Summable Double Sequences Research Articles

Tauberian theorems for $$(\overline{N},p,q)$$ summable double sequences of fuzzy numbers

In this paper, we define the weighted mean method $$(\overline{N},p,q)$$ of double sequences of fuzzy numbers and give necessary and sufficient Tauberian conditions under which convergence in Pringsheim’s sense of a double sequence of fuzzy numbers follows from its $$(\overline{N},p,q)$$ summability. These conditions are weaker than the weighted analogues of Landau’s conditions and Schmidt’s slow oscillation condition in some senses for two-dimensional case.

An alternative proof of the generalized Littlewood Tauberian theorem for Cesàro summable double sequences

In this paper, we first examine the relationships between a double sequence and its arithmetic means in different senses (i. e. $(C,1,0)$, $(C,0,1)$ and $(C,1,1)$ means) in terms of slow oscillation in certain senses and investigate some properties of oscillatory behaviors of the difference sequence between the double sequence and its arithmetic means in different senses. Next, we give an alternative proof of the generalized Littlewood Tauberian theorem for Cesàro summability method as an application of the results obtained in the first part.

Characterization of matrix classes involving some sets of summable double sequences of fuzzy numbers

Characterization of matrix classes involving some sets of summable double sequences of fuzzy numbers

On the Characterization of Some Classes of Four-Dimensional Matrices and AlmostB-Summable Double Sequences

We firstly summarize the related literature aboutBr,s,t,u-summability of double sequence spaces and almostBr,s,t,u-summable double sequence spaces. Then we characterize some new matrix classes ofLs′:Cf,BLs′:Cf, andLs′:BCfof four-dimensional matrices in both cases of0<s′≤1and1<s′<∞, and we complete this work with some significant results.

Tauberian theorems for statistically (C,1,1) summable double sequences of fuzzy numbers

Abstract In this paper, we prove that a bounded double sequence of fuzzy numbers which is statistically convergent is also statistically (C, 1, 1) summable to the same number. We construct an example that the converse of this statement is not true in general. We obtain that the statistically (C, 1, 1) summable double sequence of fuzzy numbers is convergent and statistically convergent to the same number under the slowly oscillating and statistically slowly oscillating conditions in certain senses, respectively.

A Tauberian theorem for $(C,1,1)$ summable double sequences of fuzzy numbers

In this paper, we determine necessary and sufficient Tauberian conditions under which convergence in Pringsheim's sense of a double sequence of fuzzy numbers follows from its $(C,1,1)$ summability. These conditions are satisfied if the double sequence of fuzzy numbers is slowly oscillating in different senses. We also construct some interesting double sequences of fuzzy numbers.

Solutions to a dyadic recurrent system and a certain action on [formula omitted] induced by shifts

Let l 2 ( Z ) be the Hilbert space of square summable double sequences of complex numbers with standard basis { e n : n ∈ Z } , and let us consider a bounded matrix A on l 2 ( Z ) satisfying the following system of equations 1. 〈 A e 2 j , e 2 i 〉 = p i j + a 〈 A e j , e i 〉 ; 2. 〈 A e 2 j , e 2 i − 1 〉 = q i j + b 〈 A e j , e i 〉 ; 3. 〈 A e 2 j − 1 , e 2 i 〉 = v i j + c 〈 A e j , e i 〉 ; 4. 〈 A e 2 j − 1 , e 2 i − 1 〉 = w i j + d 〈 A e j , e i 〉 for all i , j , where P = ( p i j ) , Q = ( q i j ) , V = ( v i j ) , W = ( w i j ) are bounded matrices on l 2 ( Z ) and a , b , c , d ∈ C . It is clear that the solutions of the above system of equations introduces a new class of infinite matrices whose entries are related “dyadically”. In this paper, we will show that while the task of constructing these matrices explicitly using purely algebraic methods may appear to be very complicated and tedious, if not impossible, it can be carried out alternatively in a systematical and relatively simple way by applying the theory of Hardy classes of operators through a certain action on B ( H ) (space of bounded operators on H ) induced by a shift.

Uniformly summable double sequences

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.

Tauberian theorems for statistically convergent double sequences

In [F. Moricz, Tauberian theorems for Cesàro summable double sequences, Studia Math. 110 (1994) 83–96], Moricz has proved some Tauberian theorems for Cesàro summable double sequences and deduced the Tauberian theorems of Landau [E. Landau, Über die Bedeutung einiger neuerer Grenzwertsätze der Herren Hardy and Axer, Prac. Mat.-Fiz. 21 (1910) 97–177] and Hardy [G.H. Hardy, Divergent Series, Univ. Press, Oxford 1956] type. In [J.A. Fridy, M.K. Khan, Statistical extension of some classical Tauberian theorems, Proc. Amer. Math. Soc. 128 (2000) 2347–2355], Fridy and Khan have given statistical extensions of some classical Tauberian theorems. The concept of statistical convergence for double sequences has recently been introduced by Mursaleen and Edely [M. Mursaleen, Osama H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl. 288 (2003) 223–231] and by Moricz [F. Moricz, Statistical convergence of multiple sequences, Arch. Math. 81 (2003) 82–89] independently. In this paper we give some Tauberian theorems for statistically convergent double sequences. We have also provided some examples including an example to Problem 1 of Moricz [F. Moricz, Tauberian theorems for Cesàro summable double sequences, Studia Math. 110 (1994) 83–96].

Statistical convergence of double sequences

The idea of statistical convergence was first introduced by Fast (1951) but the rapid developments were started after the papers of Šalát (1980) and Fridy (1985). Now a days it has become one of the most active area of research in the field of summability. In this paper we define and study statistical analogue of convergence and Cauchy for double sequences. We also establish the relation between statistical convergence and strongly Cesàro summable double sequences.

Power-law correlations and other models with long-range dependence on a lattice

This paper introduces long-range dependence for a stationary random field on a plane lattice, derives an exact power-law correlation model and other models with long-range dependence on the lattice, and explores the close connection between short-range dependent correlation functions and absolutely summable double sequences.

Power-law correlations and other models with long-range dependence on a lattice

This paper introduces long-range dependence for a stationary random field on a plane lattice, derives an exact power-law correlation model and other models with long-range dependence on the lattice, and explores the close connection between short-range dependent correlation functions and absolutely summable double sequences.