Uniformly summable double sequences

2007 ◽  
Vol 44 (1) ◽  
pp. 147-158 ◽  
Author(s):  
Richard Patterson ◽  
Ekrem Savaş
Keyword(s):  
Bounded Sequence ◽  
Matrix Transformations ◽  
Double Sequences ◽  
Matrix Methods ◽  
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. 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.

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.

Inclusion sets of regular summability matrices. III

1966 ◽  
Vol 62 (3) ◽  
pp. 389-394 ◽  
Author(s):  
J. W. Baker ◽  
G. M. Petersen
Keyword(s):  
Bounded Sequence ◽  
Finite Set ◽  
Summability Matrices ◽  
Image Position ◽  
Summability Matrix ◽  
Regular Summability ◽  
Bounded Sequences

Let A = (am, n) be a (regular summability) matrix. Then will denote the set of bounded sequences which are summed by A. If {Ai} (i = 1, 2, …, N) is a finite set of such matrices, and if consists of every bounded sequence then we shall say that the matrices span the bounded sequences. Ifx = {xn} belongs to then we denote the value to which A sums x by A-lim x. If y = {yn} is any sequence, then the A-transform of y (if it exists) is the sequence {Aμ(y)}, where

Matrices and norms

1961 ◽  
Vol 57 (2) ◽  
pp. 271-273
Author(s):  
G. M. Petersen
Keyword(s):  
Bounded Sequence ◽  
The Other ◽  
Image Position ◽  
Summability Matrix ◽  
Regular Summability ◽  
Bounded Sequences

We shall define the norm h(A) of a regular summability matrix A = (amn) by Two matrices are said to be b-equivalent if every bounded sequence summable by on matrix is summable by the other. If A sums all bounded sequences that are summable by B, A is said to be b-stronger than B. The norm of a method is defined as , where the inf is taken over all the matrices equivalent to for bounded sequences. These norms have been investigated by Brudno(1). One of his main results is that, if sums all bounded sequences that are summable, then In paper we shall prove the following.

Domain of Cesàro mean of order one in some spaces of double sequences

2014 ◽  
Vol 51 (3) ◽  
pp. 335-356 ◽  
Author(s):  
Mohammad Mursaleen ◽  
Feyzi Başar
Keyword(s):  
Banach Spaces ◽  
Sequence Spaces ◽  
Matrix Transformations ◽  
Double Sequences ◽  
Cesàro Mean ◽  
Dimensional Matrix ◽  
Cesaro Mean

In this study, we define the spaces \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tilde M_u ,\,\tilde C_p ,\,\tilde C_{0p} ,\,\tilde C_{bp} ,\,\tilde C_r \,{\text{and}}\,\tilde L_q$$ \end{document} of double sequences whose Cesàro transforms are bounded, convergent in the Pringsheim’s sense, null in the Pringsheim’s sense, both convergent in the Pringsheim’s sense and bounded, regularly convergent and absolutely q-summable, respectively, and also examine some properties of those sequence spaces. Furthermore, we show that these sequence spaces are Banach spaces. We determine the alpha-dual of the space \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tilde M_u$$ \end{document} and the β(bp)-dual of the space \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tilde C_r$$ \end{document}, and β(ϑ)-dual of the space \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tilde C_\eta$$ \end{document} of double sequences, where ϑ, η ∈ {p, bp, r}. Finally, we characterize the classes (\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tilde C_{bp}$$ \end{document}: Cϑ) and (μ: \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tilde C_\vartheta$$ \end{document}) for ϑ ∈ {p, bp, r} of four dimensional matrix transformations, where μ is any given space of double sequences.

scholarly journals On the paranormed space L(t) of double sequences

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 1043-1053 ◽  
Author(s):  
Hüsamettin Çapan ◽  
Feyzi Başar
Keyword(s):  
Sequence Space ◽  
Matrix Transformations ◽  
Topological Properties ◽  
Double Sequences ◽  
Paranormed Space ◽  
Dimensional Matrix ◽  
Paranormed Sequence Space ◽  
Alpha Beta

In this paper, we introduce the paranormed sequence space L(t) which is the generalization of the space Lq of all absolutely q-summable double sequences. We examine some topological properties of the space L(t) and determine its alpha-, beta- and gamma-duals. Finally, we characterize some classes of four-dimensional matrix transformations from the space L(t) into some spaces of double sequences.

scholarly journals On the paranormed space $\mathcal{M}_{u}(t)$ of double sequences

2017 ◽  
Vol 37 (3) ◽  
pp. 99-111 ◽  
Author(s):  
Feyzi Başar ◽  
Hüsamettin Çapan
Keyword(s):  
Sequence Space ◽  
Normed Space ◽  
Matrix Transformations ◽  
Topological Properties ◽  
Double Sequences ◽  
Paranormed Space ◽  
Dual Spaces ◽  
Dimensional Matrix ◽  
Paranormed Sequence Space ◽  
Alpha Beta

In this paper, we introduce the paranormed sequence space $\mathcal{M}_{u}(t)$ corresponding to the normed space $\mathcal{M}_{u}$ of bounded double sequences. We examine general topological properties of this space and determine its alpha-, beta- and gamma-duals. Furthermore, we characterize some classes of four-dimensional matrix transformations concerning this space and its dual spaces.

scholarly journals On statistical acceleration convergence of double sequences

2017 ◽  
Vol 35 (2) ◽  
pp. 257
Author(s):  
Bipan Hazarika
Keyword(s):  
Matrix Transformations ◽  
Double Sequences ◽  
Dimensional Matrix

In this article the notion of statistical acceleration convergence of double sequences in Pringsheim's sense has been introduced. We prove the decompostion theorems for  statistical acceleration convergence of double sequences and some theorems related to that concept have been established using the four dimensional matrix transformations. We provided some examples, where the results of acceleration convergence fails to hold for the statistical cases.

scholarly journals Rate preservation of double sequences underl-ltype transformation

2002 ◽  
Vol 30 (10) ◽  
pp. 637-643
Author(s):  
Richard F. Patterson
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Double Sequences ◽  
Type Mapping ◽  
Dimensional Matrix ◽  
Necessary And Sufficient

Following the concepts of divergent rate preservation for ordinary sequences, we present a notion of rates preservation of divergent double sequences underl-ltype transformations. Definitions for Pringsheim limit inferior and superior are also presented. These definitions and the notion of asymptotically equivalent double sequences, are used to present necessary and sufficient conditions on the entries of a four-dimensional matrix such that, the rate of divergence is preserved for a given double sequences underl-ltype mapping wherel=:{xk,l:∑k,l=1,1∞,∞|xk,l|<∞}.

scholarly journals On matrix methods of convergence of order α in (ℓ)-groups

Filomat ◽  
2015 ◽  
Vol 29 (9) ◽  
pp. 2069-2077 ◽  
Author(s):  
Antonio Boccuto ◽  
Pratulananda Das
Keyword(s):  
Matrix Method ◽  
Statistical Convergence ◽  
Cauchy Type ◽  
Matrix Methods ◽  
Type Criterion ◽  
Summability Matrix ◽  
Convergent Sequences

We introduce a concept of convergence of order ?, with 0 < ? ? 1, with respect to a summability matrix method A for sequences (which generalizes the notion of statistical convergence of order ?), taking values in (?)-groups. Some main properties and differences with the classical A-convergence are investigated. A Cauchy-type criterion and a closedness result for the space of convergent sequences according our notion is proved.

scholarly journals Gap Tauberian theorems

1993 ◽  
Vol 47 (3) ◽  
pp. 385-393 ◽  
Author(s):  
Jeff Connor
Keyword(s):  
Tauberian Theorem ◽  
Strong Summability ◽  
Tauberian Theorems ◽  
Weighted Mean ◽  
Tauberian Condition ◽  
Support Set ◽  
Tauberian Conditions ◽  
Summability Matrix ◽  
Regular Summability ◽  
Bk Spaces

In the first section we establish a connection between gap Tauberian conditions and isomorphic copies of Co for perfect coregular conservative BK spaces and in the second we give a characterisation of gap Tauberian conditions for strong summability with respect to a nonnnegative regular summability matrix. These results are used to show that a gap Tauberian condition for strong weighted mean summability is also a gap Tauberian condition for ordinary weighted mean summability. We also make a remark regarding the support set of a matrix and give a Tauberian theorem for a class of conull spaces.

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