The uniform norm of hyperinterpolation on the unit sphere in an arbitrary number of dimensions, Constructive Approximation

Abstract. In this paper we study the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere Sr¡1 ‰ Rr. The hyperinterpolation approximation Ln f, where f 2 C.Sr¡1/, is derived from the exact L2 orthogonal projection 5n f onto the space Prn.Sr¡1 / of spherical polynomials of degree n or less, with the Fourier coefficients approximated by a positive weight quadrature rule that integrates exactly all polynomials of degree • 2n. We extend to arbitrary r the recent r D 3 result of Sloan and Womersley [9], by proving that under an additional “quadrature regularity ” assumption on the quadrature rule, the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere is O.nr=2¡1/, which is the same as that of the orthogonal projection 5n, and best possible among all linear projections onto Prn.Sr¡1/. 1