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A characterization of Fourier transforms

100%
Jaming P.
Colloquium Mathematicum
|
2010
|
tom 118
|
nr 2
569-580
EN
The aim of this paper is to show that, in various situations, the only continuous linear (or not) map that transforms a convolution product into a pointwise product is a Fourier transform. We focus on the cyclic groups ℤ/nℤ, the integers ℤ, the torus 𝕋 and the real line. We also ask a related question for the twisted convolution.
2
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Harmonic functions on the real hyperbolic ball I: Boundary values and atomic decomposition of Hardy spaces

100%
Jaming P.
Colloquium Mathematicum
|
1999
|
tom 80
|
nr 1
63-82
EN
We study harmonic functions for the Laplace-𝔹eltrami operator on the real hyperbolic space $𝔹_n$. We obtain necessary and sufficient conditions for these functions and their normal derivatives to have a boundary distribution. In doing so, we consider different behaviors of hyperbolic harmonic functions according to the parity of the dimension of the hyperbolic ball $𝔹_n$. We then study the Hardy spaces $H^p(𝔹_n)$, 0
3
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Uncertainty principles for integral operators

63%
Ghobber S., Jaming P.
Studia Mathematica
|
2014
|
tom 220
|
nr 3
197-220
EN
The aim of this paper is to prove new uncertainty principles for integral operators 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 $f ∈ L²(ℝ^{d},μ)$ is highly localized near a single point then 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 $f ∈ L²(ℝ^{d},μ)$ and its integral transform 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 𝓣. We apply our results to obtain new uncertainty principles for the Dunkl and Clifford Fourier transforms.
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