Needlet approximation for isotropic random fields on the sphere

TL;DR

This paper develops a multiscale needlet-based approximation method for isotropic random fields on spheres, proving convergence and providing numerical validation with Gaussian fields.

Contribution

It introduces a fully discrete needlet approximation for isotropic random fields on spheres and establishes convergence properties comparable to semidiscrete methods.

Findings

Semidiscrete needlet decomposition converges in mean and pointwise for weakly isotropic fields.

Fully discrete needlet approximation achieves similar convergence order as semidiscrete.

Numerical experiments confirm effectiveness for Gaussian random fields.

Abstract

In this paper we establish a multiscale approximation for random fields on the sphere using spherical needlets --- a class of spherical wavelets. We prove that the semidiscrete needlet decomposition converges in mean and pointwise senses for weakly isotropic random fields on $\mathbb{S}^{d}$, $d\ge2$. For numerical implementation, we construct a fully discrete needlet approximation of a smooth $2$-weakly isotropic random field on $\mathbb{S}^{d}$ and prove that the approximation error for fully discrete needlets has the same convergence order as that for semidiscrete needlets. Numerical examples are carried out for fully discrete needlet approximations of Gaussian random fields and compared to a discrete version of the truncated Fourier expansion.