Browsing by Author "Patterson, Richard F."

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Lacunary Statistical Convergence of Multiple Sequences
(Pergamon-elsevier Science Ltd, 2006) Savas, Ekrem; Patterson, Richard F.
Quite recently, Mursaleen and Edely [M. Mursaleen, O.H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl. (in press)], defined the statistical analogue for double sequences x = {(xk,1)} as follows: A real double sequence x = {x(k,l)} is said to be P-statistically convergent to L provided that for each epsilon > 0 P - lim(m,n) 1/mn {numbers of (j, k) : j < m and k < n, vertical bar x(j,k) - L vertical bar >=epsilon}. In this paper we introduce and study lacunary statistical convergence for double sequences and we shall also present some inclusion theorems. (c) 2005 Published by Elsevier Ltd.
Multidimensional Matrix Characterizations of the Banach and Pringsheim Core
(Pergamon-elsevier Science Ltd, 2007) Patterson, Richard F.; Savas, Ekrem
In this paper we shall present a multidimensional invariant Pringsheim core theorem. Conditions on a four-dimensional matrix transformation that will ensure that the transformed Pringsheim core of a bounded double sequence [x] is contained in the double Banach core of [x] shall also be presented. (C) 2006 Published by Elsevier Ltd.
Some Σ-Double Sequence Spaces Defined by Orlicz Function
(Academic Press inc Elsevier Science, 2006) Savas, Ekrem; Patterson, Richard F.
In this paper we introduce some new double sequence spaces using the notions of invariant mean and Orlicz function. We also examine some properties of the resulting sequence spaces. (c) 2005 Elsevier Inc. All rights reserved.
Uniformly Summable Double Sequences
(Akademiai Kiado Zrt, 2007) Patterson, Richard F.; Savas, Ekrem
In 1945 Brudno presented the following important theorem: If A and B are regular summability matrix methods such that every bounded sequence summed by A is also summed by B, then it is summed by B to the same value. In 1960 Petersen extended Brudno's theorem by using uniformly summable methods. The goal of this paper is to extend Petersen's theorem to double sequences by using four dimensional matrix transformations and notion of uniformly summable methods for double sequences. In addition to this extension we shall also present an accessible analogue of this theorem.