View source for Stepanov almost-periodic functions

A class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087720/s0877201.png" /> of functions that are measurable and summable together with their <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087720/s0877202.png" />-th power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087720/s0877203.png" /> on every finite interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087720/s0877204.png" /> and that can be approximated in the metric of the Stepanov space (see below) by finite sums

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087720/s0877205.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087720/s0877206.png" /> are complex coefficients and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087720/s0877207.png" /> are real numbers. The distance in the Stepanov space is defined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087720/s0877208.png" /></td> </tr></table>

Functions of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087720/s0877209.png" /> can also be defined using the concept of an [[Almost-period|almost-period]].

Functions of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087720/s08772010.png" /> possess a number of properties also possessed by [[Bohr almost-periodic functions|Bohr almost-periodic functions]]. For example, functions of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087720/s08772011.png" /> are bounded and uniformly continuous (in the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087720/s08772012.png" />), the limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087720/s08772013.png" /> of a convergent sequence of Stepanov almost-periodic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087720/s08772014.png" /> (in the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087720/s08772015.png" />) belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087720/s08772016.png" />. If a function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087720/s08772017.png" /> 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 [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. [V.V. Stepanov] Stepanoff,   "Sur quelques généralisations des fonctions presque périodiques"  ''C.R. Acad. Sci. Paris'' , '''181'''  (1925)  pp. 90–92</TD></TR></table>



====Comments====
See also [[Almost-periodic function|Almost-periodic function]].

The different spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087720/s08772018.png" />, each with its metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087720/s08772019.png" />, are topologically equivalent.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. [V.V. Stepanov] Stepanoff,   "Ueber einige Verallgemeinerungen der fastperiodischen Funktionen"  ''Math. Ann.'' , '''45'''  (1925)  pp. 473–498</TD></TR></table>