Semi-continuous and discrete wavelet frames on $n$-dimensional spheres

TL;DR

This paper demonstrates that certain wavelet systems on n-dimensional spheres form semi-continuous and discrete frames under mild conditions, using kernel localization properties.

Contribution

It establishes the frame properties of spherical wavelets derived from approximate identities and Poisson wavelets on n-dimensional spheres.

Findings

Semi-continuous frames are formed by spherical wavelets under mild conditions.

Poisson wavelets form discrete frames on dense grids.

Localization properties of kernels are key to the proofs.

Abstract

The paper shows that under some mild conditions $n$-dimensional spherical wavelets derived from approximate identities build semi-continuous frames. Moreover, for sufficiently dense grids Poisson wavelets on $n$-dimensional spheres constitute a discrete frame. In the proof we only use the localization properties of the reproducing kernel and its gradient.