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''on uniform approximation by entire functions''
 
''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
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In order that there exist for any continuous complex-valued function $f(z)$ on a continuum $E$ and any rapidly-decreasing positive function $\epsilon(r)$, $0\leq r$ (as $r\to\infty$), having a positive lower bound on any finite interval, an entire function $g(z)$ 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>
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$$|f(z)-g(z)|<\epsilon(|z|),\quad z\in E,$$
  
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" />.
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it is necessary and sufficient that $E$ has no interior points and that there exists a function $\eta(t)$, $0<t<+\infty$, that increases to $+\infty$ and is such that any point $z$ of the complement $CE$ can be joined to $\infty$ by a Jordan curve situated outside $E$ and outside the disc $|\zeta|<\eta(|z|)$.
  
 
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]]]).
 
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]]]).

Latest revision as of 18:27, 13 August 2015

on uniform approximation by entire functions

In order that there exist for any continuous complex-valued function $f(z)$ on a continuum $E$ and any rapidly-decreasing positive function $\epsilon(r)$, $0\leq r$ (as $r\to\infty$), having a positive lower bound on any finite interval, an entire function $g(z)$ such that

$$|f(z)-g(z)|<\epsilon(|z|),\quad z\in E,$$

it is necessary and sufficient that $E$ has no interior points and that there exists a function $\eta(t)$, $0<t<+\infty$, that increases to $+\infty$ and is such that any point $z$ of the complement $CE$ can be joined to $\infty$ by a Jordan curve situated outside $E$ and outside the disc $|\zeta|<\eta(|z|)$.

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

References

[1] M.V. Keldysh, M.A. Lavrent'ev, "Sur un problème de M. Carleman" Dokl. Akad. Nauk SSSR , 23 : 8 (1939) pp. 746–748
[2] 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


Comments

References

[a1] D. Gaier, "Lectures on complex approximation" , Birkhäuser (1987) (Translated from German)
How to Cite This Entry:
Keldysh-Lavrent'ev theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Keldysh-Lavrent%27ev_theorem&oldid=17048
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article