Uncertainty principles for integral operators

Saifallah Ghobber ¹   ; Philippe Jaming ²

¹ Département de Mathématiques Appliquées Institut Préparatoire aux Études d'Ingénieurs de Nabeul Université de Carthage Campus Universitaire Merazka, 8000, Nabeul, Tunisie
² Université Bordeaux IMB, UMR 5251 F-33400 Talence, France and CNRS, IMB, UMR 5251 F-33400 Talence, France

Studia Mathematica, Tome 220 (2014) no. 3, pp. 197-220

Résumé

The aim of this paper is to prove new uncertainty principles for integral operators ${\mathcal T}$ with bounded kernel for which there is a Plancherel Theorem. The first of these results is an extension of Faris's local uncertainty principle which states that if a nonzero function $f\in L^2({\mathbb {R}}^d,\mu )$ is highly localized near a single point then ${\mathcal T} (f)$ cannot be concentrated in a set of finite measure. The second result extends the Benedicks–Amrein–Berthier uncertainty principle and states that a nonzero function $f\in L^2({\mathbb {R}}^d,\mu )$ and its integral transform ${\mathcal T} (f)$ cannot both have support of finite measure. From these two results we deduce a global uncertainty principle of Heisenberg type for the transformation ${\mathcal T}$. We apply our results to obtain new uncertainty principles for the Dunkl and Clifford Fourier transforms.

Détail

DOI : 10.4064/sm220-3-1

Keywords: paper prove uncertainty principles integral operators mathcal bounded kernel which there plancherel theorem first these results extension fariss local uncertainty principle which states nonzero function mathbb highly localized near single point mathcal cannot concentrated set finite measure second result extends benedicks amrein berthier uncertainty principle states nonzero function mathbb its integral transform mathcal cannot have support finite measure these results deduce global uncertainty principle heisenberg type transformation mathcal apply results obtain uncertainty principles dunkl clifford fourier transforms