Stepanov almost-periodic functions
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].
References
| [1] | W. [V.V. Stepanov] Stepanoff, "Sur quelques généralisations des fonctions presque périodiques" C.R. Acad. Sci. Paris , 181 (1925) pp. 90–92 |
Comments
See also Almost-periodic function.
The different spaces 
, each with its metric 
, are topologically equivalent.
References
| [a1] | W. [V.V. Stepanov] Stepanoff, "Ueber einige Verallgemeinerungen der fastperiodischen Funktionen" Math. Ann. , 45 (1925) pp. 473–498 |
Stepanov almost-periodic functions. B.M. Bredikhin (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Stepanov_almost-periodic_functions&oldid=11586