Matrix transformations of paranormed sequence spaces through de la Vallée-Pousin mean

In this paper we define the sequence space related to the concept of invariant mean and de la Vallée-Pousin mean. We also determine the necessary and sufficient conditions to characterize the matrices which transform paranormed sequence spaces into the spaces and , where denotes the space of all , -convergent sequences.


Definitions, notations and preliminaries
By , we shall denote the space of all real-valued sequences.Any vector subspace of is called a 'sequence space'.If ∈ , then we write instead of x=(x k ) k=0 ∞ .We shall write , ℓ , and for the spaces of all finite, bounded, convergent and null sequences, respectively.Further, we shall use the conventions that e=(1,1,1,…) and is the sequence whose only non-zero term is 1 in the th place for each ∈ , where 0,1,2, … .A sequence space with a linear topology is called a -space if each of the maps : → defined by is continuous for all ∈ .A -space is called an -space if is complete linear metric space; a -space is a normed space.
A linear topological space over the real field is said to be a paranormed space if there is a subadditive function : → such that 0, , and scalar multiplication is continuous, i.e., | | → 0 and → 0 imply → 0 for all 's in and 's in , where is the zero vector in the linear space .Assume here and after that be a sequence such that 0 for all ∈ and be the bounded sequence of strictly positive real numbers with sup and max 1, .Then, the sequence spaces Let be a one-to-one mapping from the set of natural numbers into itself.A continuous linear functional on the space ℓ is said to be an or a σ -mean if and only if (i) 0 if 0 (i.e.0 for all ), (ii) 1, where e=(1,1,1,…), (iii) for all ∈ ℓ .Throughout this paper we consider the mapping which has no finite orbits, that is, for all integer 0 and 1, where denotes the iterate of at .Note that, a -mean extends the limit functional on the space in the sense that lim for all ∈ , (MURSALEEN, 1983).Consequently, ⊂ , the set of bounded sequences all of whose -means are equal.We say that a sequence isif and only if ∈ .Using this concept, Schaefer (1972) defined and characterized -conservative, -regular and -coercive matrices.If is translation then is reduced to the set of almost convergent sequences (LORENTZ, 1948).As an application of almost convergence, Mohiuddine (2011) established some approximation theorems for sequences of positive linear operators through this concept.The idea of -convergence for double sequences was introduced in (ÇAKAN et al., 2006) and further studied recently in (MURSALEEN;MOHIUDDINE, 2007).Çakan et al. (2009), Mohiuddine and Alotaibi (2013;2014), Mursaleen and Mohiuddine (2008;2009b;2010a;2010b;2010c;2012), studied various classes of four dimensional matrices, e.g.

De la
-lim .Let L=lim r→∞ L (r) .Then the -mean of is φ(x)=lim i φ(x (i) ) (since x=lim i x (i) and φ is continuous and linear).Further lim i φ(x (i) )=lim i L (i)  Now define a sequence by where 0 δ 1 and Then it is easy to see that ∈ ℓ and .Applying this sequence to (3.1.2) we get the condition (3.1.1).
This completes the proof of the theorem.
This completes the proof of the theorem.

Conclusion
Two notions -one of -mean and the other of de la Vallée-Pousin meanplay a very active role in recent research on matrix transformations.With the help of these two notions, author has defined the concept of , -bounded sequence, denoted by .He also characterized the matrix classes ℓ , and ℓ , , where ℓ and are defined in Section 1 and Section 2, respectively.