Frames of directional wavelets on $n$-dimensional spheres

TL;DR

This paper proves the existence of discrete directional wavelet frames on n-dimensional spheres and explores energy conservation properties, extending previous wavelet constructions with new theoretical guarantees.

Contribution

It establishes the existence of discrete wavelet frames on spheres and demonstrates energy conservation properties under certain spectral constraints, broadening wavelet theory.

Findings

Discrete frames of directional wavelets exist on n-dimensional spheres.

Energy conservation holds even when wavelets are not their own reconstruction family.

All previously defined wavelets satisfy the spectral constraint for energy conservation.

Abstract

The major goal of the paper is to prove that discrete frames of (directional) wavelets derived from an approximate identity exist. Additionally, a kind of energy conservation property is shown to hold in the case when a wavelet family is not its own reconstruction family. Although an additional constraint on the spectrum of the wavelet family must be satisfied, it is shown that all the wavelets so far defined in the literature possess this property.