Constructive Polynomial Approximation on the Sphere Original Research Article

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.