bounded sequences of X and x_k

→

L

(

^Sλ

(

^I

))

^{then x}k

→

L

[

V

, λ](

^I

).

(iii) S_λ

(

^I

) ∩

^m

(

^X

) = [

^V

, λ](

^I

) ∩

^m

(

^X

).

Proof. (i) Let

ϵ >

^{0 and x}k

→

L

[

V

, λ](

^I

)

^{. We have}

−

kϵIn

‖

x_k

−

L

‖ ≥ −

k∈In&‖xk−L‖>ε

‖

x_k

−

L

‖ ≥ ϵ.|{

^k

∈

I_n

: ‖

x_k

−

L

‖ ≥ ϵ}|.

So for a given

δ >

⁰

,

1

λ

n

|{

k

∈

I_n

: ‖

x_k

−

L

‖ ≥ ϵ}| ≥ δ ⇒ λ

¹n

−

kϵIn

‖

x_k

−

L

‖ ≥ ϵδ

i.e.

{

n

∈

N

:

¹

λn

|{

k

∈

I_n

: ‖

x_k

−

L

‖ ≥ ϵ}| ≥ δ} ⊂ {

ⁿ

∈

N

:

¹

λn

{ ∑

kϵ^In

‖

x_k

−

L

‖ ≥ ϵ} ≥ ϵδ}.

Since x_k

→

L

[

V

, λ](

^I

)

, so the set on the right-hand side belongs to I and so it follows that x_k

→

L

(

^Sλ

(

^I

)).

To show that S_λ

(

^I

) [

V

, λ](

^I

)

, take a fixed A

∈

I. Define x

= (

^xk

)

^by x_k

=







ku for n

− [ λ

n

] +

1

≤

k

≤

n

,

ⁿ

̸∈

A ku for n

− λ

n

+

₁

≤

_k

≤

_n

,

ⁿ

∈

_A

θ

^otherwise.

where u

∈

X is a fixed element with

‖

u

‖ =

1, and

θ

is the null element of X . Then x

̸∈

m

(

^X

)

and for every

ϵ >

⁰

(

⁰

< ϵ <

¹

)

since

1

λ

n

|{

k

∈

I_n

: ‖

x_k

−

0

‖ ≥ ϵ}| = [ √ λ

n

] λ

n

→

0 as n

→ ∞

and n

̸∈

A, so for every

δ >

⁰

,



n

∈

N

:

¹

λ

n

|{

k

∈

I_n

: ‖

x_k

−

0

‖ ≥ ϵ}| ≥ δ



⊂

A

∪ {

1

,

²

, . . . ,

^m

}

for some m

∈

N. Since I is admissible, it follows that x_k

→ θ(

^Sλ

(

^I

))

. Obviously

1

λ

n

−

kϵIn

‖

x_k

− θ‖ → ∞ (

ⁿ

→ ∞ )

i.e. x_k9

θ[

^V

, λ](

^I

)

. Note that if A

∈

I is infinite then x_k9

θ(

^Sλ

)

. This example also shows that I

− λ

-statistical convergence is more general than

λ

-statistical convergence.

(ii) Suppose that x_k

→

L

(

^Sλ

(

^I

))

^{and x}

∈

l∞. Let

‖

x_k

−

L

‖ ≤

M

∀

k. Let

ϵ >

0 be given. Now 1

λ

n

−

kϵIn

‖

x_k

−

L

‖ =

¹

λ

n

−

k∈In&‖xk−L‖≥ε

‖

x_k

−

L

‖ +

¹

λ

n

−

k∈In&‖xk−L‖<ε

‖

x_k

−

L

‖

≤

^M

λ

n

|{

k

∈

I_n

: ‖

x_k

−

L

‖ ≥ ϵ}| + ϵ.

Note that

{

n

∈

N

:

_λ¹

n

|{

k

∈

I_n

: ‖

x_k

−

L

‖ ≥ ϵ}| ≥

_M^ϵ

} =

A

(ϵ)

^(say)

∈

I. If n

∈ (

^A

(ϵ))

^cthen 1

λ

n

−

kϵIn

‖

x_k

−

L

‖ <

²

ϵ.

Hence



n

∈

N

:

¹

λ

n

−

kϵIn

‖

x_k

−

L

‖ ≥

2

ϵ



⊂

A

(ϵ)

and so belongs to I. This shows that x_k

→

L

[

V

, λ](

^I

).

(iii) This readily follows from (i) and (ii).  Theorem 2.2. (i) S

(

^I

) ⊂

^Sλ

(

^I

)

^{if lim inf}n→∞λn

n

>

⁰

.

(ii) If lim inf_n→∞ λn

n

=

0

,

I-strongly (by which we mean that

∃

a subsequence

(

ⁿ

(

^j

))

^∞j=1, for which ^λ_nⁿ₍⁽_j^j₎⁾

<

¹_j

∀

j and

{

n

(

^j

) :

^j

∈

N

} ̸∈

I) then S

(

^I

)

& Sλ

(

^I

).

Proof. (i) For given

ϵ >

⁰

,

1

n

|{

k

≤

n

: ‖

x_k

−

L

‖ ≥ ϵ}| ≥

¹

n

|{

k

∈

I_n

: ‖

x_k

−

L

‖ ≥ ϵ}|

≥ λ

n

n 1

λ

n

|{

k

∈

I_n

: ‖

x_k

−

L

‖ ≥ ϵ}|.

İf lim inf_n→∞λn

n

=

a then from definition

{

n

∈

N

:

^λn

n

<

₂^a

}

is finite. For

δ >

⁰

,



n

∈

_N

:

¹

λ

n

|{

_k

∈

_In

: ‖

_xk

−

_L

‖ ≥ ϵ}| ≥ δ



⊂



n

∈

_N

:

¹

n

|{

_k

∈

_In

: ‖

_xk

−

_L

‖ ≥ ϵ}| ≥

^a 2

δ



∪



n

∈

N

: λ

n

n

<

^a 2

 .

Since I is admissible, the set on the right-hand side belongs to I and this completed the proof of (i).

(ii) Define a sequence x

= (

^xj

)

^by x_i

=



u if i

∈

I_n(j)

,

^j

=

1

,

²

, . . . θ

^otherwise.

where as before u

∈

X

, ‖

^u

‖ =

1 and

θ

is the zero element of X . Then x is statistically convergent and so x

∈

S

(

^I

) (

Since I is admissible

)

^{. But x}

̸∈ [

V

, λ](

^I

)

^{and so by}Theorem 2.1(ii) x

̸∈

S_λ

(

^I

).

^

Theorem 2.3. If

λ ∈ △

be such that lim_n^λ_nⁿ

=

1, then S_λ

(

^I

) ⊂

^S

(

^I

).

Proof. Let

δ >

0 be given. Since lim_n^λ_nⁿ

=

1, we can choose m

∈

N such that

|

^λn

n

−

1

| <

₂^δ, for all n

≥

m

.

Now observe that, for

ε >

⁰

1

n

|{

k

≤

n

: ‖

x_k

−

L

‖ ≥ ϵ}| =

¹

n

|{

k

≤

n

− λ

n

: ‖

x_k

−

L

‖ ≥ ϵ}| +

¹

n

|{

k

∈

I_n

: ‖

x_k

−

L

‖ ≥ ϵ}|

≤

ⁿ

− λ

n

n

+

¹

n

|{

k

∈

I_n

: ‖

x_k

−

L

‖ ≥ ϵ}|

≤

1

−



1

− δ

2

 +

¹

n

|{

k

∈

I_n

: ‖

x_k

−

L

‖ ≥ ϵ}|

= δ

2

+

¹

n

|{

k

∈

I_n

: ‖

x_k

−

L

‖ ≥ ϵ}|,

for all n

≥

m

.

^Hence



n

∈

N

:

¹

n

|{

k

≤

n

: ‖

x_k

−

L

‖ ≥ ϵ}| ≥ δ



⊂



n

∈

N

:

¹

n

|{

k

∈

I_n

: ‖

x_k

−

L

‖ ≥ ϵ}| ≥ δ

2



∪ {

1

,

²

,

³

, . . . ,

^m

} .

If I

−

S_λ

−

limx

=

L then the set on the right-hand side belongs to I and so the set on the left-hand side also belongs to I.

This shows that x

= (

^xk

)

is I-statistically convergent to L. 

Remark 1. We do not know whether the condition inTheorem 2.3is necessary and leave it as an open problem.

Remark 2. Taking the sequence

(λ

n

)

^where

λ

n

=

1 for n

=

1 to 10 and

λ

n

=

n

−

10 for all n

≥

10, if we construct the sequence as inTheorem 2.1.(i) and take I

=

I_d(the ideal of density zero sets of N) then for A

= {

1²

,

²²

,

³²

,

⁴²

,

⁵²

, . . .}

^{, the} sequence x

= (

^xk

)

is an example of a sequence which is I-statistically convergent (byTheorem 2.3) but is not statistically convergent.

Theorem 2.4. S_λ

(

^I

) ∩

^m

(

^X

)

is a closed subset of m

(

^X

)

if X is a Banach space.

Proof. Suppose that

(

^xn

) ⊂

^Sλ

(

^I

) ∩

^m

(

^X

)

is a convergent sequence and it converges to x

∈

m

(

^X

)

. We need to show that x

∈

S_λ

(

^I

) ∩

^m

(

^X

)

. Assume that xⁿ

→

L_n

(

^Sλ

(

^I

)) ∀

ⁿ

∈

N. Take a sequence

{ ϵ

n

}

_n∈N of strictly decreasing positive numbers converging to zero. We can find an n

∈

N such that

‖

_x

−

_x^j

‖

_∞

<

^ϵ₄^{n for all j}

≥

n. Choose 0

< δ <

¹₅

.

Now A

=



m

∈

N

:

¹

λ

m









k

∈

I_m

: ‖

xⁿ_k

−

L_n

‖ ≥ ϵ

n

4





 < δ



∈

F

(

^I

)

and

B

=



m

∈

N

:

¹

λ

m









k

∈

I_m

: ‖

xⁿ_k⁺¹

−

L_n+1

‖ ≥ ϵ

n

4





 < δ



∈

F

(

^I

).

Since A

∩

B

∈

F

(

^I

)

^and

∅ ̸∈

F

(

^I

)

, we can choose m

∈

A

∩

B. Then 1

λ

m









k

∈

I_m

: ‖

xⁿ_k

−

L_n

‖ ≥ ϵ

n

4

∨ ‖

xⁿ_k⁺¹

−

L_n+1

‖ ≥ ϵ

n

4





 ≤

2

δ <

¹

.

Since

λ

m

→ ∞

and A

∩

B

∈

F

(

^I

)

is infinite, we can actually choose the above m so that

λ

m

>

5 (say). Hence there must exist a k

∈

I_mfor which we have simultaneously,

‖

xⁿ_k

−

L_n

‖ <

^ϵ₄^{n and}

‖

xⁿ_k⁺¹

−

L_n+1

‖ <

^ϵ₄ⁿ

.

Then it follows that

‖

L_n

−

L_n+1

‖ ≤ ‖

L_n

−

xⁿ_k

‖ + ‖

xⁿ_k

−

xⁿ_k⁺¹

‖ + ‖

xⁿ_k⁺¹

−

L_n+1

‖

≤ ‖

xⁿ_k

−

L_n

‖ + ‖

xⁿ_k⁺¹

−

L_n+1

‖ + ‖

x

−

xⁿ

‖

_∞

+ ‖

x

−

xⁿ⁺¹

‖

_∞

< ϵ

n

4

+ ϵ

n

4

+ ϵ

n

4

+ ϵ

n

4

= ϵ

n

.

This implies that

{

L_n

}

_n∈Nis a Cauchy sequence in X and let L_n

→

L

∈

X as n

→ ∞

. We shall prove that x

→

L

(

^Sλ

(

^I

))

^. Choose

ε >

0 and choose n

∈

N such that

ϵ

n

<

^ϵ₄

, ‖

^x

−

xⁿ

‖

_∞

<

^ϵ₄

, ‖

^Ln

−

L

‖ <

₄^ϵ. Now since

1

λ

γ

|{

k

∈

I_γ

: ‖

x_k

−

L

‖ ≥ ϵ}| ≤

¹

λ

γ

|{

k

∈

I_γ

: ‖

x_k

−

xⁿ_k

‖ + ‖

xⁿ_k

−

L_n

‖ + ‖

L_n

−

L

‖ ≥ ϵ}|

≤

¹

λ

γ









k

∈

I_γ

: ‖

xⁿ_k

−

L_n

‖ ≥ ϵ

2





 ,

it follows that



γ ∈

^N

:

¹

λ

γ

|{

k

∈

I_γ

: ‖

x_k

−

L

‖ ≥ ϵ}| ≥ δ



⊂



γ ∈

^N

:

¹

λ

γ









k

∈

I_γ

: ‖

xⁿ_k

−

L

‖ ≥ ϵ

2





 ≥ δ



for any given

δ >

0. This shows that x

→

L

(

^Sλ

(

^I

))

and this completes the proof of the theorem.  References

[1] H. Fast, Sur la convergence ststistique, Colloq. Math. 2 (1951) 241–244.

[2] I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959) 361–375.

[3] J.A. Fridy, On statistical convergence, Analysis 5 (1985) 301–313.

[4] T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980) 139–150.

[5] P. Kostyrko, T. Salat, W. Wilczynski, I-convergence, Real Anal. Exchange 26 (2) (2000/2001) 669–686.

[6] B.K. Lahiri, Pratulananada Das, I and I^∗-convergence in topological spaces, Math. Bohem. 130 (2005) 153–160.

[7] Pratulananda Das, P. Kostyrko, W. Wilczynski, P. Malik, I and I^∗-convergence of double sequences, Math. Slovaca 58 (5) (2008) 605–620.

[8] M. Mursaleen,λ-statistical convergence, Math. Slovaca 50 (2000) 111–115.

[9] L. Leindler, Über die de la Vallée–Pousnsche Summierbarkeit allge meiner orthogonalreihen, Acta Math. Acad. Sci. Hungarica 16 (1965) 375–387.

[10] E. Kolk, The statistical convergence in Banach spaces, Acta Comment. Univ. Tartu 928 (1991) 41–52.

[11] Pratulananda Das, Ekrem Savas, S. Ghosal, A new approach to certain summability methods using ideal, communicated.