scholarly journals On inclusion relations for absolute summability

2002 ◽  
Vol 32 (3) ◽  
pp. 129-138 ◽  
Author(s):  
B. E. Rhoades ◽  
Ekrem Savaş
Keyword(s):  
Sufficient Conditions ◽  
Triangular Matrix ◽  
Absolute Summability ◽  
Lower Triangular Matrix ◽  
Weighted Mean ◽  
Inclusion Relations ◽  
Lower Triangular

We obtain necessary and (different) sufficient conditions for a series summable|N¯,pn|k,1<k≤s<∞, to imply that the series is summable|T|s, where(N¯,pn)is a weighted mean matrix andTis a lower triangular matrix. As corollaries of this result, we obtain several inclusion theorems.

scholarly journals Matrix transformations of starshaped sequences

2001 ◽  
Vol 28 (4) ◽  
pp. 189-200
Author(s):  
Chikkanna R. Selvaraj ◽  
Suguna Selvaraj
Keyword(s):  
Sufficient Conditions ◽  
Triangular Matrix ◽  
Necessary And Sufficient Conditions ◽  
Matrix Transformations ◽  
Lower Triangular Matrix ◽  
Lower Triangular ◽  
Necessary And Sufficient

We deal with matrix transformations preserving the starshape of sequences. The main result gives the necessary and sufficient conditions for a lower triangular matrixAto preserve the starshape of sequences. Also, we discuss the nature of the mappings of starshaped sequences by some classical matrices.

scholarly journals CONNECTEDNESS OF SELF-AFFINE SETS WITH PRODUCT DIGIT SETS

Fractals ◽  
2017 ◽  
Vol 25 (06) ◽  
pp. 1750053 ◽  
Author(s):  
JING-CHENG LIU ◽  
JUN JASON LUO ◽  
KE TANG
Keyword(s):  
Triangular Matrix ◽  
Sufficient Condition ◽  
Lower Triangular Matrix ◽  
Necessary And Sufficient Condition ◽  
Lower Triangular ◽  
Necessary And Sufficient ◽  
Affine Set

Let [Formula: see text] be an expanding lower triangular matrix and [Formula: see text]. Let [Formula: see text] be the associated self-affine set. In the paper, we generalize some connectedness results on self-affine tiles to self-affine sets and provide a necessary and sufficient condition for [Formula: see text] to be connected.

scholarly journals A summability factor theorem for absolute summability involving almost increasing sequences

2004 ◽  
Vol 2004 (69) ◽  
pp. 3793-3797 ◽  
Author(s):  
B. E. Rhoades ◽  
Ekrem Savaş
Keyword(s):  
Sufficient Conditions ◽  
Triangular Matrix ◽  
Absolute Summability ◽  
Summability Factor ◽  
Factor Theorem ◽  
Increasing Sequences ◽  
Almost Increasing Sequences

We obtain sufficient conditions for the series∑anλnto be absolutely summable of orderkby a triangular matrix.

scholarly journals The inverse along a lower triangular matrix

2012 ◽  
Vol 219 (3) ◽  
pp. 886-891 ◽  
Author(s):  
Xavier Mary ◽  
Pedro Patrício
Keyword(s):  
Triangular Matrix ◽  
Lower Triangular Matrix ◽  
Lower Triangular

scholarly journals Elementary triangular matrices and inverses of k-Hessenberg and triangular matrices

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Luis Verde-Star
Keyword(s):  
Triangular Matrix ◽  
Identity Matrix ◽  
Lower Triangular Matrix ◽  
Triangular Matrices ◽  
Hessenberg Matrices ◽  
Banded Matrices ◽  
Lower Triangular ◽  
And Inversion ◽  
Block Hessenberg Matrices ◽  
Explicit Expressions

AbstractWe use elementary triangular matrices to obtain some factorization, multiplication, and inversion properties of triangular matrices. We also obtain explicit expressions for the inverses of strict k-Hessenberg matrices and banded matrices. Our results can be extended to the cases of block triangular and block Hessenberg matrices. An n × n lower triangular matrix is called elementary if it is of the form I + C, where I is the identity matrix and C is lower triangular and has all of its nonzero entries in the k-th column,where 1 ≤ k ≤ n.

An Inclusion Theorem for Bohr-Hardy Summability Factors

1971 ◽  
Vol 23 (4) ◽  
pp. 653-658 ◽  
Author(s):  
B. Thorpe
Keyword(s):  
Convergent Sequence ◽  
Triangular Matrix ◽  
Partial Sums ◽  
Its Sequence ◽  
Summability Factors ◽  
Lower Triangular Matrix ◽  
Inclusion Theorem ◽  
Sequence Transformation ◽  
Lower Triangular ◽  
Regular Transformation

1. Let A denote a sequence to sequence transformation given by the normal matrix A = (ank)(n, k = 0, 1, 2, …), i.e., a lower triangular matrix with ann ≠ 0 for all n. For B = (bnk) we write B ⇒ A if every B limitable sequence is A limitable to the same limit, and say that B is equivalent to A if B ⇒ A and A ⇒ B. If B is normal, then it is well known that the inverse of B exists (we denote it by B-l) and that B ⇒ A if and only if F = AB-1 is a regular transformation, i.e., transforms every convergent sequence into a sequence converging to the same limit. We say that a series ∑ an† is summable A if its sequence of partial sums is A-limitable.

A Hardy-Davies-Petersen Inequality for a Class of Matrices

1978 ◽  
Vol 30 (03) ◽  
pp. 458-465 ◽  
Author(s):  
P. D. Johnson ◽  
R. N. Mohapatra
Keyword(s):  
Triangular Matrix ◽  
Diagonal Matrix ◽  
Lower Triangular Matrix ◽  
Lower Triangular ◽  
Class Of Matrices

Let ω be the set of all real sequences a = ﹛an﹜ n ≧0. Unless otherwise indicated operations on sequences will be coordinatewise. If any component of a has the entry oo the corresponding component of a-1 has entry zero. The convolution of two sequences s and q is given by s * q . The Toeplitz martix associated with sequence s is the lower triangular matrix defined by tnk = sn-k (n ≧ k), tnk = 0 (n &lt; k). It can be seen that Ts(q) = s * q for each sequence q and that Ts is invertible if and only if s0 ≠ 0. We shall denote a diagonal matrix with diagonal sequence s by Ds.

Sign in / Sign up

Export Citation Format

Share Document