Multiresolution on n-dimensional spheres

TL;DR

This paper develops multiscale polynomial wavelet systems on n-dimensional spheres, analyzing their properties and providing algorithms for wavelet transform, with applications to zonal functions and uncertainty principles.

Contribution

It introduces a novel multiresolution framework of polynomial wavelets on n-dimensional spheres, including construction, properties, and algorithms.

Findings

Wavelet systems exhibit good localization and positive definiteness.

Decomposition and reconstruction algorithms are effectively formulated.

Uncertainty products for zonal functions are derived and applied.

Abstract

In the present paper, multiscale systems of polynomial wavelets on an n-dimensional sphere are constructed. Scaling functions and wavelets are investigated,and their reproducing and localization properties and positive definiteness are examined. Decomposition and reconstruction algorithms for the wavelet transform are presented. Formulae for variances in space and momentum domain, as well as for the uncertainty product, of zonal functions over n-dimensional spheres are derived and applied to the scaling functions.