Runge's theorem

In the theory of functions, the Runge- theory deals with the question of when may be approximated by holomorphic functions on a larger territory on a branch holomorphic functions. It was considerably developed by Carl Runge, who published in 1885 his approximation theorem.

Runge- theory for compacta

For a lot of the algebra is the rational functions that have only poles.

The Runge'sche approximation theorem for compact bone now states: Let be a compact. Does every bounded component, then every holomorphic function is to be approximated uniformly by functions.

As an important special case, one obtains the little set of Runge: If the complement is connected to a compact set, then each is uniformly approximable by polynomials on holomorphic function.

Runge- theory for spaces

The set of Runge rational approximation is as follows: Let be a field and a set whose conclusion is true in every hole. Then the algebra is dense in the algebra of holomorphic functions. The hole in this case a compact component is called.

Two areas Runge'sches hot couple if each of holomorphic function can be approximated uniformly by holomorphic functions on compacta on. From the above approximation theorem finally follows with the help of the theorem of Behnke -Stein characterization:

Applications

• The set of Mittag-Leffler can be derived from the Runge'schen sets.
• There are pointwise convergent polynomial sequences that do not converge locally uniformly on all compact sets.
• The unit disk can be holomorphic and actually embed in. ( In fact, even in what not but follows directly from the Runge'schen records.)
• Each area of a domain of holomorphy, ie each area there is a holomorphic function defined on it, which can not be extended holomorphically about this area also.

Generalizations

• On Riemann surfaces, by Behnke and Stein 1948
• Set of Mergelyan by Mergelyan 1951; additionally treated, among other problems with continuous continuation on the edge