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Divergence of Fourier series

Published online by Cambridge University Press:  17 April 2009

Masako Izuml  and
Shin-Ichi Izumi
Masako Izuml
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, ACT.
Shin-Ichi Izumi
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, ACT.
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Abstract

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Carleson has proved that the Fourier series of functions belonging to the class L2 converge almost everywhere.

Improving his method, Hunt proved that the Fourier series of functions belonging to the class Lp (p > 1) converge almost everywhere. On the other hand, Kolmogoroff proved that there is an integrable function whose Fourier series diverges almost everywhere. We shall generalise Kolmogoroff's Theorem as follows: There is a function belonging to the class L(logL)p (p > 0) whose Fourier series diverges almost everywhere. The following problem is still open: whether “almost everywhere” in the last theorem can be replaced by “everywhere” or not. This problem is affirmatively answered for the class L by Kolmogoroff and for the class L(log logL)p (0 < p < 1) by Tandori.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 8 , Issue 2 , April 1973 , pp. 289 - 304
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Carleson, Lennart, “On convergence and growth of partial sums of Fourier series”, Acta Math. 116 (1966).Google Scholar
[2]Chen, Yung-ming, “On Kolmogoroff's divergent Fourier series”, Arch. Math. 14 (1963), 116–119.Google Scholar
[3]Hunt, Richard A., “On the convergence of Fourier series”, Orthogonal expansions and their continuous analogues, 235–255 (Proceedings Conf. Edwardsville, Illinois, 1967. Southern Illinois Univ. Press, Carbondale, Illinois, 1968); (cf. MR38#6296).Google Scholar
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