Numerische Berechnung extremaler Fundamentalsysteme für Polynomräume über der Vollkugel. (Calculation of extremal fundamental systems for polynomial spaces over the unit ball) (Q1262069)

Multivariate polynomial interpolation is the subject of this paper. The authors develop an exchange algorithm for calculation of extremal fundamental systems (nodal systems with a Lebesgue constant not greater than dimension) as dimension and complexity increase rapidly with the algebraic degree. This method is based on the authors' earlier studies on polynomial spaces over the sphere and will be transferred by isomorphic reflections to polynomial spaces over the unit ball \(B^ r\). The authors document for \(B^ 3\) and the algebraic degree 6 an extremal fundamental system with a Lebesgue constant \(<26\) (remark that the dimension is 84) numerically. The corresponding cubature is demonstrated on examples of the type \[ \int_{x\in B^ 3}f(x_ 1)\cdot \frac{dx_ 1dx_ 2dx_ 3}{\sqrt{1-x^ 2_ 1-x^ 2_ 2-x^ 2\quad_ 3}}. \]