The Uniform Error of Hyperinterpolation on the Sphere

This paper considers the problem of approximation of a continuous function on the unit sphere S ⊆ IR by a spherical polynomial from the space IPn of all spherical polynomials of degree ≤ n. For r = 3 it was shown in [16] that the hyperinterpolation approximation Lnf (which is the linear projection obtained by approximating the Fourier coefficients in the exact L2 orthogonal projection by a positive-weight quadrature rule that integrates exactly all polynomials of degree≤ 2n) has the exact order ‖Ln‖ ≍ n for its uniform norm, provided the underlying quadrature rule satisfies an additional ‘quadrature regularity’ assumption. This rate of growth is the same as that of ‖Pn‖, where Pnf is the L2 orthogonal projection, and is optimal among all linear projections on IPn for r = 3. For r ≥ 3 an upper bound on ‖Ln‖ of non-optimal asymptotic order O(n) also holds, without any special assumption on the quadrature rule. This paper first surveys these recent theoretical results, and then for the first time presents numerical results, both on the growth of the uniform norm of the hyperinterpolation operator up to degree 200, and on the uniform error in the approximation for a set of test functions.