65 research outputs found
Uncertainty principles for integral operators
The aim of this paper is to prove new uncertainty principles for an integral
operator with a 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 is
highly localized near a single point then 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
and its integral transform cannot both have support of finite
measure. From these two results we deduce a global uncertainty principle of
Heisenberg type for the transformation . We apply our results to obtain a
new uncertainty principles for the Dunkl and Clifford Fourier transforms
New Sign Uncertainty Principles
We prove new sign uncertainty principles which vastly generalize the recent
developments of Bourgain, Clozel & Kahane and Cohn & Gon\c{c}alves, and apply
our results to a variety of spaces and operators. In particular, we establish
new sign uncertainty principles for Fourier and Dini series, the Hilbert
transform, the discrete Fourier and Hankel transforms, spherical harmonics, and
Jacobi polynomials, among others. We present numerical evidence highlighting
the relationship between the discrete and continuous sign uncertainty
principles for the Fourier and Hankel transforms, which in turn are connected
with the sphere packing problem via linear programming. Finally, we explore
some connections between the sign uncertainty principle on the sphere and
spherical designs.Comment: 45 pages, 2 figures, 3 tables, v3: typos corrected, numerics extende
Fourier Transforms
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