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Difference between revisions of "Stepanov almost-periodic functions"

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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
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A class $S_l^p$ of functions that are measurable and summable together with their $p$-th power $(p\geq1)$ on every finite interval $[x,x+l]$ 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>
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$$\sum_{n=1}^Na_ne^{i\lambda_nx},$$
  
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
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where $a_n$ are complex coefficients and $\lambda_n$ 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>
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$$D_{S_l^p}[f(x),g(x)]=\sup_{-\infty<x<\infty}\left[\frac1l\int\limits_x^{x+l}|f(x)-g(x)|^pdx\right]^{1/p}.$$
  
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]].
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Functions of the class $S_l^p$ 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]]].
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Functions of the class $S^p=S_1^p$ possess a number of properties also possessed by [[Bohr almost-periodic functions|Bohr almost-periodic functions]]. For example, functions of the class $S^p$ are bounded and uniformly continuous (in the metric $D_{S_l^p}$), the limit $f$ of a convergent sequence of Stepanov almost-periodic functions $\{f_n\}$ (in the metric of $S^p$) belongs to $S^p$. If a function in $S^p$ 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====
 
====References====
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See also [[Almost-periodic function|Almost-periodic function]].
 
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.
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The different spaces $S_l^p$, each with its metric $D_{S_l^p}$, are topologically equivalent.
  
 
====References====
 
====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>
 
<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>

Latest revision as of 08:36, 25 August 2014

A class $S_l^p$ of functions that are measurable and summable together with their $p$-th power $(p\geq1)$ on every finite interval $[x,x+l]$ and that can be approximated in the metric of the Stepanov space (see below) by finite sums

$$\sum_{n=1}^Na_ne^{i\lambda_nx},$$

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

$$D_{S_l^p}[f(x),g(x)]=\sup_{-\infty<x<\infty}\left[\frac1l\int\limits_x^{x+l}|f(x)-g(x)|^pdx\right]^{1/p}.$$

Functions of the class $S_l^p$ can also be defined using the concept of an almost-period.

Functions of the class $S^p=S_1^p$ possess a number of properties also possessed by Bohr almost-periodic functions. For example, functions of the class $S^p$ are bounded and uniformly continuous (in the metric $D_{S_l^p}$), the limit $f$ of a convergent sequence of Stepanov almost-periodic functions $\{f_n\}$ (in the metric of $S^p$) belongs to $S^p$. If a function in $S^p$ 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 $S_l^p$, each with its metric $D_{S_l^p}$, are topologically equivalent.

References

[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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stepanov_almost-periodic_functions&oldid=11586
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article