A Sequence-to-Function Analogue of the Hausdorff Means for Double Sequences: The [ J, f(x, y)] Means

1975 ◽  
Vol 48 (2) ◽  
pp. 403 ◽  
Author(s):  
Mourad El-Houssieny Ismail
Keyword(s):  
Double Sequences ◽  
Hausdorff Means

The Hausdorff Means for Double Sequences

1967 ◽  
Vol 10 (3) ◽  
pp. 347-352 ◽  
Author(s):  
F. Ustina
Keyword(s):  
Basic Theory ◽  
Dimensional Case ◽  
Double Sequences ◽  
One Dimensional ◽  
The One ◽  
Hausdorff Means

The basic theory of the Hausdorff means for double sequences was developed some thirty - three years ago by C.R. Adams [1], and independently by F. Hallenbach [3], Yet today, many of the properties of these means remain largely uninvestigated. The calculations here, although clearly more complex, for the most part break down into obvious modifications of the calculations in the one dimensional case.

Gibbs phenomenon for the Hausdorff means of double sequences

1970 ◽  
Vol 11 (2) ◽  
pp. 169-185 ◽  
Author(s):  
Fred Ustina
Keyword(s):  
Fourier Series ◽  
Weight Function ◽  
Gibbs Phenomenon ◽  
Partial Sums ◽  
Double Sequences ◽  
Image Position ◽  
Hausdorff Means

let g (u) be a regular Hausdorff weight function, and let hm (Ψ x m) denote the mthe corresponding Hausdorff transform, evaluated at x m, of the sequence of partial sums of the Fourier series of Ψ (x), where . In [3], Szász investigated the Gibbs phenomenon for Ψ(x) for these means. His main results are contained in the following two theorems: . (3) THEOREM 2. Taking the limit superior as. If this maximum is attained for τ = τ′ then.

Classical Tauberian Theorems for the (C; 1; 1) Summability Method

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
Ümit Totur
Keyword(s):  
Tauberian Theorem ◽  
Summability Method ◽  
Subject Classification ◽  
Tauberian Theorems ◽  
Double Sequences ◽  
Littlewood Theorem

Abstract In this paper we generalize some classical Tauberian theorems for single sequences to double sequences. One-sided Tauberian theorem and generalized Littlewood theorem for (C; 1; 1) summability method are given as corollaries of the main results. Mathematics Subject Classification 2010: 40E05, 40G0

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.

Double sequences, almost Cauchyness and BD-N

2011 ◽  
Vol 20 (1) ◽  
pp. 349-354 ◽  
Author(s):  
J. Berger ◽  
D. Bridges ◽  
E. Palmgren
Keyword(s):  
Double Sequences
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