# Almost-periodic function on a group

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. , ). 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. ). The existence of sufficiently many linear representations of compact Lie groups has been proved . In this case, invariant integration (and consequently, the mean) can be established directly. Invariant integration on an abstract compact group has been constructed  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. ) 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.

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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=45812
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