# Weyl almost-periodic functions

From Encyclopedia of Mathematics

The class of complex-valued functions , , summable to degree on each bounded interval of the real axis and such that for every there is an for which has a relatively-dense set of -almost-periods (cf. Almost-period). The class was defined by H. Weyl [1]. The class of Weyl almost-periodic functions is an extension of the class of Stepanov almost-periodic functions.

Weyl almost-periodic functions are related to the metric

If is a null function in the metric , i.e.

and is a Stepanov almost-periodic function, then

(*) |

is a Weyl almost-periodic function. There also exist Weyl almost-periodic functions which cannot be represented in the form (*); cf. [3].

#### References

[1] | H. Weyl, "Integralgleichungen und fastperiodische Funktionen" Math. Ann. , 97 (1927) pp. 338–356 |

[2] | B.M. Levitan, "Almost-periodic functions" , Moscow (1953) (In Russian) |

[3] | B.M. Levitan, V.V. Stepanov, "Sur les fonctions presque périodiques apportenant au sens strict à la classe " Dokl. Akad. Nauk SSSR , 22 : 5 (1939) pp. 220–223 |

**How to Cite This Entry:**

Weyl almost-periodic functions. E.A. Bredikhina (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Weyl_almost-periodic_functions&oldid=16250

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098