Colloquium: Needlet approximation on the sphere - a fully discrete version
Spherical needlets are highly localized radial polynomials on the sphere Sd Rd+1, d 2, with centers at the nodes of a suitable quadrature rule. The original spherical needlet approximation as proposed by Narcowich, Petru- shev and Ward has coefficients dened by inner product integrals. In this talk we rst review the needlet construction, and then report on recent joint work with Yu Guang Wang, Q Thong Le Gia and Robert Womersley, which uses an appropriate quadrature rule to construct a fully discrete (and hence constructible) needlet approximation. We prove that the global fully dis- crete approximation is equivalent to ltered hyperinterpolation, that is to a ltered Fourier-Laplace series partial sum with inner products replaced by quadrature rules of appropriate polynomial accuracy . With Heping Wang we have established Lp-error bounds and rates of convergence in Sobolev spaces Ws p(Sd), for 2 p 1, that for s d=p are exactly the same as for the original needlet approximation. Numerical experiments conrm this result. The discrete needlet construction is also shown to have potential for local approximation by a numerical experiment that uses low-level needlets globally together with high-level needlets in a local region.