Multidimensional matrix characterization of equivalent double sequences

2012 ◽  
Vol 49 (2) ◽  
pp. 269-281
Author(s):  
Richard Patterson ◽  
Ekrem Savaş
Keyword(s):  
Rate Of Convergence ◽  
Double Sequence ◽  
Double Sequences ◽  
Multidimensional Matrix ◽  
Summability Matrix

In 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 that with [x] ∈ P0, and [y] ∈ Pδ for some δ > 0 then µ(Ax) µ(Ay). In addition other implications and variations shall also be presented.

scholarly journals On the Characterization of Some Classes of Four-Dimensional Matrices and Almost B-Summable Double Sequences

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Orhan Tug
Keyword(s):  
Double Sequence ◽  
Sequence Spaces ◽  
Double Sequences ◽  
Matrix Classes ◽  
Related Literature ◽  
Double Sequence Spaces

We firstly summarize the related literature about Br,s,t,u-summability of double sequence spaces and almost Br,s,t,u-summable double sequence spaces. Then we characterize some new matrix classes of Ls′:Cf, BLs′:Cf, and Ls′:BCf of four-dimensional matrices in both cases of 0<s′≤1 and 1<s′<∞, and we complete this work with some significant results.

scholarly journals Four dimensional characterization of bounded double sequences

2004 ◽  
Vol 35 (2) ◽  
pp. 129-134 ◽  
Author(s):  
Richard F. Patterson
Keyword(s):  
Bounded Sequence ◽  
Matrix Transformations ◽  
Double Sequences ◽  
Matrix Methods ◽  
Multidimensional Analog ◽  
Dimensional Matrix ◽  
Important Theorem ◽  
Summability Matrix ◽  
Regular Summability

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. R. G. Cooke suggested that a simpler proof would be desirable. Petersen presented such a proof. The goal of the paper is to present an accessible multidimensional analog of Brudno theorem for double sequences using four dimensional matrix transformations.

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 Regularly ideal invariant convergence of double sequences

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nimet Pancaroǧlu Akın
Keyword(s):  
Double Sequence ◽  
Double Sequences ◽  
The Relationship

AbstractIn this paper, we introduce the notions of regularly invariant convergence, regularly strongly invariant convergence, regularly p-strongly invariant convergence, regularly $(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2})$ ( I σ , I 2 σ ) -convergence, regularly $(\mathcal{I}_{\sigma }^{*},\mathcal{I}^{\sigma *}_{2})$ ( I σ ∗ , I 2 σ ∗ ) -convergence, regularly $(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2} )$ ( I σ , I 2 σ ) -Cauchy double sequence, regularly $(\mathcal{I}_{\sigma }^{*},\mathcal{I}^{\sigma *}_{2})$ ( I σ ∗ , I 2 σ ∗ ) -Cauchy double sequence and investigate the relationship among them.

scholarly journals Convergence Rates in the Implicit Renewal Theorem on Trees

2013 ◽  
Vol 50 (4) ◽  
pp. 1077-1088
Author(s):  
Predrag R. Jelenković ◽  
Mariana Olvera-Cravioto
Keyword(s):  
Fixed Point ◽  
Rate Of Convergence ◽  
Convergence Rates ◽  
Tail Asymptotics ◽  
Renewal Theorem ◽  
Branching Trees

We consider possibly nonlinear distributional fixed-point equations on weighted branching trees, which include the well-known linear branching recursion. In Jelenković and Olvera-Cravioto (2012), an implicit renewal theorem was developed that enables the characterization of the power-tail asymptotics of the solutions to many equations that fall into this category. In this paper we complement the analysis in our 2012 paper to provide the corresponding rate of convergence.

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