# Bohr almost-periodic functions

The class $U$-a.-p. of almost-periodic functions. The first definition, which was given by H. Bohr , is based on a generalization of the concept of a period: A continuous function $f(x)$ on the interval $(-\infty,\infty)$ is a Bohr almost-periodic function if for any $\epsilon>0$ there exists a relatively-dense set of $\epsilon$-almost-periods of this function (cf. Almost-period). In other words, $f(x)$ is $U$-almost-periodic (or $\in U$-a.-p.) if for any $\epsilon>0$ there exists an $L=L(\epsilon)$ such that in each interval of length $L$ there exists at least one number $\tau$ such that
$$|f(x+\tau)-f(x)|<\epsilon,\quad-\infty<x<\infty.$$
If $L(\epsilon),\epsilon\to0$, is bounded, a Bohr almost-periodic function $f(x)$ becomes a continuous periodic function. Bochner's definition (cf. Bochner almost-periodic functions), which is equivalent to Bohr's definition, is also used in the theory of almost-periodic functions. Functions in the class of $U$-almost-periodic functions are bounded and uniformly-continuous on the entire real axis. The limit $f(x)$ of a uniformly-convergent sequence of Bohr almost-periodic functions $\{f_n(x)\}$ belongs to the class of $U$-almost-periodic functions; this class is invariant with respect to arithmetical operations (in particular the Bohr almost-periodic function $f(x)/g(x)$ is $U$-almost-periodic, under the condition
$$\inf_{-\infty<x<\infty}|g(x)|>\gamma>0.$$
If $f(x)$ is $U$-almost-periodic and if $f'(x)$ is uniformly continuous on $(-\infty,\infty)$, then $f'(x)$ is $U$-almost-periodic; the indefinite integral $F(x)=\int_0^xf(t)dt$ is $U$-almost-periodic if $F(x)$ is a bounded function.