Needlets Liberated

TL;DR

This paper introduces generalized needlets on the sphere by replacing exact integration rules with QMC designs, enabling easier numerical construction while maintaining convergence properties, and provides error analysis and numerical validation.

Contribution

It relaxes the exactness requirement of needlet construction by using QMC designs, creating more practical generalized needlets with preserved convergence.

Findings

Generalized needlets achieve similar asymptotic convergence as traditional needlets.

Hybrid schemes combine traditional and generalized needlets for different levels.

Numerical experiments validate the theoretical error analysis.

Abstract

Spherical needlets were introduced by Narcowich, Petrushev, and Ward to provide a multiresolution sequence of polynomial approximations to functions on the sphere. The needlet construction makes use of integration rules that are exact for polynomials up to a given degree. The aim of the present paper is to relax the exactness of the integration rules by replacing them with QMC designs as introduced by Brauchart, Saff, Sloan, and Womersley (2014). Such integration rules (generalised here by allowing non-equal cubature weights) provide the same asymptotic order of convergence as exact rules for Sobolev spaces $\mathbb{H}^s$, but are easier to obtain numerically. With such rules we construct ``generalised needlets''. The paper provides an error analysis that allows the replacement of the original needlets by generalised needlets, and more generally, analyses a hybrid scheme in which the needlets for the lower levels are of the traditional kind, whereas the new generalised needlets are used for some number of higher levels. Numerical experiments complete the paper.