Almost periodic function

In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-periods". The concept was first studied by Harald Bohr and later generalized by Vyacheslav Stepanov, Hermann Weyl and Abram Samoilovitch Besicovitch, amongst others. There is also a notion of almost periodic functions on locally compact abelian groups, first studied by John von Neumann.

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.

There are several inequivalent definitions of almost periodic functions. The first was given by 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

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