J Austral Math Soc Ser A 46 pp212--219, 1989.

Fourier Series with Small Gaps

P. Isaza and D. Waterman

Abstract

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 behaviour on a large enough subinterval I, this will have consequences for the behaviour 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 magnotude of the Fourier coefficients. A generalized bounded variation condition coupled with a Zygmund-type condition on the modulus of continuity of the restriction of the function to I implies absolute convergence of the Fourier series.

1980 AMS Subject Classification (1985 Revision): 42A16, 42A55, 26A45

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Authors

P. Isaza

D. Waterman

Department of Mathematics, Syracuse University, Syracuse, New York 13244-1150, U.S.A.