scholarly journals On absolute summability of integrals with multipliers

2021 ◽  
Vol 15 ◽  
pp. 57
Author(s):  
L.G. Bojtsun ◽  
A.I. Khaliuzova
Keyword(s):  
Sufficient Conditions ◽  
Absolute Summability

We establish sufficient conditions for summability of integrals with multiplier by $|\overline{W}, p(y)|_k$, $k \geqslant 1$, method.

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 Absolute Summability (A) of Fourier Series

1934 ◽  
Vol 4 (1) ◽  
pp. 13-17 ◽  
Author(s):  
L. S. Bosanquet
Keyword(s):  
Fourier Series ◽  
Sufficient Conditions ◽  
Absolute Summability ◽  
The Absolute

Sufficient conditions for the absolute summability (A)1 of a Fourier series have been given by J. M. Whittaker and B. N. Prasad. They obtained theorem 1 below in the cases α = 0 and α = 1 respectively. Theorem 1 is contained in theorem 2, which was given by Prasad in the case α = 0.

scholarly journals Sufficient Conditions for Absolute Cesàro Summable Factor

2019 ◽  
Vol 4 (3) ◽  
pp. 627-634
Author(s):  
Smita Sonker ◽  
Alka Munjal
Keyword(s):  
Impulse Response ◽  
Infinite Series ◽  
Sufficient Conditions ◽  
Absolute Summability ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Minimal Set ◽  
Bibo Stability ◽  
Necessary And Sufficient ◽  
Bounded Input

Quasi-f-power increasing sequence has been used for infinite series to establish a theorem on a minimal set of sufficient conditions for absolute Cesàro φ-|〖C,α;δ;l|〗_k summable factor. Further, a set of new and well-known arbitrary results have been obtained by using the main theorem. The presented main result has been validated by the previous result under suitable conditions. In this way, the Bounded Input Bounded Output (BIBO) stability of impulse response has been improved by finding a minimal set of sufficient conditions for absolute summability because absolute summable is the necessary and sufficient condition for BIBO stability.

Absolute Summability Factors in a Sequence

1984 ◽  
Vol 27 (1) ◽  
pp. 16-30
Author(s):  
S. Baron
Keyword(s):  
Sufficient Conditions ◽  
Absolute Summability ◽  
Necessary And Sufficient Conditions ◽  
Summability Factors ◽  
Absolute Summability Factors ◽  
Necessary And Sufficient ◽  
Cesàro Method

AbstractLet α≥0 and β>— 1. The main result gives necessary and sufficient conditions for the sequence (εn) in order that the sequence (εnUn) will be absolutely summable by the Cesàro method Cβ for each sequence (Un) which is bounded or summable by the method CαAnother theorem is proven when Cα and Cβ are replaced by triangular methods A = (ank) and B=(bnk) satisfying , where (ξnk) = (ank)-1.

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.

Matrix summability of Fourier series based on inclusion theorems

1986 ◽  
Vol 100 (3) ◽  
pp. 545-557 ◽  
Author(s):  
B. E. Rhoades
Keyword(s):  
Fourier Series ◽  
Sufficient Conditions ◽  
Absolute Summability ◽  
Fourier Series Expansion ◽  
Matrix Summability ◽  
The Matrix ◽  
Derived Series ◽  
Summability Of Fourier Series ◽  
Weaker Conditions ◽  
Cesàro Method

Letdenote the Fourier series expansion of a function. Féver's celebrated theorem states that the series is summable (C, 1) at each point of continuity off, where (C, 1) denotes the Cesàro method of summability of order 1. Riesz extended this result to (C, α) for each α > 0. Since then many authors have established sufficient conditions on various methods of summability to guarantee similar results. Over the years the pattern has been to strive for weaker conditions on the matrix, and to replace the condition of continuity on the function by a less stringent one. Theorems have been proved not only for the summability off, but the summability of the derived series, and other series, related tof. Beginning with the work of Hille and Tamarkin[13], many of the theorems have been extended to absolute summability.

scholarly journals Generalization of some absolute summability methods

2001 ◽  
Vol 25 (6) ◽  
pp. 389-395
Author(s):  
W. T. Sulaiman
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Absolute Summability ◽  
Summability Methods

Mazhar (1971) gave the characterization for the series∑anϵnto be summable|N,pn|whenever∑anis summable|C,α|k,α≥0,k≥1. Here we prove two theorems, the first concerns the sufficient conditions and the second the necessary conditions satisfied by{ϵn}in order to have∑anϵnsummable|N¯,pn|kwhenever∑anis summable|C,α|k,k≥1.

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