Refined support and entropic uncertainty inequalities

TL;DR

This paper extends entropic and support uncertainty principles to frames, introduces a generalized coherence for sharper inequalities, and explores minimizers and $\,\ell^p$-norms in finite dimensions.

Contribution

It presents generalized entropic and support uncertainty inequalities for frames, with a new coherence measure and analysis of minimizers in finite-dimensional settings.

Findings

Sharpened support inequality with generalized coherence

Minimizers are constant functions on their support under certain conditions

Introduces $\,\ell^p$-norm inequalities as byproducts

Abstract

Generalized versions of the entropic (Hirschman-Beckner) and support (Elad-Bruckstein) uncertainty principle are presented for frames representations. Moreover, a sharpened version of the support inequality has been obtained by introducing a generalization of the coherence. In the finite dimensional case and under certain conditions, minimizers of this inequalities are given as constant functions on their support. In addition, $\ell^p$-norms inequalities are introduced as byproducts of the entropic inequalities.