Stepanov almost-periodic functions

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A class of functions that are measurable and summable together with their -th power on every finite interval and that can be approximated in the metric of the Stepanov space (see below) by finite sums

where are complex coefficients and are real numbers. The distance in the Stepanov space is defined by the formula

Functions of the class can also be defined using the concept of an almost-period.

Functions of the class possess a number of properties also possessed by Bohr almost-periodic functions. For example, functions of the class are bounded and uniformly continuous (in the metric ), the limit of a convergent sequence of Stepanov almost-periodic functions (in the metric of ) belongs to . If a function in is uniformly continuous (in the ordinary sense) on the whole real axis, then it is a Bohr almost-periodic function. Introduced by V.V. Stepanov [1].


[1] W. [V.V. Stepanov] Stepanoff, "Sur quelques généralisations des fonctions presque périodiques" C.R. Acad. Sci. Paris , 181 (1925) pp. 90–92


See also Almost-periodic function.

The different spaces , each with its metric , are topologically equivalent.


[a1] W. [V.V. Stepanov] Stepanoff, "Ueber einige Verallgemeinerungen der fastperiodischen Funktionen" Math. Ann. , 45 (1925) pp. 473–498
How to Cite This Entry:
Stepanov almost-periodic functions. B.M. Bredikhin (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098