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Mutual estimates of time-frequency representations and uncertainty principles

Angela A. Albanese, Claudio Mele, Alessandro Oliaro

Abstract

In this paper we give different estimates between Lebesgue norms of quadratic time-frequency representations. We show that, in some cases, it is not possible to have such bounds in classical $L^p$ spaces, but the Lebesgue norm needs to be suitably weighted. This leads to consider weights of polynomial type, and, more generally, of ultradifferentiable type, and this, in turn, gives rise to use as functional setting the ultradifferentiable classes. As applications of such estimates we deduce uncertainty principles both of Donoho-Stark type and of local type for representations.

Mutual estimates of time-frequency representations and uncertainty principles

Abstract

In this paper we give different estimates between Lebesgue norms of quadratic time-frequency representations. We show that, in some cases, it is not possible to have such bounds in classical spaces, but the Lebesgue norm needs to be suitably weighted. This leads to consider weights of polynomial type, and, more generally, of ultradifferentiable type, and this, in turn, gives rise to use as functional setting the ultradifferentiable classes. As applications of such estimates we deduce uncertainty principles both of Donoho-Stark type and of local type for representations.
Paper Structure (4 sections, 22 theorems, 95 equations)

This paper contains 4 sections, 22 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Definitions and preliminary results
  3. Estimates on time-frequency representations and Donoho-Stark uncertainty inequalities
  4. Local uncertainty principles for representations in ${\mathcal{S}}_\omega({\mathbb R}^N)$

Key Result

Theorem 1.1

Let $T,\Omega\subset\mathbb{R}^N$ be measurable sets, and $\varepsilon_T,\varepsilon_\Omega\geq 0$ with $\varepsilon_T+\varepsilon_\Omega\leq 1$. If there exists $f\in L^2(\mathbb{R}^N)$, $f\neq 0$, such that $f$ is $\varepsilon_T$-concentrated on $T$ and $\hat{f}$ is $\varepsilon_\Omega$-concentrat where $m(\cdot)$ indicates the Lebesgue measure of the corresponding set.

Theorems & Definitions (47)

  • Theorem 1.1: Donoho-Stark uncertainty principle DS
  • Theorem 1.2
  • Theorem \ref{DSVgfintro}$'$
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Definition 2.5
  • Proposition 2.6: AC2
  • Theorem 2.7
  • ...and 37 more