This paper explores the quality of polynomial interpolation approximations over the sphere Sr−1⊂Rr in the uniform norm, principally for r=3. Reimer [17] has shown there exist fundamental systems for which the norm ‖Λn‖ of the interpolation operator Λn, considered as a map from C(Sr−1) to C(Sr−1), is bounded by dn, where dn is the dimension of the space of all spherical polynomials of degree at most n. Another bound is dn1/2(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(n1/2), which is optimal among all linear projections. How much can the gap between interpolation and hyperinterpolation be closed?
['Robert S. Womersley', 'Ian H. Sloan']
How good can polynomial interpolation on the sphere be