scholarly journals A theorem on the absolute Cesàro summability of factored fourier series

1962 ◽  
Vol 59 (1) ◽  
pp. 11-25
Author(s):  
S. M. Mazhar
Keyword(s):  
Fourier Series ◽  
Cesàro Summability ◽  
Cesaro Summability ◽  
The Absolute

scholarly journals The Absolute Cesaro Summability of the Successively Derived Allied Series of a Fourier Series

1950 ◽  
Vol 8 (4) ◽  
pp. 163-176
Author(s):  
R. Mohanty
Keyword(s):  
Fourier Series ◽  
Cesàro Summability ◽  
Cesaro Summability ◽  
The Absolute ◽  
Image Position

We suppose that f(t) is integrable in the Lebesgue sense m (π, π) and is periodic with period 2π. We denote its Fourier series byThen the allied series is

The absolute generalized harmonic-Cesàro summability of a Fourier series

1994 ◽  
Vol 20 (3) ◽  
pp. 173-183
Author(s):  
Prem Chandra ◽  
G. D. Dikshit
Keyword(s):  
Fourier Series ◽  
Cesàro Summability ◽  
Cesaro Summability ◽  
The Absolute

scholarly journals Partial best approximations and the absolute Cesaro summability of multiple Fourier series

2021 ◽  
Vol 103 (3) ◽  
pp. 4-12
Author(s):  
S. Bitimkhan ◽  
◽  
D.T. Alibieva ◽  
Keyword(s):  
Fourier Series ◽  
Sufficient Conditions ◽  
Multiple Fourier Series ◽  
Dimensional Case ◽  
Cesàro Summability ◽  
Best Approximations ◽  
One Dimensional ◽  
Cesaro Summability ◽  
The Absolute ◽  
The One

The article is devoted to the problem of absolute Cesaro summability of multiple trigonometric Fourier series. Taking a central place in the theory of Fourier series this problem was developed quite widely in the one-dimensional case and the fundamental results of this theory are set forth in the famous monographs by N.K. Bari, A. Zigmund, R. Edwards, B.S. Kashin and A.A. Saakyan [1–4]. In the case of multiple series, the corresponding theory is not so well developed. The multidimensional case has own specifics and the analogy with the one-dimensional case does not always be unambiguous and obvious. In this article, we obtain sufficient conditions for the absolute summability of multiple Fourier series of the function f ∈ Lq(Is) in terms of partial best approximations of this function. Four theorems are proved and four different sufficient conditions for the |C; β¯|λ-summability of the Fourier series of the function f are obtained. In the first theorem, a sufficient condition for the absolute |C; β¯|λ- summability of the Fourier series of the function f is obtained in terms of the partial best approximation of this function which consists of s conditions, in the case when β1 = ... = βs = 1/q'. Other sufficient conditions are obtained for double Fourier series. Sufficient conditions for the |C; β1; β2|λ-summability of the Fourier series of the function f ∈ Lq(I2) are obtained in the cases β1 = 1/q', −1 < β2 < 1/q'(in the second theorem), 1/q'< β1 < +∞, β2 = 1/q', (in the third theorem), −1 < β1 < 1/q', 1/q' < β2 < +∞ (in the fourth theorem).

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