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''for an integrable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087690/s0876901.png" /> on a bounded segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087690/s0876902.png" />''
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''for an integrable function  $  f $
 +
on a bounded segment $  [ a, b] $''
  
 
The function
 
The function
  
<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/s087690/s0876903.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
f _ {h} ( t)  =
 +
\frac{1}{h}
 +
\int\limits _ { t- } h/2 ^ { t+ }  h/2 f( u)  du  = \
 +
 
 +
\frac{1}{h}
 +
\int\limits _ { - } h/2 ^ { h/2 }  f( t+ v) dv.
 +
$$
  
 
Functions of the form (*), as well as the iteratively defined functions
 
Functions of the form (*), as well as the iteratively defined functions
  
<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/s087690/s0876904.png" /></td> </tr></table>
+
$$
 +
f _ {h,r} ( t)  = \
 +
 
 +
\frac{1}{h}
 +
\int\limits _ { t- } h/2 ^ { t+ }  h/2 f _ {h,r-} 1 ( u)  du ,\ \
 +
r = 2, 3 \dots
 +
$$
 +
 
 +
$$
 +
f _ {h,1} ( t)  = f _ {h} ( t),
 +
$$
 +
 
 +
were first introduced in 1907 by V.A. Steklov (see [[#References|[1]]]) in solving the problem of expanding a given function into a series of eigenfunctions. The Steklov function  $  f _ {h} $
 +
has derivative
  
<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/s087690/s0876905.png" /></td> </tr></table>
+
$$
 +
f _ {h} ^ { \prime } ( t)  = \
  
were first introduced in 1907 by V.A. Steklov (see [[#References|[1]]]) in solving the problem of expanding a given function into a series of eigenfunctions. The Steklov function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087690/s0876906.png" /> has derivative
+
\frac{1}{h}
 +
\left \{ f \left ( t+
 +
\frac{h}{2}
 +
\right ) - f \left ( t-
 +
\frac{h}{2}
 +
\right ) \right \}
 +
$$
  
<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/s087690/s0876907.png" /></td> </tr></table>
+
almost everywhere. If  $  f $
 +
is uniformly continuous on the whole real axis, then
  
almost everywhere. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087690/s0876908.png" /> is uniformly continuous on the whole real axis, then
+
$$
 +
\sup _ {t \in (- \infty , \infty ) }  | f( t) - f _ {h} ( t) |  \leq  \omega \left (
 +
\frac{h}{2}
  
<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/s087690/s0876909.png" /></td> </tr></table>
+
, f  \right ) ,
 +
$$
  
<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/s087690/s08769010.png" /></td> </tr></table>
+
$$
 +
\sup _ {t \in (- \infty , \infty ) }  | f _ {h} ^ { \prime } ( t) |  \leq 
 +
\frac{1}{h}
 +
\omega ( h, f  ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087690/s08769011.png" /> is the modulus of continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087690/s08769012.png" />. Similar inequalities hold in the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087690/s08769013.png" />, provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087690/s08769014.png" />.
+
where $  \omega ( \delta , f  ) $
 +
is the modulus of continuity of $  f $.  
 +
Similar inequalities hold in the metric of $  L _ {p} (- \infty , \infty ) $,  
 +
provided $  f \in L _ {p} (- \infty , \infty ) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Steklov,  "On the asymptotic representation of certain functions defined by a linear differential equation of the second order, and their application to the problem of expanding an arbitrary function into a series of these functions" , Khar'kov  (1957)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.I. [N.I. Akhiezer] Achiezer,  "Theory of approximation" , F. Ungar  (1956)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Steklov,  "On the asymptotic representation of certain functions defined by a linear differential equation of the second order, and their application to the problem of expanding an arbitrary function into a series of these functions" , Khar'kov  (1957)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.I. [N.I. Akhiezer] Achiezer,  "Theory of approximation" , F. Ungar  (1956)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 08:23, 6 June 2020


for an integrable function $ f $ on a bounded segment $ [ a, b] $

The function

$$ \tag{* } f _ {h} ( t) = \frac{1}{h} \int\limits _ { t- } h/2 ^ { t+ } h/2 f( u) du = \ \frac{1}{h} \int\limits _ { - } h/2 ^ { h/2 } f( t+ v) dv. $$

Functions of the form (*), as well as the iteratively defined functions

$$ f _ {h,r} ( t) = \ \frac{1}{h} \int\limits _ { t- } h/2 ^ { t+ } h/2 f _ {h,r-} 1 ( u) du ,\ \ r = 2, 3 \dots $$

$$ f _ {h,1} ( t) = f _ {h} ( t), $$

were first introduced in 1907 by V.A. Steklov (see [1]) in solving the problem of expanding a given function into a series of eigenfunctions. The Steklov function $ f _ {h} $ has derivative

$$ f _ {h} ^ { \prime } ( t) = \ \frac{1}{h} \left \{ f \left ( t+ \frac{h}{2} \right ) - f \left ( t- \frac{h}{2} \right ) \right \} $$

almost everywhere. If $ f $ is uniformly continuous on the whole real axis, then

$$ \sup _ {t \in (- \infty , \infty ) } | f( t) - f _ {h} ( t) | \leq \omega \left ( \frac{h}{2} , f \right ) , $$

$$ \sup _ {t \in (- \infty , \infty ) } | f _ {h} ^ { \prime } ( t) | \leq \frac{1}{h} \omega ( h, f ), $$

where $ \omega ( \delta , f ) $ is the modulus of continuity of $ f $. Similar inequalities hold in the metric of $ L _ {p} (- \infty , \infty ) $, provided $ f \in L _ {p} (- \infty , \infty ) $.

References

[1] V.A. Steklov, "On the asymptotic representation of certain functions defined by a linear differential equation of the second order, and their application to the problem of expanding an arbitrary function into a series of these functions" , Khar'kov (1957) (In Russian)
[2] N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)

Comments

Steklov's fundamental paper was first published in French (1907) in the "Communications of the Mathematical Society of Kharkov" ; [1] is the Russian translation, together with additional comments by N.S. Landkof.

References

[a1] E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982)
[a2] M.W. Müller, "Approximationstheorie" , Akad. Verlagsgesellschaft (1978)
How to Cite This Entry:
Steklov function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steklov_function&oldid=17657
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article