A variation of the $L^p$ uncertainty principles for the Weinstein transform

TL;DR

This paper generalizes uncertainty principles related to the Weinstein transform within $L_eta^p$-norm spaces, extending classical results like Heisenberg's inequality to broader contexts in mathematical analysis.

Contribution

It introduces a generalized framework for uncertainty principles for the Weinstein transform in $L_eta^p$ spaces, including new $L_eta^p$ versions of classical inequalities.

Findings

Extended Heisenberg-Pauli-Weyl uncertainty principle to general $L_eta^p$-norms.

Established three continuous concentration uncertainty principles.

Generalized Donoho-Stark-type inequalities for the Weinstein transform.

Abstract

The Weinstein operator has several applications in pure and applied Mathematics especially in Fluid Mechanics and satisfies some uncertainty principles similar to the Euclidean Fourier transform. The aim of this paper is establish a generalization of uncertainty principles for Weinstein transform in $L_\alpha^p$-norm. Firstly, we extend the Heisenberg-Pauli-Weyl uncertainty principle to more general case. Then we establish three continuous uncertainty principles of concentration type. The first and the second uncertainty principles are $L_\alpha^p$ versions and depend on the sets of concentration $\Omega$ and $\Sigma$, and on the time function $\varphi$. However, the third uncertainty principle is also $L_\alpha^p$ version depends on the sets of concentration and he is independent on the band limited function $\varphi$. These $L_\alpha^p$-Donoho-Stark-type inequalities generalize the results obtained in the case $p=q=2$.