The purpose of this note is to define and to investigate the generalized Nakano sequence space A(p) and to show that the sequence space A(p) eąuipped with the Luxemburg norm is rotund and posses property-H when p = (pk) is bounded with pk > 1 for all k is an element of N.
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In this paper we study some connections between strong (A,φ)-summability of sequences and lacunary statistical convergence or lacunary strong convergence with respect to a modulus functions.
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For a given space T of all real sequences, non-negative matrix A = (a(nm)) and two sequences of convex Phi - functions (Phi p = ((Phim) and Psi = (Psi/m) we considered two modular spaces of sequences T Phi, and T*GgPsi. This note contains theorems which determinate relationship betveen these spaces.
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The purpose of this paper is to introduce and study an idea of lacunary strong (A,phi)-convergence with respect to a modulus function. In coures of these investigations we study some connections between (A, phi)-strong summability of sequences and lacunary strong convergence with respect to a modulus or lacunary statistical convergence.
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By means of terms of a sequence (pn), where pn, n = l,2,..., are pseudomodulars, and by means of an infinite matrix A = [amn ] of non-negative numbers we shall construct the modular spaces XpAos' and Xp^os. Then we shall approximate elements of these spaces by means of terms of a sequence (p.), where p, i = l,2,..., are pseudomodulars. In particular, we will investigate the special cases when pn and pt are singular integrals.
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