Converse problems of Fourier expansion and their applications Original Research Article

Let View the MathML source have a countable frequency set Freq(f) and satisfy Parsevalʹs 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. 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.