Poisson wavelets on $n$-dimensional spheres

TL;DR

This paper introduces Poisson wavelets on n-dimensional spheres, detailing their mathematical properties, explicit representations, localization, and inversion formulas, advancing wavelet analysis on spherical domains.

Contribution

It provides a comprehensive characterization of Poisson wavelets on spheres, including their expansions, recursive formulas, localization, and Euclidean limits, which were not previously established.

Findings

Derived Gegenbauer expansions for Poisson wavelets.

Established recursive formulas for explicit wavelet representations.

Provided inversion formulas and analyzed space localization and Euclidean limits.

Abstract

In this paper, Poisson wavelets on $n$-dimensional spheres, derived from Poisson kernel, are introduced and characterized. We compute their Gegenbauer expansion with respect to the origin of the sphere, as well as with respect to the field source. Further, we give recursive formulae for their explicit representations and we show how the wavelets are localized in space. Also their Euclidean limit is calculated explicitly and its space localization is described. We show that Poisson wavelets can be treated as wavelets derived from approximate identities and we give two inversion formulae.