Uintell Mathematics Encyclopedia

The real numbers $\lambda_n$ in the Fourier series corresponding to the given almost-periodic function $f$:

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

where the $a_n$ are the Fourier coefficients of $f$ (cf. Fourier coefficients of an almost-periodic function; Fourier series of an almost-periodic function). The set of Fourier indices of a function $f$ is called its spectrum. In contrast to the periodic case, the spectrum of an almost-periodic function can have finite limit points and can even be everywhere dense. Therefore, the behaviour of the Fourier series of an almost-periodic function depends in an essential way on the arithmetic structure of its spectrum.