View source for Steklov function

''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" />''

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>

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>

<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>

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

<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 <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

<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>

<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>

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" />.

====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>



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.W. Cheney,   "Introduction to approximation theory" , Chelsea, reprint  (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.W. Müller,   "Approximationstheorie" , Akad. Verlagsgesellschaft  (1978)</TD></TR></table>