Bohr almost-periodic functions

uniform almost-periodic functions

The class $U$-a.-p. of almost-periodic functions. The first definition, which was given by H. Bohr [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). 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.

References

[1]	H. Bohr, "Zur Theorie der fastperiodischen Funktionen I" Acta Math. , 45 (1925) pp. 29–127
[2]	B.M. Levitan, "Almost-periodic functions" , Moscow (1953) (In Russian)

Comments

Bohr's treatise [a1] is a good reference. An up-to-date reference is [a2].

References

[a1]	H. Bohr, "Almost periodic functions" , Chelsea, reprint (1947) (Translated from German)
[a2]	C. Corduneanu, "Almost periodic functions" , Wiley (1968)