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
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where are complex coefficients and
are real numbers. The distance in the Stepanov space is defined by the formula
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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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stepanov_almost-periodic_functions&oldid=33130