polynomially convex hull
If is a polynomially convex set, then all the components of the interior of are simply connected.
One of the reasons for this definition is the following result.
For example if we take (the unit circle) then (the closed unit disc). The fact that the inside of the disc belongs to follows from the maximum modulus principle as polynomials are analytic functions. The fact that does not contain anything outside the closed unit disc follows by looking at the polynomial which has always greater modulus outside of the unit disc then anywhere on the unit circle. So if we have a function defined on a neighbourhood of the unit circle and which we can approximate uniformly on compact subsets of this neighbourhood by polynomials, then we can extend this function analytically to the whole unit disc. So this for example implies that cannot be approximated uniformly on compact subsets by polynomials on a neighbourhood of the unit circle.
The reason why we call a “hull” of some is that the conventional convex hull of can be defined as the set of points such that for all linear functions we have . This coincides with conventional definition because if is not in the conventional convex hull, then there is a linear functional that separates from the hull (by Hahn-Banach theorem in general, or more elementarily in by Farkas’s lemma), and conversely if is in the convex hull of then such linear function does not exist for the same reason. So, intuitively the conventional convex hull is set of point that are inseparable from from by linear functions. Polynomially convex hull is the same thing, but with polynomials. Of course similar definitions can be made with respect to other classes of functions. For example, hulls with respect to plurisubharmonic functions are very useful in multivariate complex analysis.
- 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
|Title||polynomially convex hull|
|Date of creation||2013-03-22 14:21:15|
|Last modified on||2013-03-22 14:21:15|
|Last modified by||jirka (4157)|