Uncertainty product of the spherical Abel-Poisson wavelet

TL;DR

This paper computes the variances and uncertainty product limit of the spherical Abel-Poisson wavelet, revealing its similarity to the spherical Poisson wavelet with a specific parameter substitution.

Contribution

It provides the first detailed analysis of the uncertainty product for the spherical Abel-Poisson wavelet and compares it to the spherical Poisson wavelet.

Findings

Variances in space and momentum domains are calculated.

The uncertainty product limit as scale parameter approaches zero is derived.

Values closely match those of the spherical Poisson wavelet with a parameter substitution.

Abstract

In the present paper, the variances in the space and momentum domains of the spherical Abel-Poisson wavelet, as well as the limit of the uncertainty product for $\rho\to0$, where $\rho$ is the scale parameter, are computed. The values of these quantities coincide with a certain accuracy with those of the spherical Poisson wavelet with $\tfrac{1}{2}$ substituted for the order parameter.