Some multiplier difference sequence spaces defined by a sequence ...

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174 Hemen Dutta1 IntroductionThroughout the paper w, l ∞ , c and c 0 denote the spaces of all, bounded,convergent and null sequences x = (x k ) with complex terms respectively.The zero sequence is denoted by θ = (0, 0, . . . ).The notion of difference sequence space was introduced by Kizmaz [4],who studied the difference sequence spaces l ∞ (∆), c(∆) and c 0 (∆). Thenotion was further generalized by Et and Colak [1] by introducing the spacesl ∞ (∆ n ), c(∆ n ) and c 0 (∆ n ). Another type of generalization of the differencesequence spaces is due to Tripathy and Esi [10], who studied the spacesl ∞ (∆ m ), c(∆ m ) and c 0 (∆ m ). Tripathy, Esi and Tripathy [11] generalizedthe above notions and unified these as follows:haveLet m, n be non-negative integers, then for Z a given sequence space weZ(∆ n m) = {x = (x k ) ∈ w : (∆ n mx k ) ∈ Z},where ∆ n mx = (∆ n mx k ) = (∆ n−1m x k − ∆ n−1m x k+m ) and ∆ 0 mx k = x k for allk ∈ N, which is equivalent to the following binomial representation:∆ n mx k =n∑( ) n(−1) v x k+mv .vv=0Taking m = 1, we get the spaces l ∞ (∆ n ), c(∆ n ) and c 0 (∆ n ) studied by Etand Colak [1]. Taking n = 1, we get the spaces l ∞ (∆ m ), c(∆ m ) and c 0 (∆ m )studied by Tripathy and Esi [10]. Taking m = n = 1, we get the spacesl ∞ (∆), c(∆) and c 0 (∆) introduced and studied by Kizmaz [4].Let Λ = (λ k ) be a sequence of non-zero scalars. Then for E a sequencespace, the multiplier sequence space E(Λ) associated with the multiplier
Some multiplier difference sequence spaces 175sequence Λ is defined asE(Λ) = {(x k ) ∈ w : (λ k x k ) ∈ E}The scope for the studies on sequence spaces was extended by using thenotion of associated multiplier sequences. Goes and Goes [2] defined thedifferentiated sequence space dE and integrated sequence space ∫ E for agiven sequence space E, using the multiplier sequences (k −1 ) and (k) respectively.A multiplier sequence can be used to accelerate the convergence ofthe sequences in some spaces. In some sense, it can be viewed as a catalyst,which is used to accelerate the process of chemical reaction. Sometimes theassociated multiplier sequence delays the rate of convergence of a sequence.A function f : [0, ∞) −→ [0, ∞) is called a modulus if(a) f(x) = 0 if and only if x = 0,(b) f(x + y) ≤ f(x) + f(y), for x ≥ 0, y ≥ 0,(c) f is increasing,(d) f is continuous from the right at 0.Hence f is continuous everywhere in [0, ∞).The following inequality will be used throughout the article. Let p =(p k ) be a positive sequence of real numbers with 0 < p ≤ sup p k = G,D = max(1, 2 G−1 ). Then for all a k , b k ∈ C for all k ∈ N, we have|a k + b k | p k≤ D{|a k | p k+ |b k | p k}and for λ ∈ C,|λ| p k≤ max(1, |λ| G )
Page 1: General Mathematics Vol. 17, No. 3
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