Fourier series of an almost-periodic function

A series of the form

$$f(x)\sim\sum_na_ne^{i\lambda_nx},\label{*}\tag{*}$$

where the $\lambda_n$ are the Fourier indices, and the $a_n$ are the Fourier coefficients of the almost-periodic function $f$ (cf. Fourier indices of an almost-periodic function; Fourier coefficients of an almost-periodic function). A series of the form \eqref{*} can be associated with any real- or complex-valued almost-periodic function. The behaviour of the Fourier series depends crucially on the structure of the set of Fourier indices of this function and also on the restrictions imposed on the Fourier coefficients of this function.

For example, the following theorems hold. If

$$\sum_{n=0}^\infty|a_n|^2<\infty,$$

then there is a Besicovitch almost-periodic function (cf. Besicovitch almost-periodic functions) for which the trigonometric series \eqref{*} is its Fourier series. If $a_n>0$ for a uniform almost-periodic function, then the series

$$\sum_{n=0}^\infty a_n$$

converges. If the Fourier indices of a uniform almost-periodic function are linearly independent, then the Fourier series of this function converges absolutely. If a uniform almost-periodic function has a lacunary Fourier series, then this series converges uniformly.

Comments

Uniform(ly) almost-periodic functions are also known as Bohr almost-periodic functions. Cf. Lacunary trigonometric series for the notion of a lacunary Fourier series.

References

[1]	B.M. Levitan, "Almost-periodic functions" , Moscow (1953) (In Russian)
[2]	N.P. Kuptsov, "Direct and converse theorems of approximation theory and semigroups of operators" Russian Math. Surveys , 23 : 4 (1968) pp. 115–177 Uspekhi Mat. Nauk , 23 : 4 (142) (1968) pp. 117–178
[3]	V.F. Gaposhkin, "Lacunary series and independent functions" Russian Math. Surveys , 21 : 6 (1966) pp. 1–82 Uspekhi Mat. Nauk , 21 : 6 (132) (1966) pp. 3–82
[a1]	C. Corduneanu, "Almost periodic functions" , Wiley (1968)