Editing Almost periodic function (section)

{{Short description|A function that "converges" to periodicity}}
{{Distinguish|Quasiperiodic function}}
{{inline|date=March 2025}}
In [[mathematics]], an '''almost periodic function''' is, loosely speaking, a [[function (mathematics)|function]] of a [[real number|real]] variable that is [[periodic function|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 group]]s, first studied by [[John von Neumann]].

'''Almost periodicity''' is a property of [[dynamical system]]s that appear to retrace their paths through [[phase space]], but not exactly. An example would be a [[planetary system]], with [[planet]]s in [[orbit]]s moving with [[Orbital period|period]]s that are not [[commensurability (mathematics)|commensurable]] (i.e., with a period vector that is not [[Proportionality (mathematics)|proportional]] to a vector of [[integer]]s). A [[Kronecker's theorem on diophantine approximation|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, say, a [[second of arc]] to the positions they once were in.