Keldysh theorem

Keldysh' theorem on approximating continuous functions by polynomials. Let $ f ( z) $ be a function of a complex variable $ z $ that is holomorphic in a domain $ G $ and continuous in the closed domain $ \overline{G}\; $. Then in order that for any $ \epsilon > 0 $ a polynomial $ P ( z) $ exists such that

$$ | f ( z) - P ( z) | < \epsilon ,\ \ z \in \overline{G}\; , $$

it is necessary and sufficient that the complement $ C \overline{G}\; $ consists of a single domain $ G ^ {*} $ 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, established by M.V. Keldysh in 1938–1941.

a) Let $ D $ be a bounded domain in the Euclidean space $ \mathbf R ^ {n} , $ $ n \geq 2 $, with boundary $ \Gamma = \partial D $. Then there exists on $ \Gamma $ a countable set of irregular boundary points (cf. Irregular boundary point) $ \{ y _ {k} \} _ {k = 1 } ^ \infty $, such that the Dirichlet problem is solvable in $ D $ with a continuous boundary function $ f ( y) $ on $ \Gamma $ if and only if this problem is solvable at $ y _ {k} $, $ k = 1 , 2 \dots $ that is, if and only if

$$ \lim\limits _ {\begin{array}{c} x \rightarrow y _ {k} \\ x \in D \end{array} } \ u ( x) = f ( y _ {k} ) ,\ \ k = 1 , 2 \dots $$

where $ u ( x) $ is the generalized solution of the Dirichlet problem in the sense of Wiener–Perron (see Perron method, and also [3], [4]).

b) Let $ A $ be an operator acting from the space $ C ( \Gamma ) $ of continuous functions on $ \Gamma $ into the space of bounded harmonic functions (cf. Harmonic function) in $ D $ and satisfying the following conditions: $ \alpha $) $ A ( \alpha f + \beta g ) = \alpha A ( f ) + \beta A( g) $, $ f , g \in C ( \Gamma ) $, where $ \alpha , \beta $ are real numbers; that is, $ A $ is linear; $ \beta $) if $ f ( y) \geq 0 $, $ f \in C( \Gamma ) $, then $ A ( f )( x) \geq 0 $; and $ \gamma $) if the Dirichlet problem is solvable for an $ f \in C ( \Gamma ) $, then $ A ( f ) $ gives the solution of this problem. Under these conditions $ A $ is unique for $ f \in C ( \Gamma ) $, and $ A ( f ) $ gives a generalized solution of the Dirichlet problem in the sense of Wiener–Perron (see [5]–[7]).

c) In order that each solvable Dirichlet problem in $ D $ be stable in $ \overline{D}\; $, it is necessary and sufficient that the set of irregular boundary points of $ C \overline{D}\; $ coincide with the set of irregular boundary points of $ CD $. The Dirichlet problem is stable in the interior of $ D $ with respect to any function $ f \in C ( \Gamma ) $ if and only if the set of irregular boundary points of $ C \overline{D}\; $ belonging to $ \Gamma $ has zero harmonic measure in $ D $( see [4] or [6]).

References

[1]	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
[2]	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
[3]	M.V. Keldysh, "Sur la résolubilité et la stabilité du problème de Dirichlet" Dokl. Akad. Nauk SSSR , 18 (1938) pp. 315–318
[4]	M.V. Keldysh, "On the solvability and stability of the Dirichlet problem" Uspekhi Mat. Nauk , 8 (1941) pp. 171–231 (In Russian)
[5]	M.V. Keldysh, "Sur le problème de Dirichlet" Dokl. Akad. Nauk SSSR , 32 (1941) pp. 308–309
[6]	N.S. Landkof, "Foundations of modern potential theory" , Springer (1972) (Translated from Russian)
[7]	M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959)

Comments

For Keldysh' approximation theorem see also [a2], Chapt. 30.

The operator $ A $ in b) is called a Keldysh operator. See [a1] for a treatment of Keldysh operators in axiomatic potential theory.

References

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