On Some New Double Sequence Spaces Defined by a Modulus

Abstract

In this paper our purpose is to extend some results known in the literature for ordinary (single) sequences to multiply sequences of real and complex numbers. This will be accomplished by presenting the following sequence spaces: {x is an element of s" : P - lim(m,n)Sigma(infinity infinity)(k,l=0,0)a(m,n,k,l)f(vertical bar x(sigma k(p),sigma l(q))vertical bar)(Pk,l) = 0}, {x is an element of s" : P - lim(m,n)Sigma(infinity infinity)(k,l=0,0)a(m,n,k,l)f(vertical bar x(sigma k(p),sigma l(p)) - Le vertical bar)(Pk,l) = 0, for some L}, and {x is an element of s" : sup(m,n,p,q)Sigma(infinity infinity)(k,l=0,0)a(m,n,k,l)f(vertical bar x(sigma(p)sigma l(q))vertical bar)(Pk,l) < infinity} where f is a modulus function, uniformly in (p, q) and A is a nonnegative RH-regular summability matrix method. In addition, we shall give double sigma-statistical convergence. (c) 2006 Elsevier Inc. All rights reserved.