the polynomial convex hull of . It is the supremum on .
A subset is called a polynomial convex if also holds for every compact subset .
Remarks
Are and compact subsets of , then obviously
From follows
.
This justifies the term envelope in analogy to the convex envelope. This analogy can be carried further: Note that the closed convex hull of a compact subset is equal to the set of all vectors , so that for all linear functionals . In the above definition, the linear functionals are replaced by polynomials. This analogy motivates the terms polynomial convex hull and polynomial convex .
Furthermore, this consideration shows that a compact set is polynomial convex if and only if . In particular, a compact set is polynomial convex if and only if there is a polynomial for each with
for everyone .
Examples
If polynomial convex, then is connected . In this case , the reverse applies, in this case the reverse is false.
The union of two disjoint convex sets im is polynomial convex; this does not generally apply to three sets.
is not polynomial convex.
A sentence from Oka
Polynomial convex sets play an important role in the approximation of holomorphic functions by polynomials. The one-dimensional case is exactly Runge's approximation theorem .
The Oka theorem can be reproduced in the following versions:
Let be a compact, polynomial convex set. Then every holomorphic function defined in a neighborhood of can be approximated uniformly by polynomials.
Let be a polynomial convex domain. Then every holomorphic function defined for it can be approximated compactly and uniformly by polynomials.
literature
Gunning - Rossi : Analytic functions of several complex variables . Prentice Hall 1965
Lars Hörmander : An Introduction to Complex Analysis in Several Variables , North-Holland Mathematical Library 1973