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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Almost periodic function ： ウィキペディア英語版 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. ==Motivation== 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 :$e^\backslash ,$ 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 :$n^\backslash sigma\; e^.\backslash ,$ 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Almost periodic function」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.