Bohr almost-periodic functions
uniform almost-periodic functions
The class -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) |
Bohr almost-periodic functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bohr_almost-periodic_functions&oldid=33123