# Almost-periodic function on a group

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

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

1) for and for , ;

2) , for all .

A unitary matrix function , defined on , is called a unitary representation of if ( is the identity element of and is the identity matrix of order ) and if for all , . The number is called the dimension of the representation . The matrix entries are almost-periodic functions on . In the theory of almost-periodic functions on a group they play the same role as the functions in the theory of almost-periodic functions on .

Two representations and are said to be equivalent if a constant matrix exists such that . A representation is said to be irreducible if the family of the matrices , , does not admit a common non-trivial subspace in . 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 . Then the set of almost-periodic functions

on 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 the following equality holds:

(Thus, for only countably many , differs from zero; the series

is called the Fourier series of .

A representation is said to occur in the Fourier series of an almost-periodic function if for some , .

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

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

In the case the continuous characters are the functions , where , . 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 be an almost-periodic function on a group and let

Then the set is a normal subgroup of , is an invariant metric on the quotient group and is uniformly continuous on .

The almost-periodicity of implies that the completion of in the metric 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 the uniformly almost-periodic functions are precisely those that can be continuously extended to the Bohr compactification of .

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 is isomorphic to the Banach algebra of all continuous functions on the so-called Bohr compactification of . 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 , the approximation theorem is nothing else than the well-known Peter–Weyl theorem for compact groups, etc.). The Bohr compactification of can be characterized as the reflection of 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) |

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Almost-periodic function on a group.

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