View source for Bohr almost-periodic functions

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''uniform almost-periodic functions''

The class $U$-a.-p. of almost-periodic functions. The first definition, which was given by H. Bohr [[#References|[1]]], 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|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|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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bohr,   "Zur Theorie der fastperiodischen Funktionen I"  ''Acta Math.'' , '''45'''  (1925)  pp. 29–127</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.M. Levitan,   "Almost-periodic functions" , Moscow  (1953)  (In Russian)</TD></TR></table>



====Comments====
Bohr's treatise [[#References|[a1]]] is a good reference. An up-to-date reference is [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Bohr,   "Almost periodic functions" , Chelsea, reprint  (1947)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Corduneanu,   "Almost periodic functions" , Wiley  (1968)</TD></TR></table>