ANZIAM J. 42 (E) ppC1536--C1557, 2000.

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

Robert S. Womersley

Abstract

This paper considers the calculation of the minimum norm points for polynomial interpolation over the sphere S² Ã R³. The norm of the interpolation operator L_n, considered as a map from C(S²) to C(S²), is given by || L_n || = max_{x ‘ S²} ||B^-1 b(x)||₁, where the nonsingular matrix B and vector b are determined by the fundamental system of points x_j ‘ S², j = 1,º, d_n. The problem is to choose the fundamental system to minimise || L_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 || L_n ||, for n < 30, is less than c₀ + c₁ n, where c₀ ª 1.8 and c₁ ª 0.7.

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Authors

Robert S. Womersley

School of Mathematics, University of New South Wales, Sydney 2052, Australia. \protect\url{mailto:R.Womersley@unsw.edu.au}

Published 25 December, 2000