WIAS Preprint No. 1171, (2006)

A solution of Braess' approximation problem on powers of the distance function


  • Kraus, Christiane

2010 Mathematics Subject Classification

  • 41A25 41A10 41A60 41A63 41A17


  • Polynomial approximation in 2-space, maximal convergence, Bernstein-Walsh's type theorems




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].

Appeared in

  • Constructive Approximation, 27, p. 323-327, 2008 under the new title: Multivariate Polynomial Approximation of Powers of the Euclidian Distance Function

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