ABSOLUTE CONVERGENCE WITH SPEED AND MATRIX TRANSFORMS

ABSOLUTE CONVERGENCE WITH SPEED AND MATRIX TRANSFORMS

A. Aasma, P. N. Natarajan

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Abstract

We de ne the notion of absolute convergence with speed, where the speed is de ned by a monotonically increasing positive sequence  . Also we present the notion of absolute  -conservativity of a matrix, and the notion of improvement of  -convergence by a matrix. Let X; Y be two sequence spaces de ned by speeds of convergence. In this paper, we give necessary and su cient conditions for a matrix A (with real or complex entries) to map X into Y , if X or Y is the set of absolutely  -convergent sequences. We also present some examples of matrices being absolutely  -conservative or improving the absolute  -convergence, and consider these problems in the special cases if A is the Riesz matrix (R; pn) or the Zweier matrix Z1=2.

Keywords

Matrix transforms, convergence and absolute convergence with speed, absolute  -conservativity, improvement of  -convergence.