On the uncertainty product of spherical functions

TL;DR

This paper derives a formula linking the uncertainty product of spherical functions to their Fourier coefficients, explores its behavior for zonal wavelets, and discusses the implications for uncertainty principles on spheres.

Contribution

It provides a new formula for the uncertainty product of functions on the sphere in terms of Fourier coefficients and analyzes its behavior for specific wavelet functions.

Findings

Derived a Fourier coefficient-based formula for the uncertainty product.

Analyzed the uncertainty product of a directional derivative of a zonal wavelet.

Discussed the implications for uncertainty principles on the sphere.

Abstract

The uncertainty product of a function is a quantity that measures the trade-off between the space and the frequency localization of the function. Its boundedness from below is the content of various uncertainty principles. In the present paper, functions over the $n$-dimensional sphere are considered. A formula is derived that expresses the uncertainty product of a continuous function in terms of its Fourier coefficients. It is applied to a directional derivative of a zonal wavelet, and the behavior of the uncertainty product of this function is discussed.