Runge theory

In function theory, Runge's theory deals with the question of when holomorphic functions can be approximated in a sub-area by holomorphic functions in a larger area. It was largely developed by Carl Runge , who published his approximation theorem in 1885.

Runge theory for compacts

The set (red dots) matches every constrained component of (at least once). ${\displaystyle P}$ ${\displaystyle \mathbb {C} \setminus K}$

For a set, let the - algebra of the rational functions that have only in poles . ${\displaystyle P\subset \mathbb {C} }$ ${\displaystyle \mathbb {C} _{P}[z]}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle P}$

The Runge'sche approximation for compacta now states: Be a compact set . If every bounded component of holds , then every holomorphic function can be approximated uniformly by functions from . ${\displaystyle K\subset \mathbb {C} }$ ${\displaystyle P\subset \mathbb {C} \setminus K}$ ${\displaystyle \mathbb {C} \setminus K}$ ${\displaystyle K}$ ${\displaystyle \mathbb {C} _{P}[z]}$

As an important special case we obtain the small set of Runge : If a compact set the complement continuous , then each on uniformly approximated by polynomials holomorphic function. (Because in this case one can choose and rational functions without poles are polynomials.) ${\displaystyle K\subset \mathbb {C} }$ ${\displaystyle K}$ ${\displaystyle P=\emptyset }$

Runge theory for areas

The set of Runge on rational approximation is: Be one area and a lot whose completion in each hole of hits. Then the algebra with respect to the topology of the compact convergence lies close to the algebra of the holomorphic functions . A compact component of is referred to here as a hole . ${\displaystyle \Omega \subseteq \mathbb {C} }$ ${\displaystyle P\subset \mathbb {C} \setminus \Omega }$ ${\displaystyle {\overline {P}}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \Omega }$ ${\displaystyle \mathbb {C} _{P}[z]}$ ${\displaystyle {\mathcal {O}}(\Omega )}$ ${\displaystyle \mathbb {C} \setminus \Omega }$

Two domains are called Runge's pair , if each on holomorphic function can be approximated equally on compacta by on holomorphic functions. From the above approximation theorem, with the help of Behnke-Stein's theorem, the characterization follows: ${\displaystyle \Omega \subseteq \Omega '\subseteq \mathbb {C} }$ ${\displaystyle \Omega }$ ${\displaystyle \Omega '}$

${\displaystyle \Omega \subseteq \Omega '}$form a Runge pair if and only if has no compact components, i.e. has no holes relative to . ${\displaystyle \Omega '\setminus \Omega }$ ${\displaystyle \Omega }$ ${\displaystyle \Omega '}$

Applications

• The Mittag-Leffler's theorem can be derived from the Runge'schen sets.
• There exist pointwise convergent polynomial sequences that do not converge locally evenly on all compacts.
• The unit disk can be holomorphic and actually in embedding. (In fact even in , but this does not follow directly from Runge's theorems.) ${\displaystyle \mathbb {C} ^{3}}$ ${\displaystyle \mathbb {C} ^{2}}$
• Each area of is a holomorphic area , i. H. for each area there is a defined holomorphic function that cannot be holomorphically extended beyond this area. ${\displaystyle \mathbb {C} }$

Runge approximation on Riemannian surfaces

The approximation theorem was generalized to Riemannian surfaces by Behnke and Stein in 1948 . One cannot speak of polynomials on Riemann surfaces, but the approximability of a function by polynomials on compact sets is equivalent to the approximability of whole functions , as can be easily seen by breaking off the Taylor series , and in this form the following generalization is possible: ${\displaystyle K\subset \mathbb {C} }$

• Let be a Riemann surface and an open subset such that its complement has no compact connected components . Then each holomorphic function with respect to the topology of compact convergence can be approximated by holomorphic functions . ${\displaystyle X}$ ${\displaystyle Y\subset X}$ ${\displaystyle X\setminus Y}$ ${\displaystyle Y}$ ${\displaystyle X}$

Note that the statement for compact Riemann surfaces becomes trivial, because then it is necessary . For non-compact Riemann surfaces one obtains the non-trivial conclusion that , i. H. is called the 1st sheaf cohomology with values ​​in the sheaf of holomorphic functions vanishes. From this one can easily obtain the solvability of Mittag-Leffler problems (see Mittag-Leffler's theorem ) on non-compact Riemann surfaces. ${\displaystyle Y=X}$ ${\displaystyle H^{1}(X,{\mathcal {O}})=0}$ ${\displaystyle {\mathcal {O}}}$

generalization

• The set of Mergelyan (by Mergelyan 1951) additionally treated u. a. the problem with steady continuation on the edge.

literature

• Reinhold Remmert, Georg Schumacher: Function theory 2. Springer, Berlin 2007, ISBN 978-3-540-57052-3

Individual evidence

1. ↑ Wolfgang Fischer, Ingo Lieb: Funktionentheorie , Vieweg Verlag 1980, ISBN 3-528-07247-4 , Chapter VIII §1: The Runge approximation sets
2. ↑ H. Behnke, K. Stein: Development of analytical functions on Riemann surfaces. In: Math. Ann. , Vol. 120 (1947-1949), pp. 430-461
3. ↑ Otto Forster : Riemannsche surfaces , Springer-Verlag 1977, ISBN 3-540-08034-1 , 25.5: Runge's approximation theorem
4. ↑ Otto Forster : Riemannsche surfaces , Springer-Verlag 1977, ISBN 3-540-08034-1 , 26.1-26.3
5. ↑ SN Mergelyan: Uniform approximation to functions of a complex variable . In: Amer. Math. Soc. Translation , No. 101