On Weighted Integrability of Trigonometric Series and L 1 -Convergence of Fourier Series

1986 ◽  
Vol 96 (1) ◽  
pp. 53 ◽  
Author(s):  
William O. Bray ◽  
Caslav V. Stanojevic
Keyword(s):  
Fourier Series ◽  
Trigonometric Series ◽  
Weighted Integrability

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.

scholarly journals On the Integral Modulus of Continuity of Fourier Series

1988 ◽  
Vol 44 (2) ◽  
pp. 151-158
Author(s):  
Babu Ram ◽  
Suresh Kumari
Keyword(s):  
Fourier Series ◽  
Trigonometric Series ◽  
Modulus Of Continuity ◽  
Wide Class ◽  
Integral Modulus

AbstractFor a wide class of sine trigonometric series we obtain an estimate for the integral modulus of continuity.

The Approximate Symmetric Integral

1989 ◽  
Vol 41 (3) ◽  
pp. 508-555 ◽  
Author(s):  
D. Preiss ◽  
B. S. Thomson
Keyword(s):  
Fourier Series ◽  
Trigonometric Series ◽  
Real Numbers ◽  
Integration Procedure ◽  
Image Position ◽  
Derivation Process ◽  
Symmetric Integral

By a symmetric integral is understood an integral obtained from some kind of symmetric derivation process. Such integrals arise most naturally in the study of trigonometric series and in particular to handle the following problem. Suppose that a trigonometric seriesconverges everywhere to a function À. It is known that this may occur without À being integrable in any of the more familiar senses so that the series may not be considered as a Fourier series of À; indeed Denjoy [4] has shown that if bnis a sequence of real numbers decreasing to zero but with+00 then the function À(x) = is not Denjoy-integrable. It is natural to ask then for an integration procedure that can be applied to À in order that the series be the Fourier series of À with respect to this integral.

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