APPLIED MATHEMATICS REPORT AMR 98/19 Constructive Approximation on the Sphere

This paper considers the problem of constructive approximation of a continuous function on the unit sphere Sr−1 ⊆ Rr by a spherical polynomial from the space Pn of all spherical polynomials of degree ≤ n. In particular, for r = 3 it is shown that the hyperinterpolation approximation Lnf (in which the Fourier coefficients in the exact L2 orthogonal projection Pnf are approximated by a positive weight quadrature rule that integrates exactly all polynomials of degree ≤ 2n) has the exact order ‖Ln‖ ≍ n1/2 for its uniform norm, provided the underlying quadrature rule satisfies an additional ‘quadrature regularity’ assumption. For r = 3, this rate of growth is the same as that of ‖Pn‖, and is known to be optimal among all linear projections on Pn. For r ≥ 3 an upper bound on ‖Ln‖ of non-optimal asymptotic order O(n(r−1)/2) also holds, without any special assumption on the quadrature rule.