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Öğe (A(sigma))(Delta)-DOUBLE SEQUENCE SPACES VIA ORLICZ FUNCTIONS AND DOUBLE STATISTICAL CONVERGENCE(Springer International Publishing Ag, 2007) Savaş, Ekrem; Patterson, Richard F.The aim of this paper is to introduce and study a new concept of strong double (A(sigma))(Delta) convergence sequences with respect to ail Orlicz function, and some properties of the resulting sequence spaces were also examined. In addition, we define the (A(sigma))(Delta)-statistical convergence and establish some connections between the spaces of strong double (A(sigma))(Delta)-convergence sequences and the space of double (A(sigma))(Delta)-statistical convergence.Öğe Asymptotic equivalence of double sequences(2012) Patterson, Richard F.; Savaş, EkremThe goal of this paper is to present a four-dimensional matrix characterization of asymptotic equivalence of double sequences. This will be accomplished with the following notion of asymptotic equivalence of double sequences. Two double sequences are asymptotic equivalent if and only if P − limk,l xk,l yk,l = 1, where x and y are selected judicially. Using this notion necessary and sufficient conditions on the entries of a four-dimensional matrix are given to ensure that the transformation will preserve asymptotic equivalence.Öğe (Aσ)Δ -double sequence spaces via Orlicz functions and double statistical convergence(Shiraz University, 2007) Savaş, Ekrem; Patterson, Richard F.The aim of this paper is to introduce and study a new concept of strong double (A?)? -convergence sequences with respect to an Orlicz function, and some properties of the resulting sequence spaces were also examined. In addition, we define the (A?) ?-statistical convergence and establish some connections between the spaces of strong double(A?)?- convergence sequences and the space of double (A?)? -statistical convergence. © Shiraz University.Öğe A category theorem for double sequences(2011) Patterson, Richard F.; Savaş, EkremThe goal of this paper is to present the following multidimensional category theorem for double sequences via four-dimensional matrix transformations. The set of second category double subsequences of the double sequence (sk,l) is summable by a four-dimensional RH-regular matrix A=(am,n,k,l) if and only if (sk,l) is P-convergent. This theorem is established using multidimensional sliding hump constructions for bounded and unbounded doubles sequences. © 2011 Elsevier Ltd. All rights reserved.Öğe Characterization of asymptotic statistical equivalent double and single sequences(2013) Patterson, Richard F.; Savaş, EkremThis paper presents a series of four-dimensional matrix characterizations of statistically asymptotically equivalent double and single sequences. These characterizations begins with the presentation of definitions for asymptotically equivalent and asymptotically statistical equivalent of multiple L for double and single sequences. The type of questions we provided answers for are those of the following sort. What are the necessary and sufficient conditions on the entries A that ensure the preservation of statistically asymptotically equivalent rate of convergence. © 2013 Elsevier Inc. All rights reserved.Öğe Consistent classes of double summability methods(Elsevier Ltd, 2010) Patterson, Richard F.; Savaş, EkremIn 2000 Patterson proved that if a bounded double sequence is divergent then there are RH-regular matrix methods that sum it to various values. It is now natural to ask the following question. Is there a collection ?. of RH-regular matrix methods which are consistent and such that every bounded double sequence is summable by at least one method in the collection? Similar to Goffman and Petersen's presentation we will present a class of such a collection. In addition, it is clear from the presentation here that it is extremely difficult to find all such collections. However, we have extended this class to a countable collection of RH-regular matrix methods with bounded norm. © 2010 Published by Elsevier Ltd.Öğe Double sequence spaces characterized by lacunary sequences(2007) Savaş, Ekrem; Patterson, Richard F.In 1989, Das and Patel considered known sequence spaces to define two new sequence spaces called lacunary almost convergent and lacunary strongly almost convergent sequence spaces, and proved two inclusion theorems with respect to those spaces. In this paper, we shall extend those spaces to two new double sequence spaces and prove multidimensional analogues of Das and Patel's results. © 2007 Elsevier Ltd. All rights reserved.Öğe Double sequence spaces defined by a modulus(2011) Savaş, Ekrem; Patterson, Richard F.This paper begins with new definitions for double sequence spaces. These new definitions are constructed, in general, by combining modulus function and nonnegative four-dimensional matrix. We use these definitions to establish inclusion theorems between various sequence spaces such as: If A = (am,n,k,l) be a nonnegative four-dimensional matrix such that and let f be a modulus, then ??(A, f) ? ???(A, f) and ??0(A, f) ? ???(A, f). © 2011 Versita Warsaw and Springer-Verlag Wien.Öğe Double sequence spaces defined by Orlicz functions(Shiraz University, 2007) Savaş, Ekrem; Patterson, Richard F.In this paper we introduce some new double sequence spaces using the Orlicz function and examine some properties of the resulting sequence spaces. ©Shiraz University.Öğe (??, ?) - Double sequence spaces via Orlicz function(2008) Savaş, Ekrem; Patterson, Richard F.In this paper we define and study two concepts which arise from the notions of invariant means and de la Valle-Poussin mean namely: strongly double (Ã, a)- convergence defined by Orlicz function and uniform (??, ?-statistical convergence and establish natural characterization for the underline sequence spaces.Öğe DOUBLE sigma-CONVERGENCE LACUNARY STATISTICAL SEQUENCES(Eudoxus Press, LLC, 2009) Savaş, Ekrem; Patterson, Richard F.In this paper we introduce two notions of a-convergence for double sequences namely, a-statistically P-convergence and lacunary or-statistically P-convergence. These concepts are used to present multidimensional inclusion theorems.Öğe Doublesequence transformations that guarantee a given rate of p-convergence(Univ Nis, 2011) Patterson, Richard F.; Savaş, EkremIn this paper the following sequence space is presented. Let [t] be a positive double sequence and define the sequence space Omega '' (t) = {complex sequences x : x(k,l) = O(t(k,l))}. The set of geometrically dominated double sequences is defined as G ''-U(r,s is an element of(0,1))G(r,s) where G (r,s) - {complex sequences x : x(k,l) = O(r(k)s(l))} for each r,s in the interval (0, 1). Using this definition, four dimensional matrix characterizations of l(infinity,infinity), c '', and c(0)'' into G '' and into Omega ''(t) are presented. In addition to these definitions and characterizations it should be noted that this ensure a rate of converges of at least as fast as [t]. Other natural implications will also be presented.Öğe Four-dimensional matrix of geometrically dominated double series(Taylor and Francis Inc., 2015) Patterson, Richard F.; Savaş, EkremThe goal of this article is to present multidimensional matrix characterization of geometrically dominated double sequences. We begin this characterization with the following definition of geometrically dominated factorable double sequence space. Let G' denote a family of double complex number sequences that are dominated by a P-convergence geometric factorable double sequence, that is, This definition will be use to present the theorems similar to the following. The four-dimensional matrix is an G' - l' if and only if and Other implication will also be presented. © 2015 Taylor & Francis Group, LLC.Öğe Lacunary statistical convergence of multiple sequences (vol 19, pg 527, 2006)(Pergamon-Elsevier Science LTD, 2007) Savaş, Ekrem; Patterson, Richard F.[Abstract Not Available]Öğe ((lambda)over-bar, sigma) - Double sequence spaces via Orlicz function(Eudoxus Press, LLC, 2008) Savaş, Ekrem; Patterson, Richard F.In this paper we define and study two concepts which arise from the notions of invariant means and de la Valle-Poussin mean namely: strongly double ((lambda) over bar sigma)- convergence defined by Orlicz function and uniform ((lambda) over bar, sigma)-statistical convergence and establish natural characterization for the underline sequence spaces.Öğe MATRIX CHARACTERIZATION OF P-LIMIT THEOREMS AND COMPLETE P-CONVERGENCE(Ministry Communications & High Technologies Republic Azerbaijan, 2013) Patterson, Richard F.; Savaş, EkremThis paper examines real factorable double sequences of random variables via fourdimensional matrix transformation. These transformations are used to present Pringsheim limit theorems and matrix characterizations of complete P-convergence. To accomplish this, we have, present a four dimensional weighted mean characterization of Sigma(m,n) P{vertical bar U-m,U-n - E(X)vertical bar >= epsilon} < infinity where U-m,U-n Sigma(m,n)(k,l-1,1) a(m.n,k,l)X(k,l,i;) m,n >= 1,1 is the four-dimensional triangular transformation k,1=1,1 of the double sequence [X(k,)l(1)] of random variables. Keywords: P-convergent, P-divergent, RH-regular.Öğe Matrix summability of statistically p-convergence sequences(Univ Nis, 2011) Patterson, Richard F.; Savaş, EkremMatrix summability is arguable the most important tool used to characterize sequence spaces. In 1993 Kolk presented such a characterization for statistically convergent sequence space using nonnegative regular matrix. The goal of this paper is extended Kolk's results to double sequence spaces via four dimensional matrix transformation. To accomplish this goal we begin by presenting the following multidimensional analog of Kolk's Theorem : Let X be a section-closed double sequence space containing e '' and Y an arbitrary se, quence space. Then B is an element of (st(A)(2) boolean AND X, Y) if and only if B is an element of (c '' boolean AND X, Y) and B-[KxK] is an element of (X, Y) (delta(A) (K x K) = 0). In addition, to this result we shall also present implication and variation of this theorem.Öğe Multidimensional matrix characterization of equivalent double sequences(2012) Patterson, Richard F.; Savaş, EkremIn 1936 Hamilton presented a Silverman-Toeplitz type characterization of c?0 (i.e. the space of bounded double Pringsheim null sequences). In this paper we begin with the presentation of a notion of asymptotically statistical regular. Using this definition and the concept of maximum remaining difference for double sequence, we present the following Silverman-Toeplitz type characterization of double statistical rate of convergence: let A be a nonnegative c?0-c?0 summability matrix and let [x] and [y] be member of l? such thatwith [x] P<0, and [y] P< ? for some ? > 0 then ?(Ax)
Öğe On double sequences of continuous functions having continuous P-limits(2013) Patterson, Richard F.; Savaş, EkremThe goal of this paper includes the four-dimensional matrix charac-terization of double sequence of functions. However the main goal is to present answer to the following question. Is it necessarily the case that if sm;n(x) is a bounded for all (m; n) and x with continuous elements and P-converges to a continuous function there exists an RH-regular matrix transformation that maps (sm;n(x)) into a uniformly P-convergent double sequence?Öğe On double sequences of continuous functions having continuous p-limits ıı(Univ Nis, 2013) Patterson, Richard F.; Savaş, EkremThe goal of this paper is to relax the conditions of the following theorem: Let A be a compact closed set; let the double sequence of function s(1,1)(x), s(1,2)(x) s(1,3)(x) ... s(2,1)(x), s(2,2)(x) s(2,3)(x) ... s(3,1)(x), s(3,2)(x) s(3,3)(x) ... have the following properties: 1. for each (m, n) s(m,n)(x) is continuous in A; 2. for each x in A we have P - lim(m,n) s(m,n)(x) = s(x); 3. s(x) is continuous in A; 4. there exists M such that for all (m, n) and all x in A vertical bar s(m,n)(x)vertical bar <= M. Then there exists a T - transformation such that P - lim(m,n) sigma(m,n)(x) = s(x) uniformly in A and to that end we obtain the following. In order that the transformation be such that P - lim(s -> s0(S);t -> t0(T)) sigma(s;t;x) = 0 uniformly with respect x for every double sequence of continuous functions (s(m,n)(x)) define over A such that s(m,n)(x) is bounded over A and for all (m, n) and P - lim(m,n) s(m,n)(x) = 0 over A it is necessary and sufficient that P- lim(s -> s0(S);t -> t0(T)) Sigma(infinity,infinity)(k,l=1,1) vertical bar a(k,l)(s, t)vertical bar = 0
