A continuous minimax problem for calculating minimum norm polynomial interpolation points on the sphere

Robert S. Womersley


This paper considers the calculation of the minimum norm points for polynomial interpolation over the sphere S 2 ? R 3 . The norm of the interpolation operator ? n , considered as a map from C(S 2 ) to C(S 2 ), is given by ? ? n ? = max x ? S 2 ?B -1 b (x)? 1 , where the nonsingular matrix B and vector b are determined by the fundamental system of points x j ? S 2 , j = 1,?, d n . The problem is to choose the fundamental system to minimise ? ? n ?. Algorithms for solving this continuous minimax problem must be able to handle many local maxima close to the global maximum, and local maxima which lie close to each other along ridges. A first order dual algorithm is used to find a spherical parametrisation of a normalised fundamental system. The results suggest that for these points the growth in ? ? n ?, for n ? 30, is less than c 0 + c 1 n, where c 0 ? 1.8 and c 1 ? 0.7.

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DOI: http://dx.doi.org/10.21914/anziamj.v42i0.658

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