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# Title:Needlet approximation for isotropic random fields on the sphere

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.
 Comments: 28 pages, 8 figures, added an illustration of an advantage of needlet approximations which allow local concentration of nodes in regions of most interests Subjects: Numerical Analysis (math.NA); Probability (math.PR) MSC classes: 60G60, 42C40, 41A25, 65D32, 60G15, 33G55 Cite as: arXiv:1512.07790 [math.NA] (or arXiv:1512.07790v2 [math.NA] for this version)

## Submission history

From: Yu Guang Wang [view email]
[v1] Thu, 24 Dec 2015 11:19:26 UTC (845 KB)
[v2] Sat, 10 Dec 2016 09:22:15 UTC (1,456 KB)