How good can polynomial interpolation on the sphere be

Abstract

This paper explores the quality of polynomial interpolation approximations over the sphere S r−1⊂R r in the uniform norm, principally for r=3. Reimer [17] has shown there exist fundamental systems for which the norm ‖Λ n ‖ of the interpolation operator Λ n , considered as a map from C(S r−1) to C(S r−1), is bounded by d n , where d n is the dimension of the space of all spherical polynomials of degree at most n. Another bound is d n 1/2(λavg/λmin )1/2, where λavg and λmin are the average and minimum eigenvalues of a matrix G determined by the fundamental system of interpolation points. For r=3 these bounds are (n+1)2 and (n+1)(λavg/λmin )1/2, respectively. In a different direction, recent work by Sloan and Womersley [24] has shown that for r=3 and under a mild regularity assumption, the norm of the hyperinterpolation operator (which needs more point values than interpolation) is bounded by O(n 1/2), which is optimal among all linear projections. How much can the gap between interpolation and hyperinterpolation be closed? For interpolation the quality of the polynomial interpolant is critically dependent on the choice of interpolation points. Empirical evidence in this paper suggests that for points obtained by maximizing λmin , the growth in ‖Λ n ‖ is approximately n+1 for n<30. This choice of points also has the effect of reducing the condition number of the linear system to be solved for the interpolation weights. Choosing the points to minimize the norm directly produces fundamental systems for which the norm grows approximately as 0.7n+1.8 for n<30. On the other hand, ‘minimum energy points’, obtained by minimizing the potential energy of a set of (n+1)2 points on S 2, turn out empirically to be very bad as interpolation points. This paper also presents numerical results on uniform errors for approximating a set of test functions, by both interpolation and hyperinterpolation, as well as by non-polynomial interpolation with certain global basis functions.