Summability matrices and sequences of 0's and 1's

2005 ◽  
Vol 109 (1-2) ◽  
pp. 147-155
Author(s):  
F. Ćunjalo ◽  
M. K. Khan ◽  
H. I. Miller ◽  
J. A. Osikiewicz
Keyword(s):  
Summability Matrices

Inclusion of sets of regular summability matrices

1964 ◽  
Vol 60 (4) ◽  
pp. 705-712 ◽  
Author(s):  
J. W. Baker ◽  
G. M. Petersen
Keyword(s):  
Finite Subset ◽  
Null Sequence ◽  
Fixed Sequence ◽  
Summable Sequence ◽  
The Matrix ◽  
Summability Matrices ◽  
Image Position ◽  
Regular Summability ◽  
Bounded Sequences

1. In this paper we wish to discuss some problems which arise from a paper by Lorentz and Zeller; see (5). If {μn} is a fixed sequence monotonically increasing to infinity, and every sequence {sn} summed by both of the regular matrices A = (amn) and B = (bmn) and satisfying sn = O{μn) is summed to the same value by both matrices, the matrices are called (μn)-consistent. The two matrices are called consistent if they are (μn)-consistent for all {μn}, μn↗∞; they are b-consistent if the bounded sequences summed by both are summed to the same value by both. The matrix A is said to be (μn)-stronger than the matrix B, if every sequence {μn} that is B summable and satisfying sn = O(μn) is also A summable. The matrix A is stronger than B if every B summable sequence is A summable; A is b-stronger if every bounded B summable sequence is A summable. The symbol A -lim x denotes the value to which the sequence x = {xn} is summed by A; Am(x) is the transformationand A(x) is the sequence {Am(x)}. Let {A(i)}i ∈ I be any family, infinite or finite, of regular summability matrices. This family is called simultaneously consistent if, given any finite subset of I, say F, and any set of sequences {x(i)i ∈ F such that A(i) sums x(i) for each i in F, and such that is the null sequence, then .

scholarly journals A characterization of the Banach property for summability matrices

1986 ◽  
Vol 83 (3) ◽  
pp. 263-274 ◽  
Author(s):  
F. Móricz ◽  
K. Tandori
Keyword(s):  
Summability Matrices

Summability matrices and mean-convex sequences

2013 ◽  
Vol 8 ◽  
pp. 941-956
Author(s):  
Chikkanna R. Selvaraj ◽  
Suguna Selvaraj
Keyword(s):  
Summability Matrices ◽  
Mean Convex

scholarly journals Positive linear operators and summability

1970 ◽  
Vol 11 (3) ◽  
pp. 281-290 ◽  
Author(s):  
J. P. King ◽  
J. J. Swetits
Keyword(s):  
Linear Operators ◽  
Positive Linear Operators ◽  
Convergence Properties ◽  
Summability Matrices ◽  
Image Position

Let {Ln} be a sequence of positive linear operators defined on C[a, b] of the form where xnk ∈ [a, b] for each k = 0, 1,…, n = 1, 2,…. The convergence properties of the sequences {Ln(f)} to for each f ∈ C[a, b] have been the object of much recent research (see e.g. [4], [8], [11], [13]). In many cases positive linear operators of the form (1) give rise to interesting summability matrices A = (ank(x)) and vice- versa.

Inclusion of sets of regular summability matrices. II

1965 ◽  
Vol 61 (2) ◽  
pp. 381-394 ◽  
Author(s):  
J. W. Baker ◽  
G. M. Petersen
Keyword(s):  
The Matrix ◽  
Summability Matrices ◽  
Image Position ◽  
Regular Summability

1. In a previous paper, ((1)), we have discussed problems which arise in finding a matrix which is in some sense stronger than each of a set of regular summability matrices. We intend in this paper to clear up other problems in this subject. We shall retain the notation and definitions of (1) throughout, but shall later modify one set of definitions in the light of some of our results. In this paper, the term matrix will be reserved for regular summability matrices. Also, u will be used to denote the unit sequence, u = {un} = {1} and h(A) will be used to denote the norm of the matrix A = (amn), i.e.

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].

scholarly journals Convexity preserving summability matrices

1990 ◽  
Vol 13 (3) ◽  
pp. 501-506
Author(s):  
C. R. Selvaraj
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Summability Matrices ◽  
Necessary And Sufficient ◽  
Cesàro Method

The main result of this paper gives the necessary and sufficient conditions for the Abel matrices to preserve the convexity of sequences. Also, the higher orders of the Cesáro method are shown to be convexity-preserving matrices.

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