Almost periodic function

In mathematics, almost periodic functions are functions of a real number that are periodic up to a small error, first studied by Harald Bohr. There are generalizations to almost periodic functions on locally compact abelian groups.

Almost periodicity is a property of dynamical systems that appear to retrace their paths through phase space, but not exactly. An example would be a planetary system, with planets in orbits moving with periods that are not commensurable (i.e., with a period vector that is not proportional to a vector of integers). A theorem of Kronecker from diophantine approximation can be used to show that any particular configuration that occurs once, will recur to within any specified accuracy: if we wait long enough we can observe the planets all return to within a second of arc to the positions they once were in.

Definition and properties

There are several different inequivalent definitions of almost periodic functions. An almost periodic function is a complex-valued function of a real variable that has the properties expected of a function on a phase space describing the time evolution of such a system. There have in fact been a number of definitions given, beginning with that of Harald Bohr. His interest was initially in finite Dirichlet series. In fact by truncating the series for the Riemann zeta function ζ("s") to make it finite, one gets finite sums of terms of the type

:$e^\{(sigma+it)log\; n\},$

with "s" written as (σ+it) - the sum of its real part σ and imaginary part "it". Fixing σ, so restricting attention to a single vertical line in the complex plane, we can see this also as

:$n^sigma\; e^\{(log\; n)it\}.,$

Taking a "finite" sum of such terms avoids difficulties of analytic continuation to the region σ < 1. Here the 'frequencies' log "n" will not all be commensurable (they are as linearly independent over the rational numbers as the integers "n" are multiplicatively independent - which comes down to their prime factorizations).

With this initial motivation to consider types of trigonometric polynomial with independent frequencies, mathematical analysis was applied to discuss the closure of this set of basic functions, in various norms.

The theory was developed using other norms by Besicovitch, Stepanov, Weyl, von Neumann, Turing, Bochner and others in the 1920s and 1930s.

Uniform or Bohr or Bochner almost periodic functions

. It contains the space "S"^"p" of Stepanov almost periodic functions. It is the closure of the trigonometric polynomials under the seminorm :$||f||\_\{W,p\}=lim\_\{rmapstoinfty\}||f||\_\{S,r,p\}$Warning: there are nonzero functions "f" with ||"f"||=0, such as any bounded function of compact support, so to get a Banach space one has to quotient out by these functions.

Besikovitch almost periodic functions

The space "B"^"p" of Besicovitch almost period functions was introduced by
*springer|id=b/b016770|title=Bohr almost periodic functions|first=E.A.|last= Bredikhina
*springer|id=s/s087720|title=Stepanov almost periodic functions|first=E.A.|last= Bredikhina
*springer|id=w/w097680|title=Weyl almost periodic functions|first=E.A.|last= Bredikhina
*W. Stepanoff(=V.V. Stepanov), "Sur quelques généralisations des fonctions presque périodiques" C.R. Acad. Sci. Paris , 181 (1925) pp. 90–92
*W. Stepanoff(=V.V. Stepanov), "Ueber einige Verallgemeinerungen der fastperiodischen Funktionen" Math. Ann. , 45 (1925) pp. 473–498
*H. Weyl, "Integralgleichungen und fastperiodische Funktionen" Math. Ann. , 97 (1927) pp. 338–356

Notes