Almost convergence of double sequences and strong regularity of summability matrices

1988 ◽  
Vol 104 (2) ◽  
pp. 283-294 ◽  
Author(s):  
F. Móricz ◽  
B. E. Rhoades
Keyword(s):  
Double Sequence ◽  
Almost Convergence ◽  
Strong Regularity ◽  
Double Sequences ◽  
Real Numbers ◽  
Average Value ◽  
Summability Matrices ◽  
Image Position

A double sequence x = {xjk: j, k = 0, 1, …} of real numbers is called almost convergent to a limit s ifthat is, the average value of {xjk} taken over any rectangle {(j, k): m ≤ j ≤ m + p − 1, n ≤ k ≤ n + q − 1} tends to s as both p and q tend to ∞, and this convergence is uniform in m and n. The notion of almost convergence for single sequences was introduced by Lorentz [1].

Almost Convergence and Well-Distributed Sequences

1968 ◽  
Vol 20 ◽  
pp. 1211-1214 ◽  
Author(s):  
Alan Zame
Keyword(s):  
Characteristic Function ◽  
Almost Convergence ◽  
Real Numbers ◽  
Image Position

A sequence (xn) of real numbers is said to be well-distributed modulo 1 (abbreviated w.d.) if for each subinterval I = [a, b] of [0, 1] we have thatwhere χI is the characteristic function of I modulo 1. A sequence (rn) of positive numbers is lacunary if

scholarly journals Double Sequences and Iterated Limits in Regular Space

2016 ◽  
Vol 24 (3) ◽  
pp. 173-186
Author(s):  
Roland Coghetto
Keyword(s):  
Topological Space ◽  
Hausdorff Space ◽  
Cartesian Product ◽  
Double Sequence ◽  
Regular Space ◽  
Double Sequences ◽  
Real Numbers ◽  
Filter Convergence ◽  
Convergence In Pringsheim’S Sense ◽  
Double Limit

Abstract First, we define in Mizar [5], the Cartesian product of two filters bases and the Cartesian product of two filters. After comparing the product of two Fréchet filters on ℕ (F1) with the Fréchet filter on ℕ × ℕ (F2), we compare limF₁ and limF₂ for all double sequences in a non empty topological space. Endou, Okazaki and Shidama formalized in [14] the “convergence in Pringsheim’s sense” for double sequence of real numbers. We show some basic correspondences between the p-convergence and the filter convergence in a topological space. Then we formalize that the double sequence converges in “Pringsheim’s sense” but not in Frechet filter on ℕ × ℕ sense. In the next section, we generalize some definitions: “is convergent in the first coordinate”, “is convergent in the second coordinate”, “the lim in the first coordinate of”, “the lim in the second coordinate of” according to [14], in Hausdorff space. Finally, we generalize two theorems: (3) and (4) from [14] in the case of double sequences and we formalize the “iterated limit” theorem (“Double limit” [7], p. 81, par. 8.5 “Double limite” [6] (TG I,57)), all in regular space. We were inspired by the exercises (2.11.4), (2.17.5) [17] and the corrections B.10 [18].

scholarly journals On Some Lacunary Almost Convergent Double Sequence Spaces and Banach Limits

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Metin Başarır ◽  
Şükran Konca
Keyword(s):  
Double Sequence ◽  
Sequence Spaces ◽  
Almost Convergence ◽  
Double Sequences ◽  
Double Sequence Spaces ◽  
Inclusion Relations ◽  
Banach Limits ◽  
Sublinear Functionals

The object of this paper is to introduce some new sequence spaces related with the concept of lacunary strong almost convergence for double sequences and also to characterize these spaces through sublinear functionals that both dominate and generate Banach limits and to establish some inclusion relations.

On regularly generated double sequences

Filomat ◽  
2017 ◽  
Vol 31 (3) ◽  
pp. 809-822 ◽  
Author(s):  
Ümit Totur ◽  
İbrahim Çanak
Keyword(s):  
Double Sequence ◽  
Summability Method ◽  
Tauberian Theorems ◽  
Double Sequences ◽  
Real Numbers

In this paper, we introduce regularly generated sequences for double sequence of real numbers, and obtain some Tauberian theorems for (C; 1; 1) summability method using the concept of regularly generated sequence.

scholarly journals Almost Conservative Four-Dimensional Matrices through de la Vallée-Poussin Mean

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
S. A. Mohiuddine ◽  
Abdullah Alotaibi
Keyword(s):  
Double Sequence ◽  
Double Series ◽  
Almost Convergence ◽  
Double Sequences ◽  
Infinite Matrices ◽  
De La Vallée Poussin

The purpose of this paper is to generalize the concept of almost convergence for double sequence through the notion of de la Vallée-Poussin mean for double sequences. We also define and characterize the generalized regularly almost conservative and almost coercive four-dimensional matrices. Further, we characterize the infinite matrices which transform the sequence belonging to the space of absolutely convergent double series into the space of generalized almost convergence.

scholarly journals On the Domain of the Four-Dimensional Sequential Band Matrix in Some Double Sequence Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 789
Author(s):  
Orhan Tuğ ◽  
Vladimir Rakočević ◽  
Eberhard Malkowsky
Keyword(s):  
Double Sequence ◽  
Sequence Spaces ◽  
Topological Properties ◽  
Double Sequences ◽  
Real Numbers ◽  
Band Matrix ◽  
Double Sequence Spaces ◽  
Double Sequence Space ◽  
Inclusion Relations ◽  
Matrix Mappings

Let E represent any of the spaces M u , C ϑ ( ϑ = { b , b p , r } ) , and L q ( 0 < q < ∞ ) of bounded, ϑ -convergent, and q-absolutely summable double sequences, respectively, and E ˜ be the domain of the four-dimensional (4D) infinite sequential band matrix B ( r ˜ , s ˜ , t ˜ , u ˜ ) in the double sequence space E, where r ˜ = ( r m ) m = 0 ∞ , s ˜ = ( s m ) m = 0 ∞ , t ˜ = ( t n ) n = 0 ∞ , and u ˜ = ( u n ) n = 0 ∞ are given sequences of real numbers in the set c ∖ c 0 . In this paper, we investigate the double sequence spaces E ˜ . First, we determine some topological properties and prove several inclusion relations under some strict conditions. Then, we examine α -, β ( ϑ ) -, and γ -duals of E ˜ . Finally, we characterize some new classes of 4D matrix mappings related to our new double sequence spaces and conclude the paper with some significant consequences.

scholarly journals Note on selection principles of Kocinac

Filomat ◽  
2012 ◽  
Vol 26 (6) ◽  
pp. 1291-1295
Author(s):  
Dragan Djurcic ◽  
Malisa Zizovic ◽  
Aleksandar Petojevic
Keyword(s):  
Double Sequence ◽  
Double Sequences ◽  
Real Numbers ◽  
Selection Principle ◽  
Selection Principles

This paper investigates ?di, i? {2,3,4}, selection principles (which are modification of known selection principles of Kocinac) on a double sequence of double sequences of real numbers which converge to a point a?R in Pringsheim?s sense. A stronger result than one given in [6] will be proved for the ?d2 selection principle. Also, two more propositions will be proved for the Sd1 and S?1 selection principles, which 11 are also improvements of results given in [6].

scholarly journals Almost convergence of complex uncertain double sequences

Filomat ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 61-78
Author(s):  
Birojit Das ◽  
Binod Tripathy ◽  
Piyali Debnath ◽  
Baby Bhattacharya
Keyword(s):  
Cauchy Sequence ◽  
Double Sequence ◽  
Convergent Sequence ◽  
Uncertain Variable ◽  
Almost Convergence ◽  
Double Sequences ◽  
Uncertain Environment ◽  
Complex Sequences ◽  
Measure Distribution

Convergence of real sequences, as well as complex sequences are studied by B. Liu and X. Chen respectively in uncertain environment. In this treatise, we extend the study of almost convergence by introducing double sequences of complex uncertain variable. Almost convergence with respect to almost surely, mean, measure, distribution and uniformly almost surely are presented and interrelationships among them are studied and depicted in the form of a diagram. We also define almost Cauchy sequence in the same format and establish some results. Conventionally we have, every convergent sequence is a Cauchy sequence and the converse case is not true in general. But taking complex uncertain variable in a double sequence, we find that a complex uncertain double sequence is a almost Cauchy sequence if and only if it is almost convergent. Some suitable examples and counter examples are properly placed to make the paper self sufficient.

Some Inequalities Related to the Wald-Wolfowitz-Noether Condition*

1966 ◽  
Vol 9 (2) ◽  
pp. 161-169
Author(s):  
K. L. Mehra ◽  
J. S. W. Wong
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Double Sequence ◽  
Central Limit Theorems ◽  
Real Numbers ◽  
Image Position

If {avα: or = 1, 2, …, N }, with Nv → ∞ as v → ∞, is a double sequence of real numbers with the property that , then1.1is known in statistical literature as the Wald- Wolfowitz- Noether condition and it plays an important role in the proofs of certain types of central limit theorems (see e. g., [ l ], [2] ).

scholarly journals Summability of double sequences by weighted mean methods and Tauberian conditions for convergence in Pringsheim's sense

2004 ◽  
Vol 2004 (65) ◽  
pp. 3499-3511 ◽  
Author(s):  
Ferenc Móricz ◽  
U. Stadtmüller
Keyword(s):  
Double Sequence ◽  
Complex Numbers ◽  
Double Sequences ◽  
Real Numbers ◽  
Weighted Mean ◽  
Tauberian Conditions ◽  
Slow Decrease ◽  
Necessary And Sufficient ◽  
Convergence In Pringsheim’S Sense

After a brief summary of Tauberian conditions for ordinary sequences of numbers, we consider summability of double sequences of real or complex numbers by weighted mean methods which are not necessarily products of related weighted mean methods in one variable. Our goal is to obtain Tauberian conditions under which convergence of a double sequence follows from its summability, where convergence is understood in Pringsheim's sense. In the case of double sequences of real numbers, we present necessary and sufficient Tauberian conditions, which are so-called one-sided conditions. Corollaries allow these Tauberian conditions to be replaced by Schmidt-type slow decrease conditions. For double sequences of complex numbers, we present necessary and sufficient so-called two-sided Tauberian conditions. In particular, these conditions are satisfied if the summable double sequence is slowly oscillating.

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