Int. Journal of Math. Analysis, Vol. 8, 2014, no. 26, 1297 - 1304

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http://dx.doi.org/10.12988/ijma.2014.45128

The Rate of Weighted Norlund–Euler Statistical

Convergence

Ekrem Aljimi

Department of Mathematics, University of Tirana, Albania

Copyright © 2014 Ekrem Aljimi. This is an open access article distributed under the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any

medium, provided the original work is properly cited.

Abstract. In this paper we study the rate of weighted Norlund–Euler statistical

convergence. We also prove the regularity of method in theorem1.1.

Mathematics Subject Classification: 41A10; 40A05, 40C05

Keywords: Density; Statistical convergence; Generalized weighted Norlund–Euler

statistical convergence; Sequence spaces, Euler summability

1. BECKGROUND AND PRELIMINARIES

The idea of statistical convergence which is closely related to the concept of natural

density or asymptotic density of a subset of the set of natural numbers N, was first

introduced by Fast [7]. The idea of Norlund –Euler statistical convergence is defined

by E. Aljimi, E. Hoxha and V. Loku [2] and generalizations of these methods are

considered by E. Aljimi and V. Loku [21]. The weighted statistical convergence is

defined by V. Karakaja and T.A.Chishti [3]. A version of the statistically weighted

convergence sequences is given by Mursaleen et al., [16]. The important role in our

results play the rate of weighted Norlund–Euler statistical convergence. The concept

of statistical convergence and the rate of statistical convergence play an important

role in the summability theory and functional analysis. Some results of rate of

1298 Ekrem Aljimi

statistical convergence are studies by Syed Abdul Mohiuddine, Abdullah Alotaibi

and Mohammad Mursaleen [1]. The relationship between the summability theory

and statistical convergence has been introduced by Schoenberg [8]. Afterwards, the

statistical convergence has been studied as a summability method by many

researchers such as Fridy [9], Freedman et al. [10], Kolk [11, 12], Fridy and Miller

[12], Fridy and Orhan [13,14], Mursaleen et al., [15] , Savaş [16], Braha [3-5]. Also,

some topological properties of statistical convergence sequence spaces have been

studied by Salat [18]. Besides in [19, 20] Connor showed the relations between

statistical convergence and functional analysis.

In general, statistical convergence of weighted means is studied as a class of regular

matrix transformations. In this work, we introduce the concept of weighted Norlund-

Euler statistical convergence and the concept of rate of this method.

Let NK and KknkK n : . Then the natural density of K is defined by

n

KK

n

n lim)( if the limit exists, where the vertical bars indicate the number of

elements in the enclosed set. A sequence )( kxx of real numbers is said to be

statistically convergent provided that for every 0 the set

LxKkK n:)( has natural density zero (Fast [6]), for each 0 ,

0:1

lim Lxnjn

jn

In this case we write Lxst lim . Note that every convergent sequence is

statistically convergent but not conversely.

Let us use in consideration the following method of summability:

n

k

kk

kkkn

n

n

k

q

kkkn

n

Eqp

n Sqk

qqp

REqp

Rt

0 00

,,

)1(

111

If St Eqp

n ,, as n , then we say that the series

0n

nx or the sequence nS is

summable to S by generalized Norlund-Euler method and it is denoted by

qEqpNSSn ,,, [21].

Theorem 1.1. The method is regular.

The rate of weighted Norlund–Euler statistical convergence 1299

Proof. The is

kn

kkkkn

n

Eqp

n Sk

qqp

Rt

00

,,

)1(

11

. Also we know

that the method of Euler

Sq

n

qE n

n

n

q

n

0)1(

1 is regular. From regularity of

method 1

nE and definition of regularity we have Sxn

n

0

and

SSqn

qE n

n

n

q

n

0)1(

1. Also

SRR

SqpR

SSqpR

Sk

qqp

Rt n

n

n

k

kkn

n

n

k

kkn

n

kn

kkkkn

n

Eqp

n

111

)1(

11

0000

,,

Since Sxn

n

0

and hence the

St Eqp

n ,, then, it shows that the method

is regular.

Let us denote by qEqpN ,,, the sequence space all strongly convergent

sequences )( kxx which are qEqpN ,,, summable to L :

LsomeforLxqk

qqp

RxxqEqpN

n

k

kk

kkkn

nn

n 0||)1(

11lim:,,,

0 0

The matrix knaA , in qEqpN ,,, - summability is given by

nkif

nkifqk

qqp

Ra

n

k

kk

kkkn

nkn

,0

,)1(

11

0 0,

Now we are able to give the definition of the weighted statistical convergence related

to the qEqpN ,,, – summability method.

1300 Ekrem Aljimi

Definition1.2. A sequence )( kxx is said to be generalized weighted Norlund-

Euler statistical convergent if for every 0 .

0||)1(

1:

1lim

0

Lxqk

qqpRk

R

kk

kkknn

nn

The set of generalized weighted Norlund-Euler statistical convergence sequence is

denoted by q

NES as follows:

LsomeforLxqk

qqpRk

RxxS k

k

kkknn

nn

n

q

NE ,0||)1(

1:

1lim:)(

0

If the sequence )( kxx is q

NES -convergence, then we also use the notation

)( q

NEk SLx .

Definition 1.3. A sequence )( kxx is said to be

)0(],,,[ rsummableqEqpN r to the limit L if

0||)1(

11lim

1 0

rn

k

kk

kkkn

nn

Lxqn

k

qqp

R

, and we write it as

)],,,([ rk qEqpNLx . In this case L is called the rqEqpN ],,,[ limit of x.

Main results

2. Rate of generalized weighted Norlund-Euler statistical convergence

Let RF denote the linear space of all real-valued functions defined on R . Let

be the space of all functions f continuous on . We know that is

Banach spaces with norm

)(sup],[

xffbax

, ],[ baCf .

The rate of weighted Norlund–Euler statistical convergence 1301

In these section, we study the rate of weighted Norlund-Euler statistical convergence

of a sequence of positive linear operators defined from into .

Let be a Banach spaces with the uniform norm ‖ ‖ of all real-valued two

dimensional continuous functions on ; provided that is finite.

Suppose that . We write for ; and we say that

is a positive operator if for all .

Definition 2.1. Let be )( na a positive non-increasing sequence. We say that the

sequence )( nxx is weighted Norlund-Euler statistically convergent to the number

L with the rate )(0 na if for every 0

0||)1(

1:

1lim

0

Lxqk

qqpRk

Rak

kk

kkknn

nnn

In this case we write )(0)( n

q

k astNELx .

As usual we have the following auxiliary result whose proof is standard.

Lema 2.2. Let )( na and )( nb be two positive non-increasing sequence. Let )( nxx

and )( nyy be two sequences such that )(0)(1 n

q

k astNELx and

)(0)(2 n

q

k astNELx . Than

(i) ),(0)()( 1 n

q

k astNELx for any scalar

(ii) )(0)()()( 21 n

q

kk cstNELyLx

(iii) )(0)())(( 21 nn

q

kk bastNELyLx

where nnn bac ,max .

Now, we recall the notion of modulus of continuity. The modulus of continuity of

],[ baCf , denoted by ),( f is defined by

.|)()(|sup),(||

yfxffyx

1302 Ekrem Aljimi

It is well known that

1

||),(|)()(|

yxfyfxf . (2.1.1)

Than we have the following result.

Thorem 2.3. Let )( kT be sequence of positive linear operators from into .

Suppose that

(i) )(0)(1);1( n

q

k astNExT

,

(ii) )(0)(),( n

q

k bstNEf where );( xT xkk and

.)(2xy

x eey

Than for all , we have

)(0)()();( n

q

k cstNExfxfT

Where nnn bac ,max.

Proof. Let , and . Using (2.1.1)

);(),(1

),(1);1(1);1(),(

1);1()(),( );(1

),1(

1);1()(),( ;1

1

1);1()(),( );1(

1);1()();)()(()();(

2

2

2

2

xTffxTfxTf

xTxffxTxT

xTxffxeeT

xTxffxee

T

xTxfxxfyfTxfxfT

xkkk

kxkk

k

xy

k

k

xy

k

kkk

Put .);( xT xkk Hence we get

1);1(),(),( 1);1(

1);1(),(),( 21);1(

)();(

kk

kk

xTffxTK

pxTffxTf

xfxfT

kk

kkk

k

The rate of weighted Norlund–Euler statistical convergence 1303

where 2, max

fK . Hence

1)1(

1);1(

)1(

1),(),( 1

)1(

1);1(

)()1(

1);(

0

0

kk

0

0

kk

kkknk

kk

kkkn

kk

kkknk

kk

kkknk

qk

qqpxT

qk

qqpffq

k

qqpxTK

xfqk

qqpxfT

Now using definition 2.1 and conditions (i) and (ii), we get the desired result.

These complete the proof of the theorem.

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Received: May 3, 2014