On two new types of statistical convergence and a summability method

In this paper, we introduce and investigate relationship among r I -statistically convergent, s r I   -statistically convergent and [ , , ] s r I V     summable sequences respectively over normed linear spaces.


Introduction
The idea of convergence of a real sequence had been extended to statistical convergence by Fast (1951) and can also be found in Schoenberg (1959) If N denotes the set of natural numbers and N K  then The upper and the lower natural density of the subset are K defined by: then we say that the natural density of K exists and it is denoted by d(K).Clearly A sequence (x n ) of real numbers is said to be statistically convergent to L if for arbitrary has natural density zero.Statistical convergence turned out to be one of the most active areas of research in summability theory mainly due to Fridy (1985) and Šalàt (1980).
As a generalization of statistical convergence, the notion of ideal convergence was introduced first by Kostyrko et al. (2000Kostyrko et al. ( /2001)).This was further studied in topological spaces by Lahiri and Das (2005) Let r, s be non-negative integers, then for Z a given sequence space we have where:  Et and Çolak (1995).Taking s = 1, we get the spaces Tripathy and Esi (2006).Taking r = s = 1, we get the spaces and   0 c  introduced and studied by Kizmaz (1981).Some other works on difference sequences may be found in Karakaya and Dutta (2011), Tripathy and Dutta (2010), Tripathy and Dutta (2012), and many others.
Recently, Savas and Das (2011) made a new approach to the notions of [ , ] V  -summability and   statistical convergence by using ideals and introduce new notions, namely summability and I  -statistical convergence.In this paper, our intension is to generalize the results of Savas and Das (2011) by considering difference sequences.Throughout ( , . ) X will stand for a real normed linear space and by a sequence x = (x n ) we shall mean a sequence of elements of X. N will stand for the set of natural numbers.

A family
forms a filter in Y which is called the filter associated with I.
Definition 2.1: A sequence The collection of such a sequence  will be denoted by  .
The generalized de la Valée-Pousin mean is defined by In this case we write lim (i) [ , , ]( ) ( ( , )) , the space of all bounded sequences of X and ( ( , )) [ , , ]( ) , so the set on the right hand side belongs to I and so it follows that ( ( , )) (ii) Suppose that ( ( , )) and so belongs to I .

This shows that
[ , , ]( ) (iii) The proof follows from (i) and (ii).Theorem 2.2: , then the following hold ( , ) ( , ) Since I is admissible, the set on the right hand side belongs to I and the proof follows.
Theorem 2.3: If   be such that Proof.Let 0   be given.Since 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 . The proof follows if we can show that ( , ) ( ) using the fact that every bounded sequence is also Assume that ( ( , )) for all n j  . Choose is infinite, we can choose the above m so that for which we have simultaneously, We shall prove that ( ( , ))

Conclusion
The paper defines and studies two types of statistical convergence and a summability method for difference sequences over a normed space.Although we are able to extend some results of Savas and Das (2011), the following further suggestions remain open: Is there other conditions such that Theorem 2.2 holds?Whether the condition in Theorem 2.3 is necessary?
special case of more general  I statistical convergence studied byKolk (1991).The notion of difference sequence space was introduced byKizmaz (1981), who studied the difference sequence spacesTripathy et al. (2005)  generalized the above notions and unified these as follows: of the theorem. and