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Converse problems of Fourier expansion and their applications. (English) Zbl 1044.42006
Summary: Let $f\in 𝒞\left(ℝ,H\right)$ have a countable frequency set $Freq\left(f\right)$ and satisfy Parseval’s equality. We show that if $f$ satisfies one of the following conditions: (a) uniformly continuous and $Freq\left(f\right)$ has a unique limit point at infinity; (b) indefinite integral is Lipschitz, $Freq\left(f\right)$ 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:
 42A75 Classical almost periodic functions, mean periodic functions 43A60 Almost periodic functions on groups, etc.; almost automorphic functions 49N20 Periodic optimization