arXiv subjects

Philippe Jaming

Publications and source records attributed to Philippe Jaming.

At least 55 records · Page 3Linked to original sources

arXiv2016

The Cramer-Wold theorem on quadratic surfaces and Heisenberg uniqueness pairs

Two measurable sets $S, Λ\subseteq \mathcal{R}^d$ form a Heisenberg uniqueness pair, if every bounded measure $μ$ with support in S whose Fourier transform vanishes on Λ must be zero. We show that a quadratic hypersurface and the union of two hyperplanes in general position form a Heisenberg uniqueness pair in $\mathcal{R}^d$. As a corollary we obtain a new, surprising version of the classical Cramér-Wold theorem: a bounded measure supported on a quadratic hypersurface is uniquely determined by its projections onto two generic hyperplanes (whereas an arbitrary measure requires the knowledge of a dense set of projections). We also give an application to the unique continuation of eigenfunctions of second-order PDEs with constant coefficients .

math.CA↗

arXiv2016

Density of the span of powers of a function à la Müntz-Szasz

The aim of this paper is to establish density properties in $L^p$ spaces of the span of powers of functions $\{ψ^λ\,:λ\inΛ\}$, $Λ\subset\N$ in the spirit of the Müntz-Szász Theorem. As density is almost never achieved, we further investigate the density of powers and a modulation of powers $\{ψ^λ,ψ^λe^{iαt}\,:λ\inΛ\}$. Finally, we establish a Müntz-Szász Theorem for density of translates of powers of cosines $\{\cos^λ(t-θ\_1),\cos^λ(t-θ\_2)\,:λ\inΛ\}$. Under some arithmetic restrictions on $θ\_1-θ\_2$, we show that density is equivalent to a Müntz-Szász condition on $Λ$ and we conjecture that those arithmetic restrictions are not needed.Some links are also established with the recently introduced concept of Heisenberg Uniqueness Pairs.

math.CA↗

arXiv2015

An uncertainty principle and sampling inequalities in Besov spaces

We extend Strichartz's uncertainty principle [18] from the setting of the Sobolov space W 1,2 (R) to more general Besov spaces B 1/p p,1 (R). The main result gives an estimate from below of the trace of a function from the Besov space on a uniformly distributed discrete subset. We also prove the corresponding result in the multivariate case and discuss some applications to irregular approximate sampling in critical Besov spaces.

math.CA↗

arXiv2015

Uniqueness for discrete Schrodinger evolutions

We prove that if a solution of the discrete time-dependent Schrödinger equation with bounded real potential decays fast at two distinct times then the solution is trivial. For the free Shrödinger operator and for operators with compactly supported time-independent potentials a sharp analog of the Hardy uncertainty principle is obtained, using an argument based on the theory of entire functions. Logarithmic convexity of weighted norms is employed in the case of general real-valued time-dependent bounded potentials. In the latter case the result is not optimal.

math.AP↗

arXiv2015

On the existence of flat orthogonal matrices

In this note we investigate the existence of flat orthogonal matrices, i.e. real orthogonal matrices with all entries having absolute value close to $\frac{1}{\sqrt{n}}$. Entries of $\pm \frac{1}{\sqrt{n}}$ correspond to Hadamard matrices, so the question of existence of flat orthogonal matrices can be viewed as a relaxation of the Hadamard problem.

math.CO↗

arXiv2015

Zak Transform and non-uniqueness in an extension of Pauli's phase retrieval problem

The aim of this paper is to pursue the investigation of the phase retrieval problem for the fractional Fourier transform $\ff\_α$ started by the second author. We here extend a method of A.E.J.M Janssen to show that there is a countable set $\qq$ such that for every finite subset $å\subset \qq$, there exist two functions $f,g$ not multiple of one an other such that $|\ff\_αf|=|\ff\_αg|$ for every $α\in å$. Equivalently, in quantum mechanics, this result reformulates as follows: if $Q\_α=Q\cosα+P\sinα$ ($Q,P$ be the position and momentum observables), then $\{Q\_α,α\inå\}$ is not informationally complete with respect to pure states. This is done by constructing two functions $\ffi,ψ$ such that $\ff\_α\ffi$ and $\ff\_αψ$ have disjoint support for each $α\in å$. To do so, we establish a link between $\ff\_α[f]$, $α\in \qq$ and the Zak transform $Z[f]$ generalizing the well known marginal properties of $Z$.

math.CA↗

arXiv2015

The approximation of almost time and band limited functions by their expansion in some orthogonal polynomials bases

The aim of this paper is to investigate the quality of approximation of almost time and almost band-limited functions by its expansion in three classical orthogonal polynomials bases: the Hermite, Legendre and Chebyshev bases. As a corollary, this allows us to obtain the quality of approximation in the L 2 --Sobolev space by these orthogonal polynomials bases. Also, we obtain the rate of the Legendre series expansion of the prolate spheroidal wave functions. Some numerical examples are given to illustrate the different results of this work.

math.CA↗

arXiv2014

Approximation of almost time and band limited functions I: Hermite expansions

The aim of this paper is to investigate the quality of approximation of almost time and band limited functions by its expansion in the Hermite and scaled Hermite basis. As a corollary, this allows us to obtain the rate of convergence of the Hermite expansion of function in the $L^2$-Sobolev space with fixed compact support.

math.CA↗

arXiv2014

A dynamical system approach to Heisenberg Uniqueness Pairs

Let $Λ$ be a set of lines in $\mathbb{R}^2$ that intersect at the origin. For $Γ\subset\mathbb{R}^2$ a smooth curve, we denote by $\mathcal{A}\mathcal{C}(Γ)$ the subset of finite measures on $Γ$ that are absolutely continuous with respect to arc length on $Γ$. For such a $μ$, $\widehatμ$ denotes the Fourier transform of $μ$. Following Hendenmalm and Montes-Rodríguez, we will say that $(Γ,Λ)$ is a Heisenberg Uniqueness Pair if $μ\in\mathcal{A}\mathcal{C}(Γ)$ is such that $\widehatμ=0$ on $Λ$, then $μ=0$. The aim of this paper is to provide new tools to establish this property. To do so, we will reformulate the fact that $\widehatμ$ vanishes on $Λ$ in terms of an invariance property of $μ$ induced by $Λ$. This leads us to a dynamical system on $Γ$ generated by $Λ$. The investigation of this dynamical system allows us to establish that $(Γ,Λ)$ is a Heisenberg Uniqueness Pair. This way we both unify proofs of known cases (circle, parabola, hyperbola) and obtain many new examples. This method also allows to have a better geometric intuition on why $(Γ,Λ)$ is a Heisenberg Uniqueness Pair.

math.CA↗

arXiv2012

Uncertainty principles for integral operators

The aim of this paper is to prove new uncertainty principles for an integral operator $\tt$ 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 $f\in L^2(\R^d,μ)$ is highly localized near a single point then $\tt (f)$ 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 $f\in L^2(\R^d,μ)$ and its integral transform $\tt (f)$ cannot both have support of finite measure. From these two results we deduce a global uncertainty principle of Heisenberg type for the transformation $\tt$. We apply our results to obtain a new uncertainty principles for the Dunkl and Clifford Fourier transforms.

math.CA↗

arXiv2012

The Logvinenko-Sereda Theorem for the Fourier-Bessel transform

The aim of this paper is to establish an analogue of Logvinenko-Sereda's theorem for the Fourier-Bessel transform (or Hankel transform) $\ff_α$ of order $α>-1/2$. Roughly speaking, if we denote by $PW_α(b)$ the Paley-Wiener space of $L^2$-functions with Fourier-Bessel transform supported in $[0,b]$, then we show that the restriction map $f\to f|_Ω$ is essentially invertible on $PW_α(b)$ if and only if $Ω$ is sufficiently dense. Moreover, we give an estimate of the norm of the inverse map. As a side result we prove a Bernstein type inequality for the Fourier-Bessel transform.

math.CA↗

arXiv2012

The problem of mutually unbiased bases in dimension 6

We outline a discretization approach to determine the maximal number of mutually unbiased bases in dimension 6. We describe the basic ideas and introduce the most important definitions to tackle this famous open problem which has been open for the last 10 years. Some preliminary results are also listed.

math.OA↗

arXiv2010

Uniqueness results for the phase retrieval problem of fractional Fourier transforms of variable order

In this paper, we investigate the uniqueness of the phase retrieval problem for the fractional Fourier transform (FrFT) of variable order. This problem occurs naturally in optics and quantum physics. More precisely, we show that if $u$ and $v$ are such that fractional Fourier transforms of order $α$ have same modulus $|F_αu|=|F_αv|$ for some set $τ$ of $α$'s, then $v$ is equal to $u$ up to a constant phase factor. The set $τ$ depends on some extra assumptions either on $u$ or on both $u$ and $v$. Cases considered here are $u$, $v$ of compact support, pulse trains, Hermite functions or linear combinations of translates and dilates of Gaussians. In this last case, the set $τ$ may even be reduced to a single point (i.e. one fractional Fourier transform may suffice for uniqueness in the problem).

math.CA↗

arXiv2010

Strong annihilating pairs for the Fourier-Bessel transform

The aim of this paper is to prove two new uncertainty principles for the Fourier-Bessel transform (or Hankel transform). The first of these results is an extension of a result of Amrein-Berthier-Benedicks, it states that a non zero function $f$ and its Fourier-Bessel transform $\mathcal{F}_α(f)$ cannot both have support of finite measure. The second result states that the supports of $f$ and $\mathcal{F}_α(f)$ cannot both be $(\eps,α)$-thin, this extending a result of Shubin-Vakilian-Wolff. As a side result we prove that the dilation of a $\cc_0$-function are linearly independent. We also extend Faris's local uncertainty principle to the Fourier-Bessel transform.

math.CA↗

arXiv2010

On uncertainty principles in the finite dimensional setting

The aim of this paper is to prove an uncertainty principle for the representation of a vector in two bases. Our result extends previously known qualitative uncertainty principles into quantitative estimates. We then show how to transfer this result to the discrete version of the Short Time Fourier Transform. An application to trigonometric polynomials is also given.

math.CA↗

arXiv2010

Time-frequency concentration of generating systems

Uncertainty principles for generating systems $\{e_n\}_{n=1}^{\infty} \subset \ltwo$ are proven and quantify the interplay between $\ell^r(\N)$ coefficient stability properties and time-frequency localization with respect to $|t|^p$ power weight dispersions. As a sample result, it is proven that if the unit-norm system $\{e_n\}_{n=1}^{\infty}$ is a Schauder basis or frame for $\ltwo$ then the two dispersion sequences $Δ(e_n)$, $Δ(\bar{e_n})$ and the one mean sequence $μ(e_n)$ cannot all be bounded. On the other hand, it is constructively proven that there exists a unit-norm exact system $\{f_n\}_{n=1}^{\infty}$ in $\ltwo$ for which all four of the sequences $Δ(f_n)$, $Δ(\bar{f_n})$, $μ(f_n)$, $μ(\bar{f_n})$ are bounded.

math.CA↗

arXiv2009

A characterization of Fourier transforms

The aim of this paper is to show that, in various situations, the only continuous linear 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.

math.CA↗

arXiv2008

On the Fourier transform of the symmetric decreasing rearrangements

Inspired by work of Montgomery on Fourier series and Donoho-Strak in signal processing, we investigate two families of rearrangement inequalities for the Fourier transform. More precisely, we show that the $L^2$ behavior of a Fourier transform of a function over a small set is controlled by the $L^2$ behavior of the Fourier transform of its symmetric decreasing rearrangement. In the $L^1$ case, the same is true if we further assume that the function has a support of finite measure. As a byproduct, we also give a simple proof and an extension of a result of Lieb about the smoothness of a rearrangement. Finally, a straightforward application to solutions of the free Shrödinger equation is given.

math.CA↗