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 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.

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].

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 New topology in residuated lattices

2018 ◽  
Vol 16 (1) ◽  
pp. 1104-1127 ◽  
Author(s):  
L.C. Holdon
Keyword(s):  
Topological Space ◽  
Hausdorff Space ◽  
Residuated Lattice ◽  
Residuated Lattices ◽  
Discrete Space ◽  
Regular Space ◽  
Quotient Topology ◽  
Uniform Topology ◽  
The Relationship ◽  
Mv Algebras

AbstractIn this paper, by using the notion of upsets in residuated lattices and defining the operator Da(X), for an upset X of a residuated lattice L we construct a new topology denoted by τa and (L, τa) becomes a topological space. We obtain some of the topological aspects of these structures such as connectivity and compactness. We study the properties of upsets in residuated lattices and we establish the relationship between them and filters. O. Zahiri and R. A. Borzooei studied upsets in the case of BL-algebras, their results become particular cases of our theory, many of them work in residuated lattices and for that we offer complete proofs. Moreover, we investigate some properties of the quotient topology on residuated lattices and some classes of semitopological residuated lattices. We give the relationship between two types of quotient topologies τa/F and $\begin{array}{} \displaystyle \mathop {{\tau _a}}\limits^ - \end{array}$. Finally, we study the uniform topology $\begin{array}{} \displaystyle {\tau _{\bar \Lambda }} \end{array}$ and we obtain some conditions under which $\begin{array}{} \displaystyle (L/J,{\tau _{\bar \Lambda }}) \end{array}$ is a Hausdorff space, a discrete space or a regular space ralative to the uniform topology. We discuss briefly the applications of our results on classes of residuated lattices such as divisible residuated lattices, MV-algebras and involutive residuated lattices and we find that any of this subclasses of residuated lattices with respect to these topologies form semitopological algebras.

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].

On the domain of Riesz mean in the space Ls

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 925-940 ◽  
Author(s):  
Medine Yeşilkayagil ◽  
Feyzi Başar
Keyword(s):  
Banach Space ◽  
Sequence Space ◽  
Double Sequence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Double Sequences ◽  
Riesz Mean ◽  
Double Sequence Space ◽  
Necessary And Sufficient ◽  
Barrelled Space

Let 0 < s < ?. In this study, we introduce the double sequence space Rqt(Ls) as the domain of four dimensional Riesz mean Rqt in the space Ls of absolutely s-summable double sequences. Furthermore, we show that Rqt(Ls) is a Banach space and a barrelled space for 1 ? s < 1 and is not a barrelled space for 0 < s < 1. We determine the ?- and ?(?)-duals of the space Ls for 0 < s ? 1 and ?(bp)-dual of the space Rqt(Ls) for 1 < s < 1, where ? ? {p, bp, r}. Finally, we characterize the classes (Ls:Mu), (Ls:Cbp), (Rqt(Ls) : Mu) and (Rqt(Ls):Cbp) of four dimensional matrices in the cases both 0 < s < 1 and 1 ? s < 1 together with corollaries some of them give the necessary and sufficient conditions on a four dimensional matrix in order to transform a Riesz double sequence space into another Riesz double sequence space.

scholarly journals The Hilbert Space of Double Fourier Coefficients for an Abstract Wiener Space

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 389
Author(s):  
Jeong-Gyoo Kim
Keyword(s):  
Hilbert Space ◽  
Fourier Series ◽  
Fourier Coefficients ◽  
Double Sequence ◽  
Intermediate Stage ◽  
Wiener Space ◽  
Abstract Wiener Space ◽  
Abstract Space ◽  
Double Sequences ◽  
Two Parameter

Fourier series is a well-established subject and widely applied in various fields. However, there is much less work on double Fourier coefficients in relation to spaces of general double sequences. We understand the space of double Fourier coefficients as an abstract space of sequences and examine relationships to spaces of general double sequences: p-power summable sequences for p = 1, 2, and the Hilbert space of double sequences. Using uniform convergence in the sense of a Cesàro mean, we verify the inclusion relationships between the four spaces of double sequences; they are nested as proper subsets. The completions of two spaces of them are found to be identical and equal to the largest one. We prove that the two-parameter Wiener space is isomorphic to the space of Cesàro means associated with double Fourier coefficients. Furthermore, we establish that the Hilbert space of double sequence is an abstract Wiener space. We think that the relationships of sequence spaces verified at an intermediate stage in this paper will provide a basis for the structures of those spaces and expect to be developed further as in the spaces of single-indexed sequences.

scholarly journals A Generalization of Strassen’s Theorem on Preordered Semirings

Order ◽  
2021 ◽  
Author(s):  
Péter Vrana
Keyword(s):  
Compact Hausdorff Space ◽  
Polynomial Growth ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Equivalent Condition ◽  
Real Numbers ◽  
Asymptotic Spectrum ◽  
Ring Of Continuous Functions ◽  
Archimedean Property ◽  
Locally Compact Hausdorff Space

AbstractGiven a commutative semiring with a compatible preorder satisfying a version of the Archimedean property, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is an essentially unique compact Hausdorff space together with a map from the semiring to the ring of continuous functions. Strassen’s theorem characterizes an asymptotic relaxation of the preorder that asymptotically compares large powers of the elements up to a subexponential factor as the pointwise partial order of the corresponding functions, realizing the asymptotic spectrum as the space of monotone semiring homomorphisms to the nonnegative real numbers. Such preordered semirings have found applications in complexity theory and information theory. We prove a generalization of this theorem to preordered semirings that satisfy a weaker polynomial growth condition. This weaker hypothesis does not ensure in itself that nonnegative real-valued monotone homomorphisms characterize the (appropriate modification of the) asymptotic preorder. We find a sufficient condition as well as an equivalent condition for this to hold. Under these conditions the asymptotic spectrum is a locally compact Hausdorff space satisfying a similar universal property as in Strassen’s work.

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