Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T08:08:32.584Z Has data issue: false hasContentIssue false
  • English
  • Français

Fourier series with small gaps

Part of: Real functions Harmonic analysis in one variable

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[1]Bary, N. K., A treatise on trigonometric series, Vol II (Macmillan, New York, 1964).Google Scholar
[2]Čanturija, Z. A., ‘On the absolute convergence of Fourier series of the classes V[na]’ Fourier Analysis and Approximation Theory, Proc. Colloq., Budapest, 1976, Vol I., pp. 219–240 (Colloq. Math. Soc. János Bolyai 19, North-Holland, Amsterdam, 1978).Google Scholar
[3]Ingham, A. E., ‘Some trigonometric inequalities with applications to the theory of series’, Math. Z. 41 (1936), 367–379.Google Scholar
[4]Kennedy, P. B., ‘Fourier series with gaps’, Quart. J. Math. Oxford Ser. (2) 7 (1956), 224–230.CrossRefGoogle Scholar
[5]Noble, M. E., ‘Coefficient properties of Fourier series with gap conditions’, Math. Ann. 128 (1954), 55–62.Google Scholar
[6]Patadia, J. R., ‘On the absolute convergence of a lacunary Fourier series’, J. Math. Anal. Appl. 65 (1978), 391–398.Google Scholar
[7]Schramm, M. and Waterman, D., ‘On the magnitude of Fourier coefficients’, Proc. Amer. Math. Soc. 85 (1982), 407–410.Google Scholar
[8]Waterman, D., ‘On A-bounded variation’, Studia Math. 57 (1976), 33–45.Google Scholar
[9]Wiener, N., ‘A class of gap theorems’, Annali di Pisa 3 (1934), 367–372.Google Scholar
[10]Zygmund, A., Trigonometric series, 2nd ed. (Cambridge Univ. Press, New York, 1959).Google Scholar