# 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) |

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

#### Comments

The equivalence of left and right almost-periodicity was proved by Turing [b1].

#### References

[b1] | A.M. Turing, "Equivalence of left and right almost periodicity" J. Lond. Math. Soc. (1st ser.) 10 (1935) 284-285 DOI 10.1112/jlms/s1-10.40.284 Zbl 0012.40404 |

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Almost-periodic function on a group.

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