View source for Keldysh-Lavrent'ev theorem

''on uniform approximation by entire functions''

In order that there exist for any continuous complex-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055140/k0551401.png" /> on a continuum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055140/k0551402.png" /> and any rapidly-decreasing positive function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055140/k0551403.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055140/k0551404.png" /> (as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055140/k0551405.png" />), having a positive lower bound on any finite interval, an entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055140/k0551406.png" /> such that

<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/k/k055/k055140/k0551407.png" /></td> </tr></table>

it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055140/k0551408.png" /> has no interior points and that there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055140/k0551409.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055140/k05514010.png" />, that increases to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055140/k05514011.png" /> and is such that any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055140/k05514012.png" /> of the complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055140/k05514013.png" /> can be joined to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055140/k05514014.png" /> by a Jordan curve situated outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055140/k05514015.png" /> and outside the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055140/k05514016.png" />.

This result of M.V. Keldysh and M.A. Lavrent'ev [[#References|[1]]] summarizes numerous investigations on approximation by entire functions initiated by the [[Carleman theorem|Carleman theorem]] (Section 3; see also [[#References|[2]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.V. Keldysh,   M.A. Lavrent'ev,   "Sur un problème de M. Carleman"  ''Dokl. Akad. Nauk SSSR'' , '''23''' :  8  (1939)  pp. 746–748</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.N. Mergelyan,   "Uniform approximation to functions of a complex variable"  ''Transl. Amer. Math. Soc.'' , '''3'''  (1962)  pp. 294–391  ''Uspekhi Mat. Nauk'' , '''7''' :  2  (1952)  pp. 31–1A2</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Gaier,   "Lectures on complex approximation" , Birkhäuser  (1987)  (Translated from German)</TD></TR></table>