Bohl almost-periodic functions

A class of functions the typical property of which is that they can be uniformly approximated on the whole real axis by generalized trigonometric polynomials of the form

$$\sum a_{n_1\dots n_k}e^{i(n_1\alpha_1+\ldots+n_k\alpha_k)x},$$

where $n_1,\dots,n_k$ are arbitrary integers, while $\alpha_1,\dots,\alpha_k$ are given real numbers. This class of functions contains the class of continuous $2\pi$-periodic functions and is contained in the class of Bohr almost-periodic functions. P. Bohl specified several necessary and sufficient conditions for a function to be almost-periodic. In particular, any function of the type

$$f(x)=f_1(x)+\ldots+f_k(x),$$

where each one of the functions $f_1(x),\dots,f_k(x)$ is continuous and periodic (with possibly different periods), is a Bohl almost-periodic function.

References

[1]	P. Bohl, "Über die Darstellung von Funktionen einer Variabeln durch trigonometrische Reihen mit mehreren einer Variabeln proportionalen Argumenten" , Dorpat (1893) (Thesis)
[2]	P. Bohl, "Ueber eine Differentialgleichung der Störungstheorie" J. Reine Angew. Math. , 131 (1906) pp. 268–321
[3]	B.M. Levitan, "Almost-periodic functions" , Moscow (1953) (In Russian)

Comments

A very well-known reference for this kind of topic is [a1].

References

[a1]	H. Bohr, "Almost periodic functions" , Chelsea, reprint (1947) (Translated from German)