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Fourier series with small gaps

Part of: Harmonic analysis in one variable Real functions

Published online by Cambridge University Press:  09 April 2009

P. Isaza  and
D. Waterman
P. Isaza
Affiliation:
Department of Mathematics, Syracuse University, Syracuse, New York 13244-1150, U.S.A.
D. Waterman
Affiliation:
Department of Mathematics, Syracuse University, Syracuse, New York 13244-1150, U.S.A.
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Abstract

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A 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.

MSC classification

Secondary: 42A16: Fourier coefficients, Fourier series of functions with special properties, special Fourier series 42A55: Lacunary series of trigonometric and other functions; Riesz products 26A45: Functions of bounded variation, generalizations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 46 , Issue 2 , April 1989 , pp. 212 - 219
Copyright
Copyright © Australian Mathematical Society 1989

References

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