Abstract-ZusammenfassungThe Range of Values of a Circular Complex Polynomial over a Circular Complex Interval. Two Methods to find a small circular interval that encloses the range of values of a circular interval polynomial over a circular interval are developed. The first method uses Bernstein polynomials over the sides of a regular polygon enclosing the domain, the other method uses a mean value property of curves in the complex plane. They are then compared to each other as well as to the result obtained from the Hornerscheme and the true range.
Der Wertebereich eines Intervallpolynoms mit Kreisen als Koeffizienten.Es werden zwei Methoden entwickelt, einen m6glichst kleinen Kreisbereich zu finden, der den Wertebereich eines Kreisintervallpolynoms auf einem gegebenen Kreisbereich einschliegt. Die erste Methode beniitzt Bernsteinpolynome auf den Seiten eines umschriebenen reguliiren Polygons, die andere eine Mittelwerteigeuschaft yon Kurven in der komplexen Ebene. Die beiden werden untereinander und auch mit dem Hornerschema und dem wirklichen Wertebereich verglichen.
Let H(U) be the linear space of holomorphic functions on U = {z:|z|<1} endowed with the topology of compact convergence, and denote by H′(U) its topological dual space. Let be a compact subset of H(U) and ƒ∈F. We say ƒ is a support point of if there exists an L∈H'(U), non-constant on , such that On the other hand, ƒ is an extreme point of if ƒ is not a proper convex combination of two other points of .
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