Philippe Jaming

• PhD,Hab
• Professor (Full) at University of Bordeaux

The aim of this paper is two prove two versions of the Dynamical Uncertainty Principlefor the Schr\"odinger equation $i\partial_s u=\mathcal{L}u+Vu$, $u(s=0)=u_0$ where$\mathcal{L}$ is the sub-Laplacian on the Heisenberg group.We show two results of this type. For the first one, the potential $V=0$, we establish a dynamical version of Amrein-Berthi...

Let \(G\) be a locally compact abelian group, and let \(\widehat{G}\) denote its dual group, equipped with a Haar measure. A variant of the uncertainty principle states that for any \(S \subset G\) and \(\Sigma \subset \widehat{G}\), there exists a constant \(C(S, \Sigma)\) such that for any \(f \in L^2(G)\), the following inequality holds: \[\|f\|...

We consider the problem of reconstructing a function \(f\in L^2({\mathbb R})\) given phase-less samples of its Gabor transform, which is defined by More precisely, given sampling positions \(\Omega \subseteq {\mathbb R}^2\) the task is to reconstruct f (up to global phase) from measurements \(\{|{\mathcal {G}}f(\omega )|: \,\omega \in \Omega \}\)....

In this paper we show that, if an increasing sequence $\Lambda=(\lambda_k)_{k\in\mathbb{Z}}$ has gaps going to infinity $\lambda_{k+1}-\lambda_k\to +\infty$ when $k\to\pm\infty$, then for every $T>0$ and every sequence $(a_k)_{k\in\mathbb{Z}}$ and every $N\geq 1$, $$ A\sum_{k=0}^N\frac{|a_k|}{1+k}\leq\frac{1}{T}\int_{-T/2}^{T/2} \left|\sum_{k=0}^N...

In this paper, we give a direct quantitative estimate of L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document} norms of non-harmonic trigonometric polyn...

In this paper, we consider the question of finding an as small as possible family of operators \((T_j)_{j\in J}\) on \(L^2({\mathbb {R}})\) that does phase retrieval: every \(\varphi \) is uniquely determined (up to a constant phase factor) by the phaseless data \((|T_j\varphi |)_{j\in J}\). This problem arises in various fields of applied sciences...

The aim of this paper is to extend two results from the Paley--Wiener setting to more generalmodel spaces. The first one is an analogue of the oversampling Shannon sampling formula. The second one is a version of Donoho--Logan's Large Sieve Theorem which is a quantitative estimate of the embedding of the Paley--Wiener space into an $L^2(\R,\mu)$ sp...

In this paper, we give a direct quantitative estimate of $L^1$norms of non-harmonic trigonometric polynomials over large enough intervals. This extends the result by Konyagin and Mc Gehee, Pigno, Smith to the settingof trigonometric polynomials with non-integer frequencies.The result is a quantitative extension of a result by Nazarov and also cover...

In the classical phase retrieval problem in the Paley-Wiener class \(PW_L\) for \(L>0\), i.e. to recover \(f\in PW_L\) from |f|, Akutowicz, Walther, and Hofstetter independently showed that all such solutions can be obtained by flipping an arbitrary set of complex zeros across the real line. This operation is called zero-flipping and we denote by \...

The aim of this letter is to show that uncertainty principles for the pair \(u,{\mathcal F}[u]\) (\({\mathcal F}\) the Fourier transform) can be transferred without much effort to the pair \({\mathcal F}_\alpha [u],{\mathcal F}_\beta [u]\) (\({\mathcal F}_\alpha \) the Fractional Fourier transform) provided \(\beta -\alpha \notin \pi {\mathbb {Z}}\...

In this paper we consider the question of finding an as small as possible family of operators $(T_j)_{j\in J}$ on $L^2(R)$ that does phase retrieval: every $\varphi$ is uniquely determined (up to a constant phase factor) by the phaseless data $(|T_j\varphi|)_{j\in J}$. This problem arises in various fields of applied sciences where usually the oper...

This paper considers the problem of restricting the short-time Fourier transform to sets of nonzero measure in the plane. Thereby, we study under which conditions one has a sampling set and provide estimates of the corresponding sampling bound. In particular, we give a quantitative estimate for the lower sampling bound in the case of Hermite window...

We study the concentration problem on compact two-point homogeneous spaces for finite expansions of eigenfunctions of the Laplace–Beltrami operator using large sieve methods. We derive upper bounds for concentration in terms of the maximum Nyquist density. Our proof uses estimates of the spherical harmonics basis coefficients of certain zonal filte...

In the classical phase retrieval problem in the Paley-Wiener class $PW_L$ for $L>0$, i.e. to recover $f\in PW_L$ from $|f|$, Akutowicz, Walther, and Hofstetter independently showed that all such solutions can be obtained by flipping an arbitrary set of complex zeros across the real line. This operation is called zero-flipping and we denote by $\mat...

We analyze the problem of reconstruction of a bandlimited function f from the space–time samples of its states $$f_t=\phi _t*f$$ f t = ϕ t ∗ f resulting from the convolution with a kernel $$\phi _t$$ ϕ t . It is well-known that, in natural phenomena, uniform space–time samples of f are not sufficient to reconstruct f in a stable way. To enable stab...

The aim of this paper is to establish density properties in Lp spaces of the span of powers of functions {ψλ:λ∈Λ}, Λ⊂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 {ψλ,ψλeiαt:λ∈Λ}. Finally, we establish a Müntz-Szász Theorem for density of tran...

This study investigates the phase retrieval problem for wide-band signals. We solve the following problem: given f $\in$ L 2 (R) with Fourier transform in L 2 (R, e^{2c|x|} dx), we find all functions g $\in$ L 2 (R) with Fourier transform in L 2 (R, e^{2c|x| dx}), such that |f (x)| = |g(x)| for all x $\in$ R. To do so, we first translate the proble...

We analyze the problem of reconstruction of a bandlimited function $f$ from the space-time samples of its states $f_t=\phi_t\ast f$ resulting from the convolution with a kernel $\phi_t$. It is well-known that, in natural phenomena, uniform space-time samples of $f$ are not sufficient to reconstruct $f$ in a stable way. To enable stable reconstructi...

We study the concentration problem on compact two-point homogeneous spaces of finite expansions of eigenfunctions of the Laplace-Beltrami operator using large sieve methods. We derive upper bounds for concentration in terms of the maximum Nyquist density. Our proof uses estimates of the spherical harmonics basis coefficients of certain zonal filter...

In this paper we consider uncertainty principles for solutions of certain partial differential equations on $H$ -type groups. We first prove that, on $H$ -type groups, the heat kernel is an average of Gaussians in the central variable, so that it does not satisfy a certain reformulation of Hardy’s uncertainty principle. We then prove the analogue o...

We consider the problem of reconstructing a compactly supported function from samples of its Fourier transform taken along a spiral. We determine the Nyquist sampling rate in terms of the density of the spiral and show that, below this rate, spirals suffer from an approximate form of aliasing. This sets a limit to the amount of undersampling that c...

In this paper, we show that the expansions of functions from $L^p$-Paley-Wiener type spaces in terms of the prolate spheroidal wave functions converge almost everywhere for $1<p<\infty$, even in the cases when they might not converge in $L^p$-norm. We thereby consider the classical Paley-Wiener spaces $PW_c^p\subset L^p(\mathcal{R})$ of functions w...

We study the question under which conditions the zero set of a (cross-) Wigner distribution W(f, g) or a short-time Fourier transform is empty. This is the case when both f and g are generalized Gaussians, but we will construct less obvious examples consisting of exponential functions and their convolutions. The results require elements from the th...

We prove that if a solution of the time-dependent Schrödinger equation on an homogeneous tree with bounded potential decays fast at two distinct times then the solution is trivial. For the free Schrödinger operator, we use the spectral theory of the Laplacian and complex analysis and obtain a characterization of the initial conditions that lead to...

This paper considers the problem of restricting the short-time Fourier transform to domains of nonzero measure in the plane and studies sampling bounds of such systems. In particular, we give a quantitative estimate for the lower sampling bound in the case of Hermite windows and derive a sufficient condition for a large class of windows in terms of...

This study investigates the phase retrieval problem for wide-band signals. We solve the following problem: given f $\in$ L 2 (R) with Fourier transform in L 2 (R, e^{2c|x|} dx), we find all functions g $\in$ L 2 (R) with Fourier transform in L 2 (R, e^{2c|x| dx}), such that |f (x)| = |g(x)| for all x $\in$ R. To do so, we first translate the proble...

We study the observability of the one-dimensional Schr{\"o}dinger equation and of the beam and plate equations by moving or oblique observations. Applying different versions and adaptations of Ingham's theorem on nonharmonic Fourier series, we obtain various observability and non-observability theorems. Several open problems are also formulated at...

We study the question under which conditions the zero set of a (cross-) Wigner distribution W (f, g) or a short-time Fourier transform is empty. This is the case when both f and g are generalized Gaussians, but we will construct less obvious examples consisting of exponential functions and their convolutions. The results require elements from the t...

We consider the problem of reconstructing a compactly supported function from samples of its Fourier transform taken along a spiral. We determine the Nyquist sampling rate in terms of the density of the spiral and show that below this rate spirals suffer from an approximate form of aliasing. This sets a limit to the amount of undersampling that com...

In this paper we consider uncertainty principles for solutions of certain PDEs on H-type groups. We first prove that, contrary to the euclidean setting, the heat kernel on H-type groups is not characterized as the only solution of the heat equation that has sharp decay at 2 different times. We then prove the analogue of Hardy's Uncertainty Principl...

In this work, we establish a Plancherel–Polya inequality for functions in Besov spaces on spaces of homogeneous type as defined in Han and Yang (Diss Math 403:1–102, 2002) in the spirit of their recent counterpart for \({\mathbb {R}}^d\) established by Jaming and Malinnikova (J Fourier Anal Appl 22:768–786, 2016. The main tool is the wavelet decomp...

We establish quantitative estimates for sampling (dominating) sets in model spaces associated with meromorphic inner functions, i.e. those corresponding to de Branges spaces. Our results encompass the Logvinenko-Sereda-Panejah (LSP) Theorem including Kovrijkine's optimal sampling constants for Paley-Wiener spaces. It also extends Dyakonov's LSP the...

Some recent works have shown that the heat equation posed on the whole Euclidean space is null-controllable in any positive time if and only if the control subset is a thick set. This necessary and sufficient condition for null-controllability is linked to some uncertainty principles as the Logvinenko-Sereda theorem which give limitations on the si...

Some recent works have shown that the heat equation posed on the whole Euclidean space is null-controllable in any positive time if and only if the control subset is a thick set. This necessary and sufficient condition for null-controllability is linked to some uncertainty principles as the Logvinenko-Sereda theorem which give limitations on the si...

Prolate spheroidal wave functions have recently attracted a lot of attention in applied harmonic analysis, signal processing and mathematical physics. They are eigenvectors of the Sinc-kernel operator Qc : the time-and band-limiting operator. The corresponding eigenvalues play a key role and it is the aim of this paper to obtain precise non-asympto...

Prolate spheroidal wave functions have recently attracted a lot of attention in applied harmonic analysis, signal processing and mathematical physics. They are eigenvectors of the Sinc-kernel operator Qc : the time-and band-limiting operator. The corresponding eigenvalues play a key role and it is the aim of this paper to obtain precise non-asympto...

The aim of this paper is to establish the range of p's for which the expansion of a function f $\in$ L p in a generalized prolate spheroidal wave function (PSWFs) basis converges to f in L p. Two generalizations of PSWFs are considered here, the circular PSWFs introduced by D. Slepian and the weighted PSWFs introduced by Wang and Zhang. Both cases...

The aim of this paper is to establish the range of p's for which the expansion of a function f $\in$ L p in a generalized prolate spheroidal wave function (PSWFs) basis converges to f in L p. Two generalizations of PSWFs are considered here, the circular PSWFs introduced by D. Slepian and the weighted PSWFs introduced by Wang and Zhang. Both cases...

The aim of this paper is to establish uniqueness properties of solutions of the Helmholtz and Laplace equations. In particular, we show that if two solutions of such equations on a domain of R d agree on two intersecting d -- 1-dimensional submanifolds in generic position, then they agree everywhere.

The aim of this paper is to establish uniqueness properties of solutions of the Helmholtz and Laplace equations. In particular, we show that if two solutions of such equations on a domain of R d agree on two intersecting d -- 1-dimensional submanifolds in generic position, then they agree everywhere.

In this paper we consider the phase retrieval problem for solutions of the Helmholtz equation Δu+λ2u=0 on domains Ω⊂Rd, d⩾2. In dimension d=2, if u,v are two such solutions then |u|=|v| implies that either u=cv or u=cv¯ for some c∈C with |c|=1. In dimension d⩾3, the same conclusion holds under some restriction on u and v: either they are real value...

We establish quantitative estimates for sampling (dominating) sets in model spaces associated with meromorphic inner functions, i.e. those corresponding to de Branges spaces. Our results encompass the Logvinenko-Sereda-Panejah (LSP) Theorem including Kovrijkine's optimal sampling constants for Paley-Wiener spaces. It also extends Dyakonov's LSP the...

In this work we establish a sampling theorem for functions in Besov spaces on spaces of homogeneous type as defined in [HY] in the spirit of their recent counterpart for R d established by Jaming-Malinnikova in [JM]. The main tool is the wavelet decomposition presented by Deng-Han in [DH].

We prove that if a solution of the time-dependent Schr{\"o}dinger equation on an homogeneous tree with bounded potential decays fast at two distinct times then the solution is trivial. For the free Schr{\"o}dinger operator, we use the spectral theory of the Laplacian and complex analysis and obtain a characterization of the initial conditions that...

In this paper we consider the phase retrieval problem for Herglotz functions, that is, solutions of the Helmholtz equation $\Delta u+\lambda^2u=0$ on domains $\Omega\subset\mathbb{R}^d$, $d\geq2$. In dimension $d=2$, if $u,v$ are two such solutions then $|u|=|v|$ implies that either $u=cv$ or $u=c\bar v$ for some $c\in\mathbb{C}$ with $|c|=1$. In d...

Two measurable sets $S, \Lambda \subseteq \mathcal{R}^d$ form a Heisenberg uniqueness pair, if every bounded measure $\mu$ with support in S whose Fourier transform vanishes on {\Lambda} 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...

Two measurable sets $S, \Lambda \subseteq \mathcal{R}^d$ form a Heisenberg uniqueness pair, if every bounded measure $\mu$ with support in S whose Fourier transform vanishes on {\Lambda} 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...

We extend Strichartz’s uncertainty principle (Strichartz, J Funct Anal 84:97–114, 1989) from the setting of the Sobolev space \(W^{1,2}({\mathbb {R}})\) to more general Besov spaces \(B^{1/p}_{p,1}({\mathbb {R}})\). The main result gives an estimate from below of the trace of a function from the Besov space on a uniformly distributed discrete subse...

The aim of this paper is to establish density properties in $L^p$ spaces of the span of powers of functions $\{\psi^\lambda\,:\lambda\in\Lambda\}$, $\Lambda\subset\N$ in the spirit of the M\"untz-Sz\'asz Theorem. As density is almost never achieved, we further investigate the density of powers and a modulation of powers $\{\psi^\lambda,\psi^\lambda...

In this paper, we seek lower bounds of the dyadic Hilbert transform (Haar shift) of the form Xf L 2 (K) $\ge$ C(I, K)f L 2 (I) where I and K are two dyadic intervals and f supported in I. If I $\subset$ K such bound exist while in the other cases K I and K $\cap$ I = $\emptyset$ such bounds are only available under additional constraints on the der...

In this paper, we seek lower bounds of the dyadic Hilbert transform (Haar shift) of the form $\left\Vert S f\right\Vert_{L^2(K)}\geq C(I,K)\left\Vert f\right\Vert_{L^2(I)}$ where $I$ and $K$ are two dyadic intervals and $f$ supported in $I$. If $I\subset K$ such bound exist while in the other cases $K\subsetneq I$ and $K\cap I=\emptyset$ such bound...

We prove that if a solution of the discrete time-dependent Schr\"odinger equation with bounded real potential decays fast at two distinct times then the solution is trivial. For the free Shr\"odinger operator and for operators with compactly supported time-independent potentials a sharp analog of the Hardy uncertainty principle is obtained, using a...

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...

We prove that if a solution of the discrete time-dependent Schrödinger equation with bounded time-independent real potential decays fast at two distinct times then the solution is trivial. For the free Shrödinger operator or operators with compactly supported potential a sharp analog of the Hardy uncertainty principle is obtained. The argument is b...

The aim of this paper is to pursue the investigation of the phase retrieval problem for the fractional Fourier transform $\ff\_\alpha$ 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 $\aa\subset \qq$, there exist two functions $f,g$ not multiple o...

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 polyno...

In this paper, we investigate an extension of Pauliʼs phase retrieval problem. The original problem asks whether a function u is uniquely determined by its modulus |u||u| and the modulus of its Fourier transform |Fu||Fu| up to a constant phase factor. Here we extend this problem by considering the uniqueness of the phase retrieval problem for the f...

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.

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 $f\in L^2(\R^d,\mu)$ is highly localized near a single point then $\tt (f...

Let $\Lambda$ be a set of lines in $\mathbb{R}^2$ that intersect at the origin. For $\Gamma\subset\mathbb{R}^2$ a smooth curve, we denote by $\mathcal{A}\mathcal{C}(\Gamma)$ the subset of finite measures on $\Gamma$ that are absolutely continuous with respect to arc length on $\Gamma$. For such a $\mu$, $\widehat{\mu}$ denotes the Fourier transform...

The aim of this paper is to establish an analogue of Logvinenko-Sereda's theorem for the Fourier-Bessel transform (or Hankel transform) $\ff_\alpha$ of order $\alpha>-1/2$. Roughly speaking, if we denote by $PW_\alpha(b)$ the Paley-Wiener space of $L^2$-functions with Fourier-Bessel transform supported in $[0,b]$, then we show that the restriction...

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.

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 more quantitative estimates. We then show how to transfer this result to the discrete version of the short time Fourier 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 $\mathcal{F}_\alpha (f)$ cannot both have support of finite measure. The...

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 $\alpha$ have same modulus $|F_\alpha u|=|F_\alpha v|...

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...

The first part of this note is based on a joint paper with A. Bonami and G. Garrigós [3] in which the phase retrieval problem for the Radar Ambiguity Function (i.e. the Radar Ambiguity Problem) has been tackled. In particular it was shown that for wide classes of signals, the radar ambiguity problem has a unique solution, up to trivial transformati...

L'objectif de l'ouvrage est de présenter un panorama des théorèmes ergodiques pour les actions de groupes, en partant des résultats classiques des débuts de la théorie. Sont ensuite traités le cas des actions de groupes moyennables, les moyennes sphériques pour les actions de $R^d$, de $Z^d$, des groupes libres, et enfin le cas des actions du group...

In this paper, we give a new proof of a result of R. Jones showing almost everywhere convergence of spherical means of actions of Rd on Lp(X)-spaces are convergent for d⩾3 and p>d/(d-1).This is done by adapting the proof of the spherical maximal theorem by Rubio de Francia so as to obtain directly the ergodic theorem.

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.

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. Comment: In memory...

The aim of this paper is to investigate the cone of non-negative, radial, positive-definite functions in the set of continuous functions on ℝ d . Elements of this cone admit a Choquet integral representation in terms of the extremals. The main feature of this article is to characterize some large classes of such extremals. In particular, we show th...

We exhibit an infinite family of {\it 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 the paper we also prove that for any values of the parameters $(a,b)$, the standard basis and $F(a,b)$ {\it cannot be extended to a MUB-quartet}. T...

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 polynomi...

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 symmetr...

We observe that successive applications of known results from the theory of positive systems lead to an efficient general algorithm for positive realizations of transfer functions. We give two examples to illustrate the algorithm, one of which complements an earlier result of [L. Benvenuti, L. Farina, An example of how positivity may force realizat...

The aim of this paper is to investigate the cone of non-negative, radial, positive-definite functions in the set of continuous functions on $\R^d$. Elements of this cone admit a Choquet integral representation in terms of the extremals. The main feature of this article is to characterize some large classes of such extremals. In particular, we show...

We prove that if A is a set of exponentials mutually orthogonal with respect to any symmetric convex set K in the plane with a smooth boundary and everywhere non-vanishing cur- vature, then #(Aq A \ (0,q) d ) . q. This extends and clarifies in the plane the result of Iosevich and Rudnev. As a corrollary, we obtain the result from (IKP01) and (IKT01...

In this paper we prove that there exists a constant C such that, if S,Σ are subsets of ℝ d of finite measure, then for every function f∈L 2 (ℝ d ), ∫ ℝ d |f(x)| 2 dx≤Ce Cmin(|S||Σ|,|S| 1/d w(Σ),w(S)|Σ| 1/d ) ∫ ℝ d ∖S |f(x)| 2 dx+∫ ℝ d ∖Σ |f ^(x)| 2 dx where f ^ is the Fourier transform of f and w(Σ) is the mean width of Σ. This extends to dimension...

In this paper, we pursue the study of the radar ambiguity problem started in [Ph. Jaming, Phase retrieval techniques for radar ambiguity functions, J. Fourier Anal. Appl. 5 (1999) 313–333; G. Garrigós, Ph. Jaming, J.-B. Poly, Zéros de fonctions holomorphes et contre-exemples en théorie des radars, in: Actes des rencontres d'analyse complexe, Atlant...

The aim of this paper is to prove that if a planar set A has a difference set Δ(A) satisfying Δ(A) ⊂ ℤ+ + s for suitable s then A has at most 3 elements. This result is motivated by the conjecture that the disk has no more than 3 orthogonal exponentials. Further, we prove that if A is a set of exponentials mutually orthogonal with respect to any sy...

Cette habilitation comporte trois parties essentiellement indépendantes. Dans une première partie, nous nous intéressons au comportement au bord de fonctions harmoniques sur certains domaines homogènes. En particulier nous étudions la limite au brd au sens des distributions des dérivées normales de fonctions harmoniques ainsi que la caractérisatio...

The aim of this paper is to provide complementary quantitative extensions of two results of H.S. Shapiro on the time–frequency concentration of orthonormal sequences in L2(R). More precisely, Shapiro proved that if the elements of an orthonormal sequence and their Fourier transforms are all pointwise bounded by a fixed function in L2(R) then the se...

In this article, we give an estimate of the zero-free region around the origin of the ambiguity function of a one-dimensional signal u in terms of the moments of u. This is done by proving an uncertainty relation between the first zero of the Fourier transform of a non-negative function and the moments of the function. As a corollary, we also give...

We observe that successive applications of known results from the theory of positive systems lead to an {\it efficient general algorithm} for positive realizations of transfer functions. We give two examples to illustrate the algorithm, one of which complements an earlier result of \cite{large}. Finally, we improve a lower-bound of \cite{mn2} to in...

In this paper, we characterize the class of distributions on an homogeneous Lie group $\fN$ that can be extended via Poisson integration to a solvable one-dimensional extension $\fS$ of $\fN$. To do so, we introducte the $\ss'$-convolution on $\fN$ and show that the set of distributions that are $\ss'$-convolvable with Poisson kernels is precisely...

In this paper we prove that there exists a constant $C$ such that, if $S,\Sigma$ are subsets of $\R^d$ of finite measure, then for every function $f\in L^2(\R^d)$, $$\int_{\R^d}|f(x)|^2 dx \leq C e^{C \min(|S||\Sigma|, |S|^{1/d}w(\Sigma), w(S)|\Sigma|^{1/d})} (\int_{\R^d\setminus S}|f(x)|^2 dx + \int_{\R^d\setminus\Sigma}|\hat{f}(x)|^2 dx) $$ where...

In this survey, we present various forms of the uncertainty principle (Hardy, Heisenberg, Benedicks). We further give a new interpretation of the uncertainty principles as a statement about the time-frequency localization of elements of an orthonormal basis, which improves previous unpublished results of H. Shapiro. Finally, we show that Benedicks'...