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Zbl 1044.42006
Zhang, Chuanyi; Yao, Huili
Converse problems of Fourier expansion and their applications. (English)
[J] Nonlinear Anal., Theory Methods Appl. 56, No. 5, A, 761-779 (2004). ISSN 0362-546X

Summary: Let $f \in \cal C(\Bbb R, H)$ have a countable frequency set $Freq(f)$ and satisfy Parseval's equality. We show that if $f$ satisfies one of the following conditions: (a) uniformly continuous and $Freq(f)$ has a unique limit point at infinity; (b) indefinite integral is Lipschitz, $Freq(f)$ converges fast in some sense; (c) in the case of Euclidean space $H$, all the coefficients are positive, then $f$ is pseudo-almost-periodic. An example is given to show that the conclusion cannot be improved. The results are applied to the theory of Riesz--Fischer and the optimal control theory.

MSC 2000:: *42A75 Periodic functions and generalizations
43A60 Almost periodic functions on groups, etc.
49N20 Periodic optimization

Keywords: pseudo-almost-periodic functions; Fourier series; converse problem; Riesz-Fischer theory

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