Filtered hyperinterpolation: a constructive polynomial approximation on the sphere

Abstract

This paper considers a fully discrete filtered polynomial approximation on the unit sphere ${\mathbb{S}^{d}.}$ For ${f \in C(\mathbb{S}^{d}),V_{L,N}^{(a)} \, f}$ is a polynomial approximation which is exact for all spherical polynomials of degree at most L, so it inherits good convergence properties in the uniform norm for sufficiently smooth functions. The oscillations often associated with polynomial approximation of less smooth functions are localised by using a filter with support [0, a] for some a > 1, and with the value 1 on [0, 1]. The allowed choice of filters includes a recently introduced filter with minimal smoothness, and other smoother filters. The approximation uses a cubature rule with N points which is exact for all polynomials of degree ${t = \left\lceil{a L}\right\rceil+L-1.}$ The main theoretical result is that the uniform norm ${\|V_{L,N}^{(a)} \|}$ of the filtered hyperinterpolation operator is bounded independently of L, providing both good convergence and stability properties. Numerical experiments on ${\mathbb{S}^{2}}$ with a variety of filters, support intervals and cubature rules illustrate the uniform boundedness of the operator norm and the convergence of the filtered hyperinterpolation approximation for both an arbitrarily smooth function and a function with derivative discontinuities.