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

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- by Geraldo Soares De Souza and G. Sampson
- Proc. Amer. Math. Soc.
**120**(1994), 723-726 - DOI: https://doi.org/10.1090/S0002-9939-1994-1126194-8
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## Abstract:

An analytic function $F$ on the disc belongs to $B$ if $||F|{|_B} = \int _0^1 {\int _0^{2\pi } {|{F’}(r{e^{i\theta }})|d\theta dr < \infty } }$. Notice that $B \varsubsetneqq {H^1} \varsubsetneqq {L^1}$, where ${H^1}$ is the Hardy space of all analytic functions $F$ so that \[ ||F|{|_{{H^1}}} = \sup \limits _{0 < r < 1} \int _0^{2\pi } {|F(r{e^{i\theta }})| d\theta < \infty ,} \] ${L^1}$ 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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## Bibliographic Information

- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**120**(1994), 723-726 - MSC: Primary 42A20; Secondary 46E99
- DOI: https://doi.org/10.1090/S0002-9939-1994-1126194-8
- MathSciNet review: 1126194