A solution of Braess' approximation problem on powers of the distance function
Christiane Kraus · Weierstraß-Institut für Angewandte Analysis und Stochastik · 2006
The polynomial approximation behaviour of the class of functions $$ F_s: R^2\(x_0, y_0 ) -> R, F_s(x,y) = ( (x-x_0)^2 + (y-y_0)^2 )^(-s), s \in (0, \infty),$$ is studied in [Bra01]. There it is claimed that the obtained results can be embedded in a more general setting. This conjecture will be confirmed and complemented by a different approach than in [Bra01]. The key is to connect the approximation rate of F_s with its holomorphic continuability for which the classical Bernstein approximation theorem is linked with the convexity of best approximants. Approximation results of this kind also play a vital role in the numerical treatment of elliptic differential equations [Sau].