Directional wavelets on $n$-dimensional spheres

TL;DR

This paper introduces directional Poisson wavelets on n-dimensional spheres, explores their properties, and establishes their admissibility as a wavelet pair, providing explicit formulas and Fourier coefficient recursions.

Contribution

It presents the first construction and analysis of directional Poisson wavelets on n-dimensional spheres, including admissibility and explicit representations.

Findings

Directional Poisson wavelets are admissible with another wavelet family.

Recursive formulas for Fourier coefficients are derived.

Explicit representations and Euclidean limits are provided.

Abstract

Directional Poisson wavelets, being directional derivatives of Poisson kernel, are introduced on $n$-dimensional spheres. It is shown that, slightly modified and together with another wavelet family, they are an admissible wavelet pair according to the definition derived from the theory of approximate identities. We investigate some of the properties of directional Poisson wavelets, such as recursive formulae for their Fourier coefficients or explicit representations as functions of spherical variables (for some of the wavelets). We derive also an explicit formula for their Euclidean limits.