Some Results of Weighted Norlund-Euler Statistical Convergence

In this paper we will define the new weighted statistically summbaility method, known as the weighted Norlund-Euler statistical convergence. We will show some properties of this method and we have proved Korovkin type theorem. Mathematics Subject Classification: 40G15


INTRODUCTION
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 [3].The weighted statistical convergence is defined by V. Karakaja and T.A.Chishti [2].And the well defined version of the statistically weighted convergence sequences is given by Mursaleen et al., [15].The interesting role in our results play the product of summability methods and we call it Norlund-Euler summability method.The concept of statistical convergence plays an important role in the summability theory and functional analysis.The relationship between the summability theory and statistical convergence has been introduced by Schoenberg [7].Afterwards, the statistical convergence has been studied as a summability method by many researchers such as Fridy [8], Freedman et al. [9], Kolk [10,11], Fridy and Miller [12], Fridy and Orhan [13,14], Mursaleen et al., [15] , Savaş [16], Braha [3][4][5].Also, some topological properties of statistical convergence sequence spaces have been studied by Salat [17].Besides in [18,19], 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 and study the concept of weighted Norlund-Euler statistical convergence.The relations among ( )( ) . Then the natural density of K is defined by δ if the limit exists, where n K denotes the cardinality of n K .
A sequence ) ( k x x = of real numbers is said to be statistically convergent to provided that for every 0 has natural density zero (Fast [5] and Steinhauss [14]), for each 0 x be a given infinite series with sequence of its th And we say that this summability method is convergent if . In this case we say the series -summable to a definite number S.
(Hardy [19]) And we will write ( ) Let ( ) n p and ( ) n q be the two sequences of non-zero real constants such that For the given sequences ( ) n p and ( ) n q , convolution p q * is defined by: x or the sequence { } n S is summable to S by generalized Norlund method and it is denoted by ( )

Ekrem A. Aljimi, Elida Hoxha and Valdete Loku
Let us use in consideration the following method of summability: as n → ∞ , then we say that the series ∑ ∞ =0 n n x or the sequence { } n S is summable to S by Norlund-Euler method and it is denoted by = , then we get Euler summability method Let us denote by ( )( ) the sequence space all strongly convergent sequences ) summable to L: Now we are able to give the definition of the weighted statistical convergence related to the ( )( ) The set of weighted Norlund-Euler statistical convergence sequence is denoted by NE S as follows: is NE S -convergence, then we also use the notation , and we write it as In these paper, we establish the relation of NE S -convergence with statistical summability ( )( )

MAIN RESULTS
In our first theorem we establish the relation between NE S -statistical convergence and statistical summability ( )( ) Example 1: Let us consider that 1 n p = and 1 n q = for all n ∈N.Also we define the following sequence:

Weighted Norlund-Euler statistical convergence 1803
Then we have 0 1 1 These examples proves that converse is not true.

Next theorem gives the relation between NE
S -statistically convergent and is NE S -statistically convergent to L:

Application to approximation theorem
be the space of all functions The classical Korovkin approximation theorem states as follows [26]: Its statistical version was given by Gadjiev and Orhan [25].Such type of approximation theorems are proved by using the concept of almost convergence [2], [6], [21][22][23][24][25][26][27], λ -statistical convergence [29,31] and statistical lacunary summability [30].Boyanov and Veselinov [23] have proved the Korovkin theorem on ) , 0 [ ∞ C by using the test functions x x e e .In addition, some related papers on this topic can be found in [33][34][35].In this paper, we generalize the result of Boyanov and Veselinov by using the notion of statistical summability ( N and the same test functions x x e e .We also give an example to justify that our result is stronger than that of Boyanov and Veselinov [23].

Let
) (I C be the Banach space with the uniform norm ∞ , of all real-valued two ; and we say that L is a positive operator if 0 The following statistical version of Boyanov and Veselinov's result can be obtained as a special case of [24].  .Then there exists a constant 0 is monotone and linear.We obtain

Weighted Norlund-Euler statistical convergence
st