Fully discrete needlet approximation on the sphere

Yu Guang Wang, Quoc T. Le Gia, Ian H. Sloan, Robert S. Womersley

Published 2015-02-20, updated 2015-04-10Version 3

Spherical needlets are highly localised radial polynomials on the sphere $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$, $d\geq2$, with centers at the nodes of a suitable quadrature rule. The original semidiscrete spherical needlet approximation has coefficients defined by inner product integrals. We use an appropriate quadrature rule to construct a fully discrete version. We prove that the fully discrete spherical needlet approximation is equivalent to filtered hyperinterpolation, that is to a filtered Fourier-Laplace series partial sum with inner products replaced by appropriate quadrature sums. We establish $\mathbb{L}_{p}$-error bounds and rates of convergence, $2\leq p\leq \infty$, for the fully discrete needlet approximation of functions in Sobolev spaces $\mathbb{W}_{p}^{s}(\mathbb{S}^{d})$ for $s>d/p$. The power of the needlet approximation for local approximation is shown by a numerical experiment that uses low-level needlets globally together with high-level needlets in a local region.