Matrix transformations between certain sequence spaces over the non-Newtonian complex field.

Uğur Kadak,Hakan Efe

doi:10.1155/2014/705818

Abstract

In some cases, the most general linear operator between two sequence spaces is given by an infinite matrix. So the theory of matrix transformations has always been of great interest in the study of sequence spaces. In the present paper, we introduce the matrix transformations in sequence spaces over the field ℂ* and characterize some classes of infinite matrices with respect to the non-Newtonian calculus. Also we give the necessary and sufficient conditions on an infinite matrix transforming one of the classical sets over ℂ* to another one. Furthermore, the concept for sequence-to-sequence and series-to-series methods of summability is given with some illustrated examples.

Highlights

The theory of sequence spaces is the fundamental of summability
We introduce the matrix transformations in sequence spaces over the field C∗ and characterize some classes of infinite matrices with respect to the nonNewtonian calculus
Company and union negotiators had agreed at the beginning of 1981 that, thereafter, the wagerate would be adjusted to reflect changes in the cost of living index

Summary

Introduction

The theory of sequence spaces is the fundamental of summability. Summability is a wide field of mathematics, mainly in analysis and functional analysis, and has many applications, for instance, in numerical analysis to speed up the rate of convergence, in operator theory, the theory of orthogonal series, and approximation theory. The classical summability theory deals with the generalization of the convergence of sequences or series of real or complex numbers. The β-real number xk denotes the value of the function at k ∈ N and is called the kth term of the sequence. Define the binary operations addition (⊕) and multiplication (⊙) of ∗-complex numbers z1∗ = (a1̇ , b1̈ ) and z2∗ = (a2̇ , b2̈ ) as follows:. Following Grossman and Katz [15], we can give the definition of ∗-distance and some applications with respect to the ∗-calculus which is a kind of calculi of non-Newtonian calculus. Is called an infinite non-Newtonian series with β real terms.

Non-Newtonian Infinite Matrices

Conclusion