On the generalized absolute convergence of Fourier series

2019 ◽  
Vol 26 (1) ◽  
pp. 117-124
Author(s):  
Rusudan Meskhia
Keyword(s):  
Fourier Series ◽  
Trigonometric Series ◽  
Absolute Convergence ◽  
Sufficient Conditions ◽  
Trigonometric Fourier Series ◽  
Fourier Trigonometric Series ◽  
Variation Of A Function ◽  
The Absolute

Abstract In the present paper the sufficient conditions are obtained for the generalized r-absolute convergence ( {0<r<2} ) of the single Fourier trigonometric series in terms of the modulus of δ-variation of a function. It is proved that these conditions are unimprovable in a certain sense. The classical results of Berstein, Szasz, Zygmund and others, related to the absolute convergence of single trigonometric Fourier series, were previously generalized by [L. Gogoladze and R. Meskhia, On the absolute convergence of trigonometric Fourier series, Proc. A. Razmadze Math. Inst. 141 2006, 29–40].

On the Absolute Convergence of Fourier Series

1997 ◽  
Vol 4 (4) ◽  
pp. 333-340
Author(s):  
T. Karchava
Keyword(s):  
Fourier Series ◽  
Finite Number ◽  
Fourier Coefficients ◽  
Absolute Convergence ◽  
Sufficient Conditions ◽  
Trigonometric Fourier Series ◽  
Periodic Functions ◽  
Continuity Modulus ◽  
The Absolute ◽  
Necessary And Sufficient

Abstract The necessary and sufficient conditions of the absolute convergence of a trigonometric Fourier series are established for continuous 2π-periodic functions which in [0, 2π] have a finite number of intervals of convexity, and whose 𝑛th Fourier coefficients are O(ω(1/𝑛; 𝑓)/𝑛), where ω(δ; 𝑓) is the continuity modulus of the function 𝑓.

scholarly journals The multipliers of multiple trigonometric Fourier series

2016 ◽  
Vol 6 (1) ◽  
Author(s):  
Aizhan Ydyrys ◽  
Lyazzat Sarybekova ◽  
Nazerke Tleukhanova
Keyword(s):  
Fourier Series ◽  
Sufficient Conditions ◽  
Regular System ◽  
Multiple Fourier Series ◽  
Trigonometric Fourier Series ◽  
Lorentz Spaces ◽  
Complex Numbers ◽  
Multiple Trigonometric Fourier Series

Abstract We study the multipliers of multiple Fourier series for a regular system on anisotropic Lorentz spaces. In particular, the sufficient conditions for a sequence of complex numbers {λk}k∈Zn in order to make it a multiplier of multiple trigonometric Fourier series from Lp[0; 1]n to Lq[0; 1]n , p > q. These conditions include conditions Lizorkin theorem on multipliers.

scholarly journals Fourier series with small gaps

1989 ◽  
Vol 46 (2) ◽  
pp. 212-219
Author(s):  
P. Isaza ◽  
D. Waterman
Keyword(s):  
Fourier Series ◽  
Trigonometric Series ◽  
Bounded Variation ◽  
Modulus Of Continuity ◽  
Fourier Coefficients ◽  
Absolute Convergence ◽  
Order Condition ◽  
Circle Group ◽  
The Difference ◽  
Bounded Below

AbstractA trigonometric series has “small gaps” if the difference of the orders of successive terms is bounded below by a number exceeding one. Wiener, Ingham and others have shown that if a function represented by such a series exhibits a certain behavior on a large enough subinterval I, this will have consequences for the behavior of the function on the whole circle group. Here we show that the assumption that f is in any one of various classes of functions of generalized bounded variation on I implies that the appropriate order condition holds for the magnitude of the Fourier coefficients. A generalized bounded variation condition coupled with a Zygmundtype condition on the modulus of continuity of the restriction of the function to I implies absolute convergence of the Fourier series.

scholarly journals Generalization on some theorems of $ L^1 $-convergence of certain trigonometric series

2008 ◽  
Vol 39 (1) ◽  
pp. 63-74
Author(s):  
Zivorad Tomovski
Keyword(s):  
Fourier Series ◽  
Trigonometric Series ◽  
Sufficient Conditions ◽  
Sigma 1 ◽  
Complex Valued ◽  
Derivatives Of

In this paper we study $ L^1 $-convergence of the $ r $-th derivatives of Fourier series with complex-valued coefficients. Namely new necessary-sufficient conditions for $L^1$-convergence of the $ r $-th derivatives of Fourier series are given. These results generalize corresponding theorems proved by several authors (see [7], [10], [13], [19]). Applying the Wang-Telyakovskii class $ ({\bf B}{\bf V})_r^\sigma $, $ \>\sigma>0 $, $ \>r=0,1,2,\ldots\, $ we generalize also the theorem proved by Garrett, Rees and Stanojevi\'{c} in [5]. Finally, for $ \sigma=1 $ some corollaries of this theorem are given.

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