Needlet approximation for isotropic random fields on the sphere
This work provides a theoretical foundation and numerical validation for using needlets to approximate random fields on the sphere, which is relevant for applications in cosmology and geophysics.
The paper establishes a multiscale approximation for isotropic random fields on the sphere using spherical needlets, proving convergence in mean and pointwise senses for weakly isotropic fields. Numerical examples show that the fully discrete needlet approximation achieves the same convergence order as the semidiscrete version, outperforming truncated Fourier expansions.
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.