Mergelyan's theorem

The set of Mergelyan , named after the Armenian mathematician SN Mergelyan is, a set of the approximation theory about approximation by polynomials , at the same time the generalized approximation of Weierstrass and the set of Runge .

Formulation of the sentence

• Be compact and the uniform algebra of all continuous functions that in , the inside of , holomorphic are. If the complement has no bounded connected components , then each function can be approximated by uniformly by polynomials. ${\displaystyle K\subset \mathbb {C} }$ ${\displaystyle A(K)}$ ${\displaystyle \mathrm {int} \,K}$ ${\displaystyle K}$ ${\displaystyle \mathbb {C} \setminus K}$ ${\displaystyle A(K)}$

Remarks

The bounded connected components of are also called holes. Is so compact with no holes, so the polynomials are set according to the above tightly in . ${\displaystyle \mathbb {C} \setminus K}$ ${\displaystyle K}$ ${\displaystyle A(K)}$

Has no interior, then the algebra of continuous functions is on . Therefore follows: If compact without holes and is , then the polynomials are sealed in . A bounded closed interval is an example of such , and this gives us the classical Weierstrasse approximation theorem. ${\displaystyle K}$ ${\displaystyle A(K)=C(K)}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle \mathrm {int} \,K=\emptyset }$ ${\displaystyle C(K)}$ ${\displaystyle K}$

Runge's theorem, which asserts the approximation by polynomials of holomorphic functions defined in a neighborhood of (compact without holes), is a direct consequence of Mergelyan's theorem, because the restrictions of such functions on are obviously in . The version of the Rungean theorem about the compact uniform approximation of holomorphic functions on open sets without holes by polynomials can easily be derived by exhausting the open set with a suitable sequence of compact subsets. ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle A(K)}$

Generalization to Riemannian surfaces

For a generalization to Riemannian surfaces , the problem arises that one cannot speak of polynomials there. If, however, the polynomials in Mergelyan's theorem above are replaced by whole functions , an equivalent formation is obtained, since polynomials are whole and, conversely, whole functions are compactly and uniformly approximated by their Taylor polynomials . The holomorphic functions on a Riemann surface correspond to all the functions . In this form E. Bishop succeeded in making the following generalization, which is also known as the Mergelyan-Bishop theorem. ${\displaystyle \mathbb {C} }$ ${\displaystyle X}$

• Be a non-compact Riemann surface and be compact without holes. Then every function can be approximated uniformly by holomorphic functions . ${\displaystyle X}$ ${\displaystyle K\subset X}$ ${\displaystyle A(K)}$ ${\displaystyle X\rightarrow \mathbb {C} }$

Of course, such a theorem cannot exist on a compact Riemann surface, since all holomorphic functions are constant on it.

Individual evidence

1. ↑ SN Mergelyan: On the representation of functions by series of polynomials on closed sets , Doklady Akad. Nauk SSSR (NS), Volume 78, pages 405-408, 1951.
2. ↑ TW Gamelin: Uniform Algebras , Chelsea Publishing Company 1969, Chapter II, Theorem 9.1 (Mergelyan's Theorem)
3. ^ E. Bishop; Subalgebras of functions on a Riemann surface , Pacific J. Mathematics 1958, Volume 8, pages 29-50
4. ^ Marek Janicki, Peter Pflug: Extension of Holomorphic Functions , Walter de Gruyter-Verlag (2000), ISBN 3-11-015363-7 , Theorem 1.11.15 (Theorem of Mergelyan-Bishop)