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Uncertainty principles for integral operators

Tom 220 / 2014

Saifallah Ghobber, Philippe Jaming Studia Mathematica 220 (2014), 197-220 MSC: Primary 42A68; Secondary 42C20. DOI: 10.4064/sm220-3-1

Streszczenie

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.

Autorzy

  • Saifallah GhobberDépartement de Mathématiques Appliquées
    Institut Préparatoire aux Études d'Ingénieurs de Nabeul
    Université de Carthage
    Campus Universitaire
    Merazka, 8000, Nabeul, Tunisie
    e-mail
  • Philippe JamingUniversité Bordeaux
    IMB, UMR 5251
    F-33400 Talence, France
    and
    CNRS, IMB, UMR 5251
    F-33400 Talence, France
    e-mail

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