Uncertainty of Poisson wavelets

TL;DR

This paper analyzes the uncertainty product of Poisson wavelets on the sphere, revealing that they tend to minimal uncertainty in certain limits, indicating optimal localization properties similar to the Gaussian kernel.

Contribution

It introduces the computation of the uncertainty product for Poisson wavelets, providing new insights into their localization characteristics on the sphere.

Findings

Uncertainty product of Poisson wavelets tends to the minimal value in some limits.

Poisson wavelets exhibit optimal space-frequency localization properties.

Gauss kernel also shares this minimal uncertainty property.

Abstract

Poisson wavelets are a powerful tool in analysis of spherical signals. In order to have a deeper characterization of them, we compute their uncertainty product, a quantity introduced for the first time by Narcowich and Ward in~\cite{NW96} and used to measure the trade-off between the space and frequency localization of a function. Surprisingly, the uncertainty product of Poisson wavelets tends to the minimal value in some limiting cases. This shows that in the case of spherical functions, not only Gauss kernel has this property.