View source for Almost-periodic function on a group

A generalization of almost-periodic functions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a0119801.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a0119802.png" /> be an (abstract) group. A bounded complex-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a0119803.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a0119804.png" />, is called a right almost-periodic function if the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a0119805.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a0119806.png" /> runs through the entire group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a0119807.png" />, is (relatively) compact in the topology of uniform convergence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a0119808.png" />, i.e. if every sequence of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a0119809.png" /> contains a subsequence which is uniformly convergent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198010.png" />. A left almost-periodic function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198011.png" /> is defined similarly. It turns out that every right (left) almost-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198012.png" /> is also left (right) almost-periodic, and the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198015.png" /> independently run through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198016.png" />, is (relatively) compact. The latter property is often taken as a definition of almost-periodic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198017.png" />. The set of all almost-periodic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198018.png" /> is a Banach space with a norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198019.png" />.

The theory of almost-periodic functions on a group depends essentially on the mean-value theorem (cf. [[#References|[5]]], [[#References|[8]]]). A linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198020.png" /> defined on the space of almost-periodic functions is called a mean value if

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198021.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198023.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198025.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198026.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198027.png" />.

A unitary matrix function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198028.png" />, defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198029.png" />, is called a unitary representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198030.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198031.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198032.png" /> is the identity element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198034.png" /> is the identity matrix of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198035.png" />) and if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198037.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198038.png" /> is called the dimension of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198039.png" />. The matrix entries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198040.png" /> are almost-periodic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198041.png" />. In the theory of almost-periodic functions on a group they play the same role as the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198042.png" /> in the theory of almost-periodic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198043.png" />.

Two representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198045.png" /> are said to be equivalent if a constant matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198046.png" /> exists such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198047.png" />. A representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198048.png" /> is said to be irreducible if the family of the matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198050.png" />, does not admit a common non-trivial subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198051.png" />. The set of all irreducible unitary representations is partitioned into classes of mutually-equivalent representations. Let one representation be chosen from each equivalence class and let the set thus obtained be denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198052.png" />. Then the set of almost-periodic functions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198053.png" /></td> </tr></table>

on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198054.png" /> turns out to be an orthogonal (though, in general, uncountable) system with respect to the mean value.

Theorem 1 (the Parseval equality). For an almost-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198055.png" /> the following equality holds:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198056.png" /></td> </tr></table>

(Thus, for only countably many <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198058.png" /> differs from zero; the series

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198059.png" /></td> </tr></table>

is called the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198060.png" />.

A representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198061.png" /> is said to occur in the Fourier series of an almost-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198062.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198063.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198065.png" />.

Theorem 2 (the approximation theorem). The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198066.png" /> is dense in the space of almost-periodic functions equipped with the norm

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198067.png" /></td> </tr></table>

and every almost-periodic function can be arbitrarily well approximated by a finite linear combination of matrix entries of representations occurring in its Fourier series.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198068.png" /> is a topological group, then to the definition of an almost-periodic function should be added the requirement of its continuity. In this case, the representations occurring in its Fourier series are also continuous.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198069.png" /> is an Abelian group, then the continuous unitary representations are one-dimensional. They are called the characters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198070.png" />. The characters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198071.png" /> are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198072.png" /> and Parseval's equality reads as follows:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198073.png" /></td> </tr></table>

In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198074.png" /> the continuous characters are the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198075.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198077.png" />. Theorems 1 and 2 imply the main results in the theory of almost-periodic functions of a single or of several variables.

The proof of the main statements in the theory of almost-periodic functions is based on the consideration of integral equations on a group (cf. [[#References|[2]]]). The existence of sufficiently many linear representations of compact Lie groups has been proved [[#References|[3]]]. In this case, invariant integration (and consequently, the mean) can be established directly. Invariant integration on an abstract compact group has been constructed [[#References|[4]]] depending on an extension of the Peter–Weyl theory to this case.

The theory of almost-periodic functions on a group can be deduced (cf. [[#References|[3]]]) from the Peter–Weyl theory in the following way. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198078.png" /> be an almost-periodic function on a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198079.png" /> and let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198080.png" /></td> </tr></table>

Then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198081.png" /> is a normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198082.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198083.png" /> is an invariant metric on the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198084.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198085.png" /> is uniformly continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198086.png" />.

The almost-periodicity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198087.png" /> implies that the completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198088.png" /> in the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198089.png" /> is a compact group and Theorems 1 and 2 follow from the Peter–Weyl theory.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.M. Levitan,   "Almost-periodic functions" , Moscow  (1953)  pp. Chapt. 6  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Weyl,   "Integralgleichungen und fastperiodische Funktionen"  ''Math. Ann.'' , '''97'''  (1927)  pp. 338–356</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  F. Peter,   H. Weyl,   "Die Vollständigkeit der primitiven Darstellungen einer geschlossener kontinuierlichen Gruppe"  ''Math. Ann.'' , '''97'''  (1927)  pp. 737–755</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J. von Neumann,   "Zum Haarschen Mass in topologischen Gruppen"  ''Compositio Math.'' , '''1'''  (1934)  pp. 106–114</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. von Neumann,   "Almost periodic functions in a group I"  ''Trans. Amer. Math. Soc.'' , '''36'''  (1934)  pp. 445–492</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A. Weil,   ''C.R. Acad. Sci. Paris Sér. I Math.'' , '''200'''  (1935)  pp. 38–40</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  K. Yosida,   "Functional analysis" , Springer  (1980)  pp. Chapt. 8, Sect. 4; 5</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  W. Maak,   "Fastperiodische Funktionen" , Springer  (1950)</TD></TR></table>



====Comments====
Instead of the term  "mean value"  one often uses the term  "invariant-mean functional for almost-periodic functions43A07invariant mean"  (cf. [[#References|[a1]]], Sect. 18).

For an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198090.png" /> the uniformly almost-periodic functions are precisely those that can be continuously extended to the [[Bohr compactification|Bohr compactification]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198091.png" />.

A unified account of the theory of almost-periodic functions on groups can also be found in [[#References|[a2]]] and [[#References|[a3]]], Sect. 41. The basic observation is that the [[Banach algebra|Banach algebra]] of (continuous) almost-periodic functions on a (topological) group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198092.png" /> is isomorphic to the Banach algebra of all continuous functions on the so-called [[Bohr compactification|Bohr compactification]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198093.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198094.png" />. In this way the theory is reduced to the theory of continuous functions on a compact group (e.g., the mean-value theorem corresponds to the normalized [[Haar measure|Haar measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198095.png" />, the approximation theorem is nothing else than the well-known Peter–Weyl theorem for compact groups, etc.). The Bohr compactification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198096.png" /> can be characterized as the reflection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198097.png" /> in the subcategory of all compact groups. By considering reflections in other subcategories of the category of all topological groups (or even of all semi-topological semi-groups) one can define other classes of almost-periodic functions on groups (or semi-groups), see [[#References|[a4]]]. Weakly almost-periodic functions are of particular interest in functional-analytic applications (semi-groups of operators). See also [[#References|[7]]] and [[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Hewitt,   K.A. Ross,   "Abstract harmonic analysis" , '''1''' , Springer  (1979)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Weil,   "l'Intégration dans les groupes topologiques et ses applications" , Hermann  (1940)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.H. Loomis,   "An introduction to abstract harmonic analysis" , v. Nostrand  (1953)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.F. Berglund,   H.D. Junghen,   P. Milnes,  , ''Compact right to topological semigroups and generalizations of almost periodicity'' , ''Lect. notes in math.'' , '''663''' , Springer  (1978)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R.B. Burckel,   "Weakly almost periodic functions on semi-groups" , Gordon &amp; Breach  (1970)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  C. Corduneanu,   "Almost periodic functions" , Interscience  (1961)  pp. Chapt. 7</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  I. Glicksberg,   K. de Leeuw,   "Almost periodic functions on semigroups"  ''Acta Math.'' , '''105'''  (1961)  pp. 99–140</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  L. Amerio,   G. Prouse,   "Almost-periodic functions and functional equations" , v. Nostrand  (1971)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  J. Dixmier,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198098.png" /> algebras" , North-Holland  (1977)  (Translated from French)</TD></TR></table>