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. [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)

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, "$C^*$ algebras" , North-Holland (1977) (Translated from French)