Almost Periodic Functions: Their Limit Sets and Various Applications

In the present paper, we introduce and study the limit sets of the almost periodic functions f: \({{\mathbb {R}}}\rightarrow {{\mathbb {C}}}\). It is interesting, that \(r=\inf |f(x)|\) and \(R=\sup |f(x)|\) may be expressed in exact form. In particular, the formula for r coincides with the well known partition problem formula. We show that the ring \(r\le |z|\le R\) is the limit set of the almost periodic function f(x) (under some natural conditions on f). We obtain interesting applications to the partition problem in number theory and to trigonometric series theory. We extend classical results for periodic functions (estimation of Fourier coefficients and Bernstein’s theorem about absolute convergence of trigonometric series) to almost periodic functions. Using our results on almost periodic functions, we propose a new approach to interesting problems of the stability of motion. An application of our function theoretic results to the spectral problems in operator theory is studied as well. The results and corresponding computations are illustrated by several figures.