View source for Almost-periodic function

A function representable as a generalized Fourier series. There are several ways of defining classes of almost-periodic functions, based respectively on notions of closure, of an almost-period and of translation. Each of these classes can be obtained as a closure, with respect to some metric, of the set of all finite trigonometric sums.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a0119701.png" /> be the distance of two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a0119702.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a0119703.png" /> in a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a0119704.png" /> of real- or complex-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a0119705.png" />. In the following, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a0119706.png" /> will be one of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a0119707.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a0119708.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a0119709.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197010.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197011.png" /> is the set of continuous bounded functions on the real line with the metric

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and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197014.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197015.png" /> are the sets of functions that are measurable and whose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197016.png" />-th powers are integrable on every finite interval of the real line, the metrics being

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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197021.png" /> be the set of trigonometric polynomials

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where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197023.png" /> are arbitrary real numbers and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197024.png" /> are complex coefficients, and let the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197025.png" /> denote the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197026.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197027.png" />. The classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197028.png" />-a.p., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197029.png" />-a.p., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197030.png" />-a.p. and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197031.png" />-a.p. denote, respectively, the classes of uniformly almost-periodic functions, or [[Bohr almost-periodic functions|Bohr almost-periodic functions]], of [[Stepanov almost-periodic functions|Stepanov almost-periodic functions]], of [[Weyl almost-periodic functions|Weyl almost-periodic functions]] and of [[Besicovitch almost-periodic functions|Besicovitch almost-periodic functions]]. These classes of almost-periodic functions are invariant under addition. Together with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197032.png" />, each class also contained the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197035.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197036.png" /> is a real number. The metrics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197037.png" /> are topologically equivalent for all values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197038.png" />; therefore it may be assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197039.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197040.png" />-a.p.<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197041.png" />-a.p., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197042.png" />-a.p.<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197043.png" />-a.p., and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197044.png" />-a.p.<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197045.png" />-a.p.. Then

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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197048.png" />, then

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For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197051.png" />-a.p., the mean value

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exists. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197053.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197054.png" /> is a real number, differs from zero only on a countable set of values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197055.png" />; any enumeration of this set is called the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197057.png" /> of Fourier exponents of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197058.png" />.

The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197059.png" /> are called the Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197060.png" />. With a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197061.png" /> in any of the classes defined above one can associate its Fourier series

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For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197063.png" />-a.p. one has the Parseval equality

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The Riesz–Fischer theorem can be generalized to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197065.png" />-a.p.: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197067.png" /> be arbitrary real numbers, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197069.png" /> be complex numbers for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197070.png" />. Then there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197071.png" />-a.p. which has the trigonometric series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197072.png" /> as its Fourier series.

There is also a uniqueness theorem: If two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197074.png" /> have the same Fourier series, then

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In particular, for uniformly almost-periodic functions the uniqueness theorem states that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197076.png" /> (for Stepanov almost-periodic periodic functions: almost-everywhere). A uniqueness theorem in the same sense as for Fourier–Lebesgue series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197077.png" />-periodic functions does not hold for Weyl or Besicovitch almost-periodic functions.

The classes of uniformly almost-periodic and of Stepanov almost-periodic functions are, respectively, non-trivial extensions of the class of continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197078.png" />-periodic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197079.png" /> and the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197080.png" />-periodic integrable functions on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197081.png" />. For these classes of almost-periodic functions the uniqueness theorem remains valid.

A consequence of the definition of the classes of almost-periodic functions through the concept of closure is the approximation theorem: For every almost-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197082.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197083.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197084.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197085.png" />) and every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197086.png" /> there is a finite trigonometric polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197087.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197088.png" />, satisfying the inequality

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The approximation theorem may serve as a starting point of the definition of various classes of almost-periodic functions. The approximating polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197091.png" /> may contain  "extraneous"  exponents, i.e. exponents different from the Fourier exponents of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197092.png" />. However, important for some applications of the approximation theorem is the fact that the exponents different from the Fourier exponents of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197093.png" /> can be avoided in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197094.png" />.

In connection with the representability of almost-periodic functions by generalized Fourier series, the problem of convergence criteria for these series arises and various summation methods for generalized Fourier series (the Bochner–Fejér method, etc.) become meaningful. Thus, the following criteria have been obtained: absolute convergence of a generalized Fourier series if the Fourier exponents are linearly independent; uniform convergence of a Fourier series when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197095.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197096.png" /> or when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197097.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197098.png" />.

The importance of criteria for uniform convergence in the theory of almost-periodic functions is emphasized by the following theorem: If a trigonometric series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a01197099.png" /> is uniformly convergent on the entire real line, then it is the Fourier series of its sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a011970100.png" />-a.p.. Corollary: There exists uniformly almost-periodic functions with an arbitrary countable set of Fourier exponents. If particular, the Fourier exponents of a uniformly almost-periodic function may have finite limit points or may even be everywhere dense.

Other definitions of almost-periodic functions of the above classes rely on the concept of an [[Almost-period|almost-period]] and generalizations thereof.

Besides the concept of closure or that of an almost-period, the concept of a translation can also be used for the definition of almost-periodic functions. Thus, a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a011970101.png" /> is uniformly almost-periodic if and only if every infinite sequence of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a011970102.png" /> where the translation numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a011970103.png" /> are arbitrary real numbers, contains a uniformly convergent subsequence. This definition serves as a starting point in considering almost-periodic functions on groups.

The main results in the theory of almost-periodic functions remain valid if one considers the concept of a generalized translation. Other generalizations are possible and useful: almost-periodic functions with values in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a011970104.png" />-dimensional space or in a Banach or metric space, and analytic or harmonic almost-periodic functions.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bohr,   "Almost-periodic functions" , Chelsea, reprint  (1947)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.S. Besicovitch,   "Almost periodic functions" , Cambridge Univ. Press  (1932)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.M. Levitan,   "Almost-periodic functions" , Moscow  (1953)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.P. Kuptsov,   "Direct and converse theorems of approximation theory and semigroups of operators"  ''Russian Math. Surveys'' , '''32''' :  4  (1968)  pp. 115–177  ''Uspekhi Mat. Nauk'' , '''23''' :  4  (1968)  pp. 117–178</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  W. Rudin,   "Fourier analysis on groups" , Benjamin  (1962)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B.M. Levitan,   "Generalized translation operators and some of their applications" , Israel Program Sci. Transl.  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.M. Krasnosel'skii,   "Non-linear almost-periodic oscillations" , Wiley  (1973)  (Translated from Russian)</TD></TR></table>



====Comments====
The theory of almost-periodic functions was initiated by H. Bohr, who developed the notion of a uniformly almost-periodic function in his study of Dirichlet series. This original definition was the one using <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a011970105.png" />-almost-periods and the approximation theorem was one of the main achievements of the theory. For a modern approach see [[#References|[a4]]], Sect. 5 and [[#References|[a10]]], Chapt. 1, 6. As to the equivalence of the approach starting from a certain structural property that is a generalization of pure periodicity, and on the other hand, the approach starting from approximation by trigonometric polynomials (also in the definition of the various classes of generalized almost-periodic functions), see [[#References|[2]]], Chapt. 2. An interesting generalization not mentioned above are the Levitan almost-periodic functions (see [[#References|[a5]]], where they are called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011970/a011970107.png" />-almost periodic functions). A new approach to the theory of uniformly almost-periodic functions is given in [[#References|[a1]]]; this has lead to the study of so-called almost-automorphic functions [[#References|[a9]]], a class of functions closely related to the Levitan almost-periodic functions mentioned above [[#References|[a7]]].

Outside the realm of harmonic analysis an important application of (uniformly) almost-periodic functions lies in the theory of differential equations; see e.g. [[#References|[7]]], [[#References|[a8]]], [[#References|[a5]]], [[#References|[a2]]], Chapt. 4, and [[#References|[a10]]], Chapt. 4, 5. In this context the theory of dynamical systems (cf. [[Dynamical system|Dynamical system]]) is relevant and, in particular, the study of various types of almost-periodic and (or) recurrent motions, see [[#References|[a2]]], [[#References|[a3]]] and [[#References|[a6]]] (beware of conflicting terminology in the literature).

As a replacement for [[#References|[6]]], [[#References|[a11]]] may be used, which is of the same flavour as [[#References|[6]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Bochner,   "A new approach to almost periodicity"  ''Proc. Nat. Acad. Sci. USA'' , '''48'''  (1962)  pp. 2039–2043</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.U. Bronshtein,   "Extensions of minimal transformation groups" , Sijthoff &amp; Noordhoff  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W.H. Gottschalk,   G.A. Hedlund,   "Topological dynamics" , Amer. Math. Soc.  (1955)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  Y. Katznelson,   "An introduction to harmonic analysis" , Dover, reprint  (1968)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B.M. Levitan,   V.V. Zhikov,   "Almost periodic functions and differential equations" , Cambridge Univ. Press  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  V.V. Nemytskii,   V.V. Stepanov,   "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Reich,   "Präkompakte Gruppen und Fastperiodicität"  ''Math. Z.'' , '''116'''  (1970)  pp. 216–234</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  G.R. Sell,   "Topological dynamics and ordinary differential equations" , v. Nostrand-Reinhold  (1971)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  W.A. Veech,   "Almost automorphic functions on groups"  ''Amer. J. Math.'' , '''87'''  (1965)  pp. 719–751</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  C. Corduneanu,   "Almost periodic functions" , Interscience  (1961)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  B.M. Levitan,   "The application of generalized displacement operators to linear differential equations of the second order"  ''Amer. Math. Soc. Transl. Series'' , '''1'''  (1950)  pp. 408–541  ''Uspekhi Mat. Nauk'' , '''4''' :  1 (29)  (1949)  pp. 3–112</TD></TR></table>