Absolute Nörlund matrix summability of Fourier series based on inclusion theorems

1989 ◽  
Vol 105 (2) ◽  
pp. 319-333
Author(s):  
B. Kuttner ◽  
B. E. Rhoades
Keyword(s):  
Fourier Series ◽  
Sufficient Conditions ◽  
Matrix Summability ◽  
Nörlund Matrix ◽  
Related Series ◽  
Summability Of Fourier Series ◽  
Cesàro Matrix

In a recent paper [39] the second author established sufficient conditions for a Nörlund matrix (N, p) to be stronger than a Cesaro matrix (C, α) of order α, for some positive α. This result was then used to show that a number of known theorems dealing with the Nörlund summability of Fourier series, or related series, can be more easily established, since the conditions placed on the Nörlund matrix imply that it is stronger than some (C, α) method, for which the summability theorem has already been established.

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.

A New Criterion for Borel Summability of Fourier Series

1969 ◽  
Vol 12 (5) ◽  
pp. 573-580 ◽  
Author(s):  
B.N. Sahney ◽  
P.D. Kathal
Keyword(s):  
Fourier Series ◽  
Sufficient Conditions ◽  
Borel Summability ◽  
Conjugate Series ◽  
Summability Of Fourier Series ◽  
New Criterion

The application of Borel summability to Fourier series has been discussed by Takahashi and Wang [8] and Sahney [5]. Sahney [6] and Sinvhal [7] obtained sufficient conditions for the Borel summability of the derived Fourier series and its conjugate series, respectively. Kathal [3] obtained different conditions in the case of the conjugate series. In this paper we give a new criterion for Borel summability of Fourier series.

scholarly journals Absolute, in degree $p$, summability of Fourier series by method of Hausdorff

1987 ◽  
pp. 86
Author(s):  
N.I. Volkova ◽  
M.V. Kovtun
Keyword(s):  
Fourier Series ◽  
Sufficient Conditions ◽  
Conjugate Fourier Series ◽  
Summability Of Fourier Series

We establish sufficient conditions of absolute, in degree $p$, summability of Fourier series, conjugate Fourier series, differentiated Fourier series, and differentiated conjugate Fourier series.

scholarly journals A new theorem on absolute matrix summability of fourier series

2017 ◽  
Vol 102 (116) ◽  
pp. 107-113 ◽  
Author(s):  
Şebnem Yildiz
Keyword(s):  
Fourier Series ◽  
Summability Factors ◽  
Weighted Mean ◽  
Matrix Summability ◽  
Absolute Matrix Summability ◽  
Summability Of Fourier Series ◽  
Weaker Conditions

We generalize a main theorem dealing with absolute weighted mean summability of Fourier series to the |A,pn|k summability factors of Fourier series under weaker conditions. Also some new and known results are obtained.

scholarly journals Some absolutely effective product methods

1992 ◽  
Vol 15 (4) ◽  
pp. 641-651
Author(s):  
H. P. Dikshit ◽  
J. A. Fridy
Keyword(s):  
Fourier Series ◽  
General Result ◽  
Arithmetic Mean ◽  
Product Method ◽  
The Matrix ◽  
Related Series ◽  
Summability Of Fourier Series ◽  
Effective Product ◽  
Absolute Nörlund Summability

It is proved that the product methodA(C,1), where(C,1)is the Cesàro arithmetic mean matrix, is totally effective under certain conditions concerning the matrixA. This general result is applied to study absolute Nörlund summability of Fourier series and other related series.

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