Matrix transformation between sequence spaces and their practical applications

Abstract

The study of sequence spaces was motivated by the classical results of Summability theory in Functional Analysis. The results obtained by Cesaro, Borel, Nörlund and others at the turn of 20th century stimulated interest in general matrix transformation theory which deals with characterization of matrix mappings between sequence spaces by giving necessary and sufficient conditions on the entries of the infinite matrices. The first application of analysis to the theory of Summability was done by Mazur in 1927 when he proved now his famous Mazur’s consistency theorem. An outstanding contribution and plenty of work have been done in the field of sequence spaces in last 50+ years. Kizmaz [41] introduced the concept of difference sequence spaces. The work of Kizmaz was further generalized by Et and Cloak [66], Tripathy and Esi [19], Tripathi, Esi and Tripathi [20], Esi, Tripathy and Sarma [3] and others. In the meantime in constructing new sequence spaces the role of the infinite matrices 𝐺(𝑢, 𝑣) = (𝑔𝑛𝑘 ) = { 𝑢𝑛𝑣𝑘 , 0 ≤ 𝑘 ≤ 𝑛 0, 𝑘 > 𝑛 called generalized weighted mean; Δ= (𝛿𝑛𝑘 ) = { (−1)𝑛−𝑘 , 𝑛 − 1 ≤ 𝑘 ≤ 𝑛 0, 0 ≤ 𝑘 < 𝑛 or 𝑘 > 𝑛 called the difference operator matrix; 𝑆 = (𝑠𝑛𝑘 ) = { 1, 0 ≤ 𝑘 ≤ 𝑛 0, 𝑘 > 𝑛 ; the operator matrix Δ𝑗 which can be expressed as a sequential double band matrix given by and combination of them has been considered to represent difference operator. In this connection we have constructed new matrices 𝑆𝑛 = 𝜆 = (𝜆𝑛𝑘 ) = { 𝑛 − 𝑘 + 1, 𝑛 ≥ 𝑘 0, otherwise which is a lower unitriangular matrix and an operator sparse band matrix 𝜆𝑗 which can be expressed as a sequential double band matrix given by to introduce the new sequence spaces. This thesis consists of six chapters. Chapter one contains introduction with preliminaries and reviews. Chapter two has been divided into two parts. The sequence spaces 𝑤(𝑝), 𝑤0(𝑝) and 𝑤∞(𝑝) were introduced and studied by Maddox [45]. In [12], the authors have introduced the sequence spaces 𝑐0(𝑢, 𝑣; 𝑝), 𝑐(𝑢, 𝑣; 𝑝), 𝑙∞(𝑢, 𝑣; 𝑝) and in [29] 𝑙(𝑢, 𝑣; 𝑝) and established some properties. Following this in the first part of chapter two, we introduce a set of sequence spaces 𝑤(𝑢, 𝑣; 𝑝), 𝑤0(𝑢, 𝑣; 𝑝), 𝑤∞(𝑢, 𝑣; 𝑝) by the application of the generalized weighted mean matrix 𝐺(𝑢, 𝑣) as the operator, study some properties and find β- dual of 𝑤(𝑢, 𝑣; 𝑝) . We also characterize the matrix classes (𝑤(𝑢, 𝑣; 𝑝), 𝑙∞) , (𝑤(𝑢, 𝑣; 𝑝), 𝑐) and (𝑤(𝑢, 𝑣; 𝑝), 𝑐0) . Recently in [78] , the sequence spaces 𝑐0(𝑢, 𝑣; 𝑝, Δ), 𝑐(𝑢, 𝑣; 𝑝, Δ), 𝑙∞(𝑢, 𝑣; 𝑝, Δ) and 𝑙(𝑢, 𝑣; 𝑝, Δ) have been introduced. Following this in the second part of chapter two, we introduce the sequence spaces 𝑤(𝑢, 𝑣; 𝑝, Δ), 𝑤0(𝑢, 𝑣; 𝑝, Δ) and 𝑤∞(𝑢, 𝑣; 𝑝, Δ) by using the combination of the matrix 𝐺(𝑢, 𝑣) and the difference operator matrix 𝛥, study some properties and find β-dual of 𝑤(𝑢, 𝑣; 𝑝, Δ). We also characterize the matrix classes (𝑤(𝑢, 𝑣; 𝑝, Δ), 𝑐), (𝑤(𝑢, 𝑣; 𝑝, Δ), 𝑐0) and (𝑤(𝑢, 𝑣; 𝑝, Δ), Ω(𝑡)). Chapter three has also been divided into two parts. In [15] Choudhary and Mishra have introduced and studied the sequence space 𝑙(𝑝) which is the set of all sequences whose S- transforms are in the space 𝑙(𝑝). Following this in the first part we introduce a new sequence space 𝑙(𝑝, 𝜆) which is the set of all sequences whose 𝑆𝑛 = 𝜆 transforms are in l(𝑝) . We compute β- dual of 𝑙(𝑝, 𝜆) and characterize the matrix classes (𝑙(𝑝, 𝜆), c), (𝑙(𝑝, 𝜆), 𝑐0) and (𝑙(𝑝, 𝜆), 𝑙∞). Similarly in the second part we introduce a set of new paranormed sequence spaces 𝑙∞(𝑝, 𝜆) , 𝑐(𝑝, 𝜆) and 𝑐0(𝑝, 𝜆) which are generated by the infinite matrix 𝜆 . We also compute the basis for the spaces 𝑐(𝑝, 𝜆) and 𝑐0(𝑝, 𝜆) , obtain β- dual of them and characterize the matrix classes (𝑙∞(𝑝, 𝜆), 𝑙∞), (𝑙∞(𝑝, 𝜆), 𝑐) and (𝑙∞(𝑝, 𝜆), 𝑐0) . In Chapter four, we introduce a set of new paranormed sequence spaces 𝑙∞(𝑢, 𝑣; 𝑝, 𝜆𝑗 ) , 𝑐(𝑢, 𝑣; 𝑝, 𝜆𝑗 ) and 𝑐0(𝑢, 𝑣; 𝑝, 𝜆𝑗 ) generated by the combination sparse band matrix 𝜆𝑗 and the generalized weighted mean matrix 𝐺(𝑢, 𝑣) . We establish some topological properties, obtain the basis for 𝑐(𝑢, 𝑣; 𝑝, 𝜆𝑗 ) and 𝑐0(𝑢, 𝑣; 𝑝, 𝜆𝑗 ) and find β- duals. We characterize the matrix classes (𝑙∞(𝑢, 𝑣; 𝑝, 𝜆𝑗 ), 𝑙∞) , (𝑙∞(𝑢, 𝑣; 𝑝, 𝜆𝑗 ), 𝑐) and (𝑙∞(𝑢, 𝑣; 𝑝, 𝜆𝑗 ), 𝑐0) . Besides we give characterization theorem for the case of mapping that guarantees the given rate of convergence from the sequence space 𝑙∞(𝑝) to the new sequence space 𝑙∞(𝑢, 𝑣; 𝑝, 𝜆𝑗 ). In chapter five, we present a practical application of sequence space. In [26], the sequence spaces and function spaces on interval [0, 1] for DNA sequencing have been introduced and studied. The authors have introduced new sequence spaces by using generalized p- summation method and proved that these spaces of sequences and functions are Banach space. Based on the sequence spaces and function spaces on [0,1], we examine the behaviors of sequences generated by DNA nucleotides. We extend the results of authors [26] by introducing new basis function Σ 𝑥𝑘 𝑘! 𝜈 𝑘=1 , (𝜈 = 1,2,3, … 𝑛) which is the extension of existing basis function 𝑥𝑛 𝑛! , (𝑛 ∈ ℕ) defined in the polynomial function on [0,1]. Besides, we introduce a new sequence 𝑏 = (𝑏𝑛) = Σ 𝑎𝜈 ∞𝜈 =𝑛 which can characterize DNA sequence where 𝑎𝑛 ∈ {𝐴, 𝐶, 𝑇, 𝐺} and A: Adenine, C: Cytosine, T: Thymine and G: Guanine are four types of nucleotides. We conclude our thesis by providing conclusions and recommendations in chapter six.