Consistency and Nörlund means

1977 ◽  
Vol 82 (1) ◽  
pp. 107-109 ◽  
Author(s):  
I. J. Maddox
Keyword(s):  
Complex Numbers ◽  
Infinite Matrix ◽  
Unit Matrix ◽  
Nörlund Means ◽  
Summability Field ◽  
Convergent Sequences

Let A = (ank) be an infinite matrix of complex numbers; suppose I is the unit matrix and c is the space of convergent sequences. By (A) we denote the summability field of A, i.e. (A) = {x = (xk): Ax ∈ c}.

Generalized Nörlund means and consistency theorems

1980 ◽  
Vol 87 (2) ◽  
pp. 243-248
Author(s):  
I. J. Maddox
Keyword(s):  
Infinite Matrix ◽  
Nörlund Means ◽  
Summability Field ◽  
General Complex ◽  
Image Position ◽  
Convergent Sequences

It was recently shown in (1), corollary 1, that if (I) = (N, q) then I and (N, q) were consistent. By I we denote the unit infinite matrix, and (N, q) denotes a general complex Nörlund mean. The summability field (A) of any infinite matrix A = (ank) is defined aswhere c is the space of convergent sequences. The summability field of (N, q) is, for simplicity, written as (N, q) instead of ((N, q)).

scholarly journals Generalised regularity and compactness of matrix operators

2020 ◽  
Vol 12 (2) ◽  
pp. 57-66
Author(s):  
E. Malkowsky
Keyword(s):  
Measure Of Noncompactness ◽  
Linear Operators ◽  
Matrix Transformations ◽  
Complex Numbers ◽  
Infinite Matrix ◽  
Regular Matrix ◽  
Hausdorff Measure Of Noncompactness ◽  
Complex Sequences ◽  
Matrix Operators ◽  
Convergent Sequences

A well-known result by Cohen and Dunford ([2], 1937) characterises the class of all regular compact linear operators. It follows that a regular matrix transformation cannot be compact. This means that if c denotes the set of all complex sequences of complex numbers, then an infinite matrix that maps c into c and preserves the limits cannot be compact. We obtained this result in a different way applying the theory of BK spaces from functional analysis and summability, and using the Hausdorff measure of noncompactness. Furthermore, we present the extension of this result to matrix transformations between the spaces c and the spaces of strongly summable sequences by the Cesaro method of order 1, and of strongly convergent sequences. We present new unified proofs for our main results.

Uniform summability of power series

1972 ◽  
Vol 71 (2) ◽  
pp. 335-341 ◽  
Author(s):  
J. C. Kurtz ◽  
W. T. Sledd
Keyword(s):  
Banach Space ◽  
Power Series ◽  
Normal Matrix ◽  
Cesàro Means ◽  
Summability Factors ◽  
Cesaro Means ◽  
Space Topology ◽  
Nörlund Means ◽  
Summability Field ◽  
Cesáro Means

AbstractIt is shown that for the Cesàro means (C, α) with α > - 1, and for a certain class of more general Nörlund means, summability of the series σan implies uniform summability of the series σan zn in a Stolz angle at z = 1.If B is a normal matrix and (B) denotes the series summability field with the usual Banach space topology, then the vectors {ek} (ek = {0,0,..., 1,0,...}) are said to form a Toplitz basis for (B) relative to a method H if H — Σakek = a for each a = {ak}ε(B). It is shown for example that the above relation holds for B = (C,α), α> − 1 , and H = Abel method; also for B = (C,α) and H = (C,β) with 0 ≤ α ≤ β.Applications are made to theorems on summability factors.

scholarly journals Generalized Differenceλ-Sequence Spaces Defined by Ideal Convergence and the Musielak-Orlicz Function

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Awad A. Bakery
Keyword(s):  
Orlicz Function ◽  
Sequence Spaces ◽  
Complex Numbers ◽  
Infinite Matrix ◽  
Topological Properties ◽  
Normed Spaces ◽  
Ideal Convergence ◽  
Inclusion Relations ◽  
Difference Sequence Spaces ◽  
The Ideal

We introduced the ideal convergence of generalized difference sequence spaces combining an infinite matrix of complex numbers with respect toλ-sequences and the Musielak-Orlicz function overn-normed spaces. We also studied some topological properties and inclusion relations between these spaces.

Matrix transformations between some classes of sequences

1970 ◽  
Vol 68 (1) ◽  
pp. 99-104 ◽  
Author(s):  
C. G. Lascarides ◽  
I. J. Maddox
Keyword(s):  
Matrix Transformation ◽  
Matrix Transformations ◽  
Complex Numbers ◽  
Infinite Matrix ◽  
The Matrix ◽  
Complex Sequences ◽  
Class Of Matrices

Let A = (ank) be an infinite matrix of complex numbers ank (n, k = 1, 2,…) and X, Y two subsets of the space s of complex sequences. We say that the matrix A defines a (matrix) transformation from X into Y, and we denote it by writing A: X → Y, if for every sequence x = (xk)∈X the sequence Ax = (An(x)) is in Y, where An(x) = Σankxk and the sum without limits is always taken from k = 1 to k = ∞. The sequence Ax is called the transformation of x by the matrix A. By (X, Y) we denote the class of matrices A such that A: X → Y.

scholarly journals Strongly almost convergent generalized difference sequences associated with multiplier sequences

2007 ◽  
Vol 57 (4) ◽  
Author(s):  
Ayhan Esi ◽  
Binod Tripathy
Keyword(s):  
Statistical Convergence ◽  
Sequence Spaces ◽  
Complex Numbers ◽  
Difference Sequence ◽  
Difference Sequences ◽  
Inclusion Relations ◽  
Difference Sequence Spaces ◽  
Convergent Sequences

AbstractLet Λ = (λk) be a sequence of non-zero complex numbers. In this paper we introduce the strongly almost convergent generalized difference sequence spaces associated with multiplier sequences i.e. w 0[A,Δm,Λ,p], w 1[A,Λm,Λ,p], w ∞[A,Δm,Λ,p] and study their different properties. We also introduce ΔΛm-statistically convergent sequences and give some inclusion relations between w 1[Δm,λ,p] convergence and ΔΛm-statistical convergence.

scholarly journals A Class of Sequences Defined by Weak Ideal Convergence and Musielak-Orlicz Function

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Awad A. Bakery
Keyword(s):  
Orlicz Function ◽  
Sequence Spaces ◽  
Complex Numbers ◽  
Infinite Matrix ◽  
Topological Properties ◽  
Inclusion Relation ◽  
Normed Spaces ◽  
Ideal Convergence

We introduced the weak ideal convergence of new sequence spaces combining an infinite matrix of complex numbers and Musielak-Orlicz function over normed spaces. We also study some topological properties and inclusion relation between these spaces.

scholarly journals Some New Lacunary Strong Convergent Vector-Valued Sequence Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
M. Mursaleen ◽  
A. Alotaibi ◽  
Sunil K. Sharma
Keyword(s):  
Orlicz Function ◽  
Sequence Spaces ◽  
Complex Numbers ◽  
Infinite Matrix ◽  
Topological Properties ◽  
Inclusion Relations ◽  
Vector Valued

We introduce some vector-valued sequence spaces defined by a Musielak-Orlicz function and the concepts of lacunary convergence and strong (A)-convergence, whereA=(aik)is an infinite matrix of complex numbers. We also make an effort to study some topological properties and some inclusion relations between these spaces.

scholarly journals New Definitons about AI-Statistical Convergence with Respect to a Sequence of Modulus Functions and Lacunary Sequences

2018 ◽  
Author(s):  
Ömer Kişi ◽  
Hafize Gümüş ◽  
Ekrem Savaş
Keyword(s):  
Statistical Convergence ◽  
Summability Method ◽  
New Method ◽  
Modulus Function ◽  
Complex Numbers ◽  
Infinite Matrix ◽  
Lacunary Sequences ◽  
Lacunary Statistical Convergence ◽  
Inclusion Relations

In this paper, we generalize the concept of I-statistical convergence, which is a recently introduced summability method, with an infinite matrix of complex numbers and a modulus function. Lacunary sequences will also be included in our definitions. The name of our new method will be A^{I}-lacunary statistical convergence with respect to a sequence of modulus functions. We also define strongly A^{I}-lacunary convergence and we give some inclusion relations between these methods.

STRONGLY (VΛ,A,P) - SUMMABLE SEQUENCE SPACES DEFINED BY A MODULUS

2007 ◽  
Vol 12 (4) ◽  
pp. 419-424 ◽  
Author(s):  
Tunay Bilgin ◽  
Yilmaz Altun
Keyword(s):  
Key Words ◽  
Statistical Convergence ◽  
Sequence Spaces ◽  
Modulus Function ◽  
Complex Numbers ◽  
Infinite Matrix ◽  
Real Numbers ◽  
Positive Real ◽  
Summable Sequence ◽  
De La Vallée Poussin

We introduce the strongly (Vλ,A,p) ‐ summable sequences and give the relation between the spaces of strongly (Vλ,A,p) ‐ summable sequences and strongly (Vλ,A,p) ‐ summable sequences with respect to a modulus function when A = (α ik ) is an infinite matrix of complex numbers and ρ = (pi) is a sequence of positive real numbers. Also we give natural relationship between strongly (Vλ, A,p) ‐ convergence with respect to a modulus function and strongly Sλ (A) ‐ statistical convergence. Key words: De la Vallee‐Poussin mean, modulus function, statistical convergence.

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