- Open Access
Double almost statistical convergence of order α
© Savaş; licensee Springer 2013
Received: 16 November 2012
Accepted: 15 February 2013
Published: 20 March 2013
The goal of this paper is to define and study λ-double almost statistical convergence of order α. Further some inclusion relations are examined. We also introduce a new sequence space by combining the double almost statistical convergence and an Orlicz function.
The notion of statistical convergence was introduced by Fast  and Schoenberg  independently. Over the years and under different names, statistical convergence was discussed in the theory of Fourier analysis, ergodic theory and number theory. Later on it was further investigated from the sequence space point of view and linked with summability theory by Fridy , Connor , S̆alát , Cakalli , Miller , Maddox  and many others. However, Mursaleen  defined the concept of λ-statistical convergence as a new method and found its relation to statistical convergence, -summability and strong -summability. Recently, for , Çolak and Bektaş  have introduced the λ-statistical convergence of order α and strong -summability of order α for sequences of complex numbers.
In this paper we define and study λ-double almost statistical convergence of order α. Also, some inclusion relations have been examined.
Note that if is a finite set, then , and for any set , .
We write in case is st-statistically convergent to L.
Let be the set of all real or complex double sequences. By the convergence of a double sequence we mean the convergence in the Pringsheim sense, that is, the double sequence has a Pringsheim limit L denoted by provided that, given , there exists such that whenever . We will describe such an x more briefly as ‘P-convergent’ (see ).
The notion of almost convergence for single sequences was introduced by Lorentz  and some others.
By we denote the space of strongly almost convergent double sequences. It is easy to see that the inclusions strictly hold.
The notion of strong almost convergence for single sequences has been introduced by Maddox .
if (i.e., for all i, j),
, where with for all i, j and
, where the shift operators , , are defined by , , .
Let be the set of all Banach limits on . A double sequence is said to be almost convergent to a number L if for all (see ).
The idea of statistical convergence was extended to double sequences by Mursaleen and Edely . More recent developments on double sequences can be found in [16–18], where some more references can be found. For the single sequences, statistical convergence of order α and strong p-Cesàro summability of order α was introduced by Çolak . Quite recently, in , Çolak and Bektaş generalized this notion by using de la Valée-Poussin mean.
Let be a two-dimensional set of positive integers and let be the numbers of in K such that and .
(i.e., the set K has double natural density zero).
We now give the following definition.
for every , in which case we say that x is double statistically convergent of order α to L. In this case, we write . The set of all double statistically convergent sequences of order α will be denoted by . If we take in this definition, we can have previous definition.
where . A sequence is said to be -summable to a number L if as .
In  Mursaleen introduced the idea of λ-statistical convergence for a single sequence as follows:
In this case, we write and we denote the set of all λ-statistically convergent sequences by .
A sequence is said to be -summable to a number L, if as in the Pringsheim sense. Throughout this paper, we denote by and (, ) by .
2 Main results
In this section, we define λ-double almost statistically convergent sequences of order α. Also, we prove some inclusion theorems.
We now have the following.
where denotes the α th power of . In case is -statistically convergent of order α to L, we write . We denote the set of all -statistically convergent sequences of order α by . We write if and for .
Therefore is not uniquely determined for .
If we take , the strong -summability of order α reduces to the strong -summability.
We denote the set of all strongly -summable sequences of order α by .
We now are ready to state the following theorem.
Theorem 2.1 If , then .
for every , and finally we have that . This proves the result. □
We have the following from the previous theorem.
If a sequence is -statistically convergent of order α to L, then it is -statistically convergent to L, that is, for each ,
Using (2.1) and taking the limit as , we have . □
Theorem 2.3 Let and r be a positive real number, then .
and we get that . □
We have the following corollary which is a consequence of Theorem 2.3.
If , then .
for each and .
Theorem 2.4 Let α and β be fixed real numbers such that and . If a sequence is a strongly -summable sequence of order α to L, then it is -statistically convergent of order β to L, i.e., .
This shows that if is a strongly -summable sequence of order α to L, then it is -statistically convergent of order β to L. This completes the proof. □
We have the following corollary.
If a sequence is a strongly -summable sequence of order α to L, then it is -statistically convergent of order α to L, i.e., .
3 Some sequence spaces
In present section, we study the inclusion relations between the set of -statistically convergent sequences of order α and strongly -summable sequences of order α with respect to an Orlicz function M.
Recall in  that an Orlicz function is continuous, convex, non-decreasing function such that and for , and as .
An Orlicz function M is said to satisfy -condition for all values of u if there exists such that , .
becomes a Banach space called an Orlicz sequence space. The space is closely related to the space which is an Orlicz sequence space with for .
If , then we say that x is almost strongly double λ-summable of order α with respect to the Orlicz function M.
If , and for all , then .
If for all , then .
If for all and , then .
If , then .
We now have the following theorem.
Theorem 3.1 If and x is almost strongly λ-double convergent to with respect to the Orlicz function M, that is, , then is unique.
The proof of Theorem 3.1 is straightforward. So, we omit it.
In the following theorems, we assume that is bounded and .
Theorem 3.2 Let be real numbers such that and M be an Orlicz function, then .
Since , the left-hand side of the above inequality tends to zero as uniformly in p, q. Hence the right-hand side tends to zero as uniformly in p, q and therefore . This proves the result. □
Corollary 3.1 Let and M be an Orlicz function, then .
We conclude this paper with the following theorem.
Theorem 3.3 Let M be an Orlicz function and be a bounded sequence, then .
Therefore . This proves the result. □
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