Philippe Jaming

Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms

We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions f on Rd which may be written as P(x)exp(-(Ax,x)), with A a real symmetric definite positive matrix, are characterized by integrability conditions on the product f(x)f(y). We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenbergs inequality for this transform.

A generalized Pauli problem and an infinite family of MUB-triplets in dimension 6

We exhibit an infinite family of triplets of mutually unbiased bases (MUBs) in dimension 6. These triplets involve the Fourier family of Hadamard matrices, F(a, b). However, in the main result of this paper we also prove that for any values of the parameters (a, b), the standard basis and F(a, b) cannot be extended to a MUB-quartet. The main novelty lies in the method of proof which may successfully be applied in the future to prove that the maximal number of MUBs in dimension 6 is three.

Phase Retrieval Techniques for Radar Ambiguity Problems.

The radar ambiguity function plays a central role in the theory of radar signals. Its absolute value (¦A(u)¦) measures the correlation between the signal u emitted by the radar transmitter and its echo after reaching a moving target. It is important to know signals that give rise to ambiguity functions of given shapes. Therefore, it is also important to know to what extent ¦A(u)¦ determines the signal. This problem is called the “radar ambiguity problem” by Bueckner [5]. Using methods developed for phase retrieval problems, we give here a partial answer for some classes of time limited (compactly supported) signals. In doing so, we also obtain results for Paulis problem; in particular, we build functions that have infinitely many Pauli partners.

Uncertainty principles for integral operators

The aim of this paper is to prove new uncertainty principles for an integral operator

Uniqueness results in an extension of Pauli's phase retrieval problem

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Strong annihilating pairs for the Fourier–Bessel transform

with a bounded kernel for which there is a Plancherel theorem. The first of these results is an extension of Fariss local uncertainty principle which states that if a nonzero function

Uncertainty principles for orthonormal sequences

f\in L^2(\R^d,\mu)

On uncertainty principles in the finite dimensional setting

is highly localized near a single point then

A dynamical system approach to Heisenberg Uniqueness Pairs

\tt (f)

The problem of mutually unbiased bases in dimension 6

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