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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" />.
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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
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 +
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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" />;
+
A generalization of almost-periodic functions defined on  $  \mathbf R $.
 +
Let  $  G $
 +
be an (abstract) group. A bounded complex-valued function  $  f(x) $,
 +
$  x \in G $,
 +
is called a right almost-periodic function if the family  $  f ( x a ) $,
 +
where  $  a $
 +
runs through the entire group  $  G $,
 +
is (relatively) compact in the topology of uniform convergence on  $  G $,
 +
i.e. if every sequence of functions  $  f ( a x _ {1} ) , f ( x a _ {2} ) \dots $
 +
contains a subsequence which is uniformly convergent on  $  G $.  
 +
A left almost-periodic function on  $  G $
 +
is defined similarly. It turns out that every right (left) almost-periodic function  $  f $
 +
is also left (right) almost-periodic, and the family  $  f ( a x b ) $,
 +
where  $  a $
 +
and  $  b $
 +
independently run through  $  G $,  
 +
is (relatively) compact. The latter property is often taken as a definition of almost-periodic functions on  $  G $.  
 +
The set of all almost-periodic functions on  $  G $
 +
is a Banach space with a norm  $  \| f \| = \sup _ {x \in G }  | f (x) | $.
  
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" />.
+
The theory of almost-periodic functions on a group depends essentially on the mean-value theorem (cf. [[#References|[5]]], [[#References|[8]]]). A linear functional  $  M _ {x} \{ f (x) \} $
 +
defined on the space of almost-periodic functions is called a mean value if
  
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" />.
+
1)  $  M _ {x} \{ 1 \} = 1 , M _ {x} \{ f (x) \} \geq  0 $
 +
for  $  f (x) \geq  0 $
 +
and $  M _ {x} \{ f (x) \} > 0 $
 +
for  $  f (x) \geq  0 $,  
 +
$  f \not\equiv 0 $;
  
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
+
2)  $  M _ {x} \{ f ( x a ) \} = M _ {x} \{ f ( a x ) \} = M _ {x} \{ f ( x  ^ {-1} ) \} = M _ {x} \{ f (x) \} $,  
 +
for all  $  a \in G $.
  
<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>
+
A unitary matrix function  $  g (x) = \{ g _ {ij} (x) \} _ {i,j=1}  ^ {r} $,
 +
defined on  $  G $,
 +
is called a unitary representation of  $  G $
 +
if  $  g (e) = I _ {r} $(
 +
$  e $
 +
is the identity element of  $  G $
 +
and  $  I _ {r} $
 +
is the identity matrix of order  $  r $)
 +
and if for all  $  x , y \in G $,
 +
$  g ( x y ) = g (x) g (y) $.
 +
The number  $  r $
 +
is called the dimension of the representation  $  g $.
 +
The matrix entries  $  g _ {ij} (x) $
 +
are almost-periodic functions on  $  G $.  
 +
In the theory of almost-periodic functions on a group they play the same role as the functions  $  \mathop{\rm exp} ( i \lambda (x)) $
 +
in the theory of almost-periodic functions on  $  \mathbf R $.
  
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.
+
Two representations  $  g (x) $
 +
and  $  g  ^  \prime  (x) $
 +
are said to be equivalent if a constant matrix  $  A $
 +
exists such that  $  g  ^  \prime  (x) = A  ^ {-1} g (x) A $.  
 +
A representation  $  g $
 +
is said to be irreducible if the family of the matrices  $  g (x) $,  
 +
$  x \in G $,  
 +
does not admit a common non-trivial subspace in  $  \mathbf R  ^ {r} $.
 +
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  $  S $.  
 +
Then the set of almost-periodic functions
  
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:
+
$$
 +
H  =  \{ \phi _  \lambda  (x) \}  = \
 +
\{ {\phi _  \lambda  } : {
 +
\phi _  \lambda  = g _ {ij}  ^  \lambda  , g \in S } \}
 +
$$
  
<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>
+
on  $  G $
 +
turns out to be an orthogonal (though, in general, uncountable) system with respect to the mean value.
  
(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
+
Theorem 1 (the Parseval equality). For an almost-periodic function  $  f (x) $
 +
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/a01198059.png" /></td> </tr></table>
+
$$
 +
M _ {x} \{ | f (x) |  ^ {2} \}  = \
 +
\sum
  
is called the Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011980/a01198060.png" />.
+
\frac{| M _ {x} \{ f (x) \overline \phi \; _  \lambda  (x) \} |  ^ {2} }{M _ {x} \{ | \phi _  \lambda  (x) |  ^ {2} \} }
 +
.
 +
$$
  
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" />.
+
(Thus, for only countably many  $  \lambda $,
 +
$  M _ {x} \{ f (x) \overline \phi \; _  \lambda  (x) \} $
 +
differs from zero; the series
  
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
+
$$
 +
\sum \phi _  \lambda 
 +
\frac{M _ {x} \{ f (x)
 +
\overline \phi \; _  \lambda  (x) \} }{[ M _ {x} \{ | \phi _  \lambda  (x) |
 +
^ {2} \} ]  ^ {1/2} }
  
<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>
+
$$
 +
 
 +
is called the Fourier series of  $  f $.
 +
 
 +
A representation  $  g \in S $
 +
is said to occur in the Fourier series of an almost-periodic function  $  f $
 +
if  $  M _ {x} \{ f (x) \overline{g}\; _ {ij} (x) \} \neq 0 $
 +
for some  $  i , j $,
 +
$  1 \leq  i, j \leq  r $.
 +
 
 +
Theorem 2 (the approximation theorem). The set  $  H $
 +
is dense in the space of almost-periodic functions equipped with the norm
 +
 
 +
$$
 +
\| f \|  = \
 +
\sup _
 +
{x \in G } \
 +
| f (x) | ,
 +
$$
  
 
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.
 
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 $  G $
 +
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:
+
If $  G $
 +
is an Abelian group, then the continuous unitary representations are one-dimensional. They are called the characters of $  G $.  
 +
The characters of $  G $
 +
are denoted by $  \chi $
 +
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>
+
$$
 +
M _ {x} \{ | f (x) |  ^ {2} \}  = \
 +
\sum _ { n }
 +
| a _ {n} |  ^ {2} ,\ \
 +
a _ {n}  = M _ {x} \{ f (x) \overline \chi \; _ {n} (x) \} .
 +
$$
  
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.
+
In the case $  G = \mathbf R  ^ {n} $
 +
the continuous characters are the functions $  \chi (x) = \mathop{\rm exp} ( i \lambda \cdot x ) $,  
 +
where $  \lambda \in \mathbf R  ^ {n} $,  
 +
$  \lambda \cdot x = \lambda _ {1} x _ {1} + \dots + \lambda _ {n} x _ {n} $.  
 +
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 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
+
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 $  f $
 +
be an almost-periodic function on a group $  G $
 +
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>
+
$$
 +
\rho ( x , y )  = \
 +
\sup _ {a , b \in G } \
 +
| f ( a x b ) - f ( a y b ) | .
 +
$$
  
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" />.
+
Then the set $  E = \{ {t \in G } : {\rho ( t , e ) = 0 } \} $
 +
is a normal subgroup of $  G $,  
 +
$  \rho $
 +
is an invariant metric on the quotient group $  G / E $
 +
and $  f $
 +
is uniformly continuous on $  G / E $.
  
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.
+
The almost-periodicity of $  f $
 +
implies that the completion of $  G / E $
 +
in the metric $  \rho $
 +
is a compact group and Theorems 1 and 2 follow from the Peter–Weyl theory.
  
 
====References====
 
====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>
 
<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====
 
====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).
 
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" />.
+
For an Abelian group $  G $
 +
the uniformly almost-periodic functions are precisely those that can be continuously extended to the [[Bohr compactification|Bohr compactification]] of $  G $.
  
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]]].
+
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 $  G $
 +
is isomorphic to the Banach algebra of all continuous functions on the so-called [[Bohr compactification|Bohr compactification]] $  G _ {c} $
 +
of $  G $.  
 +
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 $  G _ {c} $,  
 +
the approximation theorem is nothing else than the well-known Peter–Weyl theorem for compact groups, etc.). The Bohr compactification of $  G $
 +
can be characterized as the reflection of $  G $
 +
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====
 
====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>
 
<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>

Revision as of 16:10, 1 April 2020


A generalization of almost-periodic functions defined on $ \mathbf R $. Let $ G $ be an (abstract) group. A bounded complex-valued function $ f(x) $, $ x \in G $, is called a right almost-periodic function if the family $ f ( x a ) $, where $ a $ runs through the entire group $ G $, is (relatively) compact in the topology of uniform convergence on $ G $, i.e. if every sequence of functions $ f ( a x _ {1} ) , f ( x a _ {2} ) \dots $ contains a subsequence which is uniformly convergent on $ G $. A left almost-periodic function on $ G $ is defined similarly. It turns out that every right (left) almost-periodic function $ f $ is also left (right) almost-periodic, and the family $ f ( a x b ) $, where $ a $ and $ b $ independently run through $ G $, is (relatively) compact. The latter property is often taken as a definition of almost-periodic functions on $ G $. The set of all almost-periodic functions on $ G $ is a Banach space with a norm $ \| f \| = \sup _ {x \in G } | f (x) | $.

The theory of almost-periodic functions on a group depends essentially on the mean-value theorem (cf. [5], [8]). A linear functional $ M _ {x} \{ f (x) \} $ defined on the space of almost-periodic functions is called a mean value if

1) $ M _ {x} \{ 1 \} = 1 , M _ {x} \{ f (x) \} \geq 0 $ for $ f (x) \geq 0 $ and $ M _ {x} \{ f (x) \} > 0 $ for $ f (x) \geq 0 $, $ f \not\equiv 0 $;

2) $ M _ {x} \{ f ( x a ) \} = M _ {x} \{ f ( a x ) \} = M _ {x} \{ f ( x ^ {-1} ) \} = M _ {x} \{ f (x) \} $, for all $ a \in G $.

A unitary matrix function $ g (x) = \{ g _ {ij} (x) \} _ {i,j=1} ^ {r} $, defined on $ G $, is called a unitary representation of $ G $ if $ g (e) = I _ {r} $( $ e $ is the identity element of $ G $ and $ I _ {r} $ is the identity matrix of order $ r $) and if for all $ x , y \in G $, $ g ( x y ) = g (x) g (y) $. The number $ r $ is called the dimension of the representation $ g $. The matrix entries $ g _ {ij} (x) $ are almost-periodic functions on $ G $. In the theory of almost-periodic functions on a group they play the same role as the functions $ \mathop{\rm exp} ( i \lambda (x)) $ in the theory of almost-periodic functions on $ \mathbf R $.

Two representations $ g (x) $ and $ g ^ \prime (x) $ are said to be equivalent if a constant matrix $ A $ exists such that $ g ^ \prime (x) = A ^ {-1} g (x) A $. A representation $ g $ is said to be irreducible if the family of the matrices $ g (x) $, $ x \in G $, does not admit a common non-trivial subspace in $ \mathbf R ^ {r} $. 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 $ S $. Then the set of almost-periodic functions

$$ H = \{ \phi _ \lambda (x) \} = \ \{ {\phi _ \lambda } : { \phi _ \lambda = g _ {ij} ^ \lambda , g \in S } \} $$

on $ G $ 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 $ f (x) $ the following equality holds:

$$ M _ {x} \{ | f (x) | ^ {2} \} = \ \sum \frac{| M _ {x} \{ f (x) \overline \phi \; _ \lambda (x) \} | ^ {2} }{M _ {x} \{ | \phi _ \lambda (x) | ^ {2} \} } . $$

(Thus, for only countably many $ \lambda $, $ M _ {x} \{ f (x) \overline \phi \; _ \lambda (x) \} $ differs from zero; the series

$$ \sum \phi _ \lambda \frac{M _ {x} \{ f (x) \overline \phi \; _ \lambda (x) \} }{[ M _ {x} \{ | \phi _ \lambda (x) | ^ {2} \} ] ^ {1/2} } $$

is called the Fourier series of $ f $.

A representation $ g \in S $ is said to occur in the Fourier series of an almost-periodic function $ f $ if $ M _ {x} \{ f (x) \overline{g}\; _ {ij} (x) \} \neq 0 $ for some $ i , j $, $ 1 \leq i, j \leq r $.

Theorem 2 (the approximation theorem). The set $ H $ is dense in the space of almost-periodic functions equipped with the norm

$$ \| f \| = \ \sup _ {x \in G } \ | f (x) | , $$

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 $ G $ 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 $ G $ is an Abelian group, then the continuous unitary representations are one-dimensional. They are called the characters of $ G $. The characters of $ G $ are denoted by $ \chi $ and Parseval's equality reads as follows:

$$ M _ {x} \{ | f (x) | ^ {2} \} = \ \sum _ { n } | a _ {n} | ^ {2} ,\ \ a _ {n} = M _ {x} \{ f (x) \overline \chi \; _ {n} (x) \} . $$

In the case $ G = \mathbf R ^ {n} $ the continuous characters are the functions $ \chi (x) = \mathop{\rm exp} ( i \lambda \cdot x ) $, where $ \lambda \in \mathbf R ^ {n} $, $ \lambda \cdot x = \lambda _ {1} x _ {1} + \dots + \lambda _ {n} x _ {n} $. 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. [2]). The existence of sufficiently many linear representations of compact Lie groups has been proved [3]. In this case, invariant integration (and consequently, the mean) can be established directly. Invariant integration on an abstract compact group has been constructed [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. [3]) from the Peter–Weyl theory in the following way. Let $ f $ be an almost-periodic function on a group $ G $ and let

$$ \rho ( x , y ) = \ \sup _ {a , b \in G } \ | f ( a x b ) - f ( a y b ) | . $$

Then the set $ E = \{ {t \in G } : {\rho ( t , e ) = 0 } \} $ is a normal subgroup of $ G $, $ \rho $ is an invariant metric on the quotient group $ G / E $ and $ f $ is uniformly continuous on $ G / E $.

The almost-periodicity of $ f $ implies that the completion of $ G / E $ in the metric $ \rho $ is a compact group and Theorems 1 and 2 follow from the Peter–Weyl theory.

References

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

Comments

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

For an Abelian group $ G $ the uniformly almost-periodic functions are precisely those that can be continuously extended to the Bohr compactification of $ G $.

A unified account of the theory of almost-periodic functions on groups can also be found in [a2] and [a3], Sect. 41. The basic observation is that the Banach algebra of (continuous) almost-periodic functions on a (topological) group $ G $ is isomorphic to the Banach algebra of all continuous functions on the so-called Bohr compactification $ G _ {c} $ of $ G $. 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 on $ G _ {c} $, the approximation theorem is nothing else than the well-known Peter–Weyl theorem for compact groups, etc.). The Bohr compactification of $ G $ can be characterized as the reflection of $ G $ 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 [a4]. Weakly almost-periodic functions are of particular interest in functional-analytic applications (semi-groups of operators). See also [7] and [a5].

References

[a1] E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 1 , Springer (1979)
[a2] A. Weil, "l'Intégration dans les groupes topologiques et ses applications" , Hermann (1940)
[a3] L.H. Loomis, "An introduction to abstract harmonic analysis" , v. Nostrand (1953)
[a4] 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)
[a5] R.B. Burckel, "Weakly almost periodic functions on semi-groups" , Gordon & Breach (1970)
[a6] C. Corduneanu, "Almost periodic functions" , Interscience (1961) pp. Chapt. 7
[a7] I. Glicksberg, K. de Leeuw, "Almost periodic functions on semigroups" Acta Math. , 105 (1961) pp. 99–140
[a8] L. Amerio, G. Prouse, "Almost-periodic functions and functional equations" , v. Nostrand (1971)
[a9] J. Dixmier, " algebras" , North-Holland (1977) (Translated from French)
How to Cite This Entry:
Almost-periodic function on a group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Almost-periodic_function_on_a_group&oldid=13583
This article was adapted from an original article by V.V. ZhikovB.M. Levitan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article