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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A function in the Dirichlet space such that its Fourier series diverges almost everywhere


Authors: Geraldo Soares De Souza and G. Sampson
Journal: Proc. Amer. Math. Soc. 120 (1994), 723-726
MSC: Primary 42A20; Secondary 46E99
MathSciNet review: 1126194
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Abstract: An analytic function $ F$ on the disc belongs to $ B$ if $ \vert\vert F\vert{\vert _B} = \int_0^1 {\int_0^{2\pi } {\vert{F'}(r{e^{i\theta }})\vert d\theta \,dr < \infty } } $. Notice that $ B \varsubsetneqq {H^1} \varsubsetneqq {L^1}$, where $ {H^1}$ is the Hardy space of all analytic functions $ F$ so that

$\displaystyle \vert\vert F\vert{\vert _{{H^1}}} = \mathop {\sup }\limits_{0 < r < 1} \int_0^{2\pi } {\vert F(r{e^{i\theta }})\vert\,d\theta < \infty ,} $

$ {L^1}$ is the Lebesgue space of integrable functions on $ [0,2\pi ]$, and the inclusion $ {H^1} \varsubsetneqq {L^1}$ is taken in the sense of boundary values, that is, $ F \in {H^1} \Rightarrow {\lim _{r \to 1}} - \Re F(r{e^{i\theta }}) \in {L^1}$.

Kolmogorov in 1923 showed that there exists an $ f$ in $ {L^1}$ so that its Fourier series diverges almost everywhere. In 1953 Sunouchi showed that there exists an $ f$ in $ {H^1}$ with an almost everywhere divergent Fourier series. The purpose of this note is to announce.

Theorem 1. There exists an $ f \in B$, whose Fourier series diverges a.e.

This problem was mentioned to the first author by Professor Guido Weiss.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1126194-8
PII: S 0002-9939(1994)1126194-8
Keywords: Fourier series, convergence of Fourier series
Article copyright: © Copyright 1994 American Mathematical Society