View source for Keldysh theorem

Keldysh' theorem on approximating continuous functions by polynomials. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k0551501.png" /> be a function of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k0551502.png" /> that is holomorphic in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k0551503.png" /> and continuous in the closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k0551504.png" />. Then in order that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k0551505.png" /> a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k0551506.png" /> exists 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/k055150/k0551507.png" /></td> </tr></table>

it is necessary and sufficient that the complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k0551508.png" /> consists of a single domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k0551509.png" /> containing the point at infinity. The theorem was established by M.V. Keldysh . It is one of the basic results in the theory of uniform approximation of functions by polynomials in the complex domain (see ).

Keldysh' theorems in potential theory are theorems on the solvability of the [[Dirichlet problem|Dirichlet problem]], established by M.V. Keldysh in 1938–1941.

a) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515010.png" /> be a bounded domain in the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515011.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515012.png" />, with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515013.png" />. Then there exists on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515014.png" /> a countable set of irregular boundary points (cf. [[Irregular boundary point|Irregular boundary point]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515015.png" />, such that the Dirichlet problem is solvable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515016.png" /> with a continuous boundary function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515017.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515018.png" /> if and only if this problem is solvable at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515020.png" /> that is, if and only if

<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/k055150/k05515021.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515022.png" /> is the generalized solution of the Dirichlet problem in the sense of Wiener–Perron (see [[Perron method|Perron method]], and also [[#References|[3]]], [[#References|[4]]]).

b) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515023.png" /> be an operator acting from the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515024.png" /> of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515025.png" /> into the space of bounded harmonic functions (cf. [[Harmonic function|Harmonic function]]) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515026.png" /> and satisfying the following conditions: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515027.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515030.png" /> are real numbers; that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515031.png" /> is linear; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515032.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515034.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515035.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515036.png" />) if the Dirichlet problem is solvable for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515037.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515038.png" /> gives the solution of this problem. Under these conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515039.png" /> is unique for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515040.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515041.png" /> gives a generalized solution of the Dirichlet problem in the sense of Wiener–Perron (see [[#References|[5]]]–[[#References|[7]]]).

c) In order that each solvable Dirichlet problem in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515042.png" /> be stable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515043.png" />, it is necessary and sufficient that the set of irregular boundary points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515044.png" /> coincide with the set of irregular boundary points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515045.png" />. The Dirichlet problem is stable in the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515046.png" /> with respect to any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515047.png" /> if and only if the set of irregular boundary points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515048.png" /> belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515049.png" /> has zero [[Harmonic measure|harmonic measure]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515050.png" /> (see [[#References|[4]]] or [[#References|[6]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.V. Keldysh,   "Sur la réprésentation par des séries de polynômes des fonctions d'une variable complexe dans des domaines fermés"  ''Mat. Sb.'' , '''16''' :  3  (1945)  pp. 249–258</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.N. Mergelyan,   "Uniform approximations to functions of a complex variable"  ''Transl. Amer. Math. Soc. (1)'' , '''3'''  (1962)  pp. 294–391  ''Uspekhi Mat. Nauk'' , '''7''' :  2  (1952)  pp. 3–122</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.V. Keldysh,   "Sur la résolubilité et la stabilité du problème de Dirichlet"  ''Dokl. Akad. Nauk SSSR'' , '''18'''  (1938)  pp. 315–318</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.V. Keldysh,   "On the solvability and stability of the Dirichlet problem"  ''Uspekhi Mat. Nauk'' , '''8'''  (1941)  pp. 171–231  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.V. Keldysh,   "Sur le problème de Dirichlet"  ''Dokl. Akad. Nauk SSSR'' , '''32'''  (1941)  pp. 308–309</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  N.S. Landkof,   "Foundations of modern potential theory" , Springer  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  M. Brélot,   "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1959)</TD></TR></table>



====Comments====
For Keldysh' approximation theorem see also [[#References|[a2]]], Chapt. 30.

The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055150/k05515051.png" /> in b) is called a Keldysh operator. See [[#References|[a1]]] for a treatment of Keldysh operators in axiomatic potential theory.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Netuka,   "The classical Dirichlet problem and its generalizations" , ''Potential theory (Copenhagen, 1979)'' , ''Lect. notes in math.'' , '''787''' , Springer  (1980)  pp. 235–266</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Gaier,   "Lectures on complex approximation" , Birkhäuser  (1987)  (Translated from German)</TD></TR></table>