This article is cited in 1 paper

An example of an orthonormal system of convergence in $C$ but not in $L^2$

A. M. Olevskii


Abstract: We prove the following theorem.
Theorem. {\it For any $p_0\in[1,\infty)$ there exists a complete uniformly bounded orthonormal system $\{\varphi_n\}$ having the following properties}:
1) For all $f\in L^p, p>p_0,$ the Fouries series $\sum c_n\varphi_n$ converges to $f$ almost everywhere.
2) {\it There exists an $F\in L^{p_0}$ whose Fourier series diverges almost everywhere.}
Bibliography: 8 titles.

UDC: 517.522.3

MSC: Primary 42A20; Secondary 42A60, 42A64

Received: 03.10.1972

 English version: Mathematics of the USSR-Sbornik, 1973, 20:1, 145–153

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