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Philippe Jaming

Publications and source records attributed to Philippe Jaming.

At least 19 recordsLinked to original sources
arXiv2026

Dynamical phase retrieval for Schr{\"o}dinger evolution on finite graphs

We study dynamical phase retrieval for Schr\''odinger evolutions on finite connected graphs. Let \[ H\_Q=\Delta\_G+Q \] be a graph Schr\''odinger operator with a real diagonal potential. We investigate when phaseless data obtained from the associated Schr\''odinger evolution \[ |e^{-itH\_Q}u\_0(j)|, \qquad 0\leq t\leq T,\ j\in V, \] determines the initial state $u\_0\in\C^V$ up to a global phase. We give a uniqueness criterion in terms of the eigenvalues and eigenvectors of $H\_Q$. The assumptions are a $B\_2$ condition on the spectrum, meaning that the sums $\lambda\_j+\lambda\_k$ determine the unordered pair $\{j,k\}$, invertibility of the squared-eigenvector matrix $\bigl(\phi\_k(j)^2\bigr)\_{j,k}$ and an overlap condition on the supports of pairs of eigenvectors. Under these hypotheses, the phaseless Schr\''odinger data determine every initial state uniquely, modulo global phase. We then show that the criterion is both realized and generic. Every finite connected graph admits an explicit real diagonal potential for which the criterion holds. Moreover, for every finite connected graph, dynamical phase retrieval holds for Lebesgue-almost every real potential $Q\in\R^V$ and every $T>0$. We also give several obstructions to uniqueness.

math.CA↗
arXiv2026

Curved Ingham inequalities and observability of the toroidal Schr{\"o}dinger equation

We prove that solutions of the toroidal Schr{\"o}dinger equation can be observed from suitably curved space-time trajectories, thus of zero Lebesgue measure. To do so, we establish new upper and lower bounds for certain trigonometric sums along curves, in the spirit of the celebrated Ingham inequality. In a second part, we establish observability properties over arbitrarily short curves of the low-and high-frequency components separately. For the low-frequency component, we establish strong restrictions on the zero sets of the trigonometric sums under consideration.

math.AP↗
arXiv2026

Approximate null-controllability of discrete heat equations with potentials on lattices

We investigate approximate null-controllability for semi-discrete heat equations on the lattice $h\mathbb{Z}^d$ with a potential. By establishing spectral inequalities for the discrete Schr{\"o}dinger operator $P_h = -\Delta_h + V$ on equidistributed sets, we derive observability estimates via the Lebeau-Robbiano method and the Hilbert Uniqueness Method. For bounded potentials, we obtain quantitative controllability results with explicit dependence on the potential and show near optimality of the geometric condition on the observation set. We also treat polynomial growth potentials, for which similar properties hold with weaker control cost estimates. These results extend discrete Carleman techniques to the full-space lattice setting and provide new spectral estimates for discrete Schr{\"o}dinger operators.

math.AP↗
arXiv2026

Convergence of Hermite expansions in modulation spaces

The aim of this paper is to give an elementary proof that Hermite expensions of a function $f$ in the modulation space $M^p(R)$ converges to $f$ in $M^p(R)$ when $1< p<+\infty$ and may diverge when $p = 1,\infty$. The result was previously established for $1< p<+\infty$ by Garling and Wojtaszczyk and for $p = 1,\infty$ by Lusky in an equivalent setting of Fock spaces by different methods. Higher dimesional results are also considered. In an appendix, we also establish upper bounds for the Zak transform of Hermite functions.

math.CA↗
arXiv2025

On Wiener's Lemma on locally compact abelian groups

We establish a general form of Wiener's lemma for measures on locally compact abelian (LCA) groups by using Fourier analysis and the theory of F{{\o}}lner sequences. Our approach provides a unified framework that that encompasses both the discrete and continuous cases. We also show a version of Wiener's lemma for Bochner-Riesz means on both R^d and T^d . Mathematics Subject Classification (2010). 43A25.

math.CA↗
arXiv2025

Sign retrieval in spaces of variable bandwidth

The aim of this paper is to get a deeper understanding of the spaces of variable bandwidth introduced by Gr{\"o}chenig and Klotz (What is variable bandwidth? Comm. Pure Appl. Math., 70 (2017), 2039-2083). In particular, we show that when the variation of the bandwidth is modeled by a step function with a finite number of jumps, then, the sign retrieval principle applies.

math.CA↗
arXiv2025

On the Three Balls Inequality for Discrete Schr{\"o}dinger Operators on Certain Periodic Graphs

We investigate quantitative unique continuation properties for discrete magnetic Schr{\"o}dinger operators in certain periodic graphs. This unique continuation property will be quantified through what is known in the literature as a Three Balls Inequality. We are able to extend this inequality to another family of periodic graph which contains the Hexagonal lattice. We also give a sketch of the proof for general star periodic graph.Our proofs are based on Carleman estimates.

math.CA↗
arXiv2025

Uncertainty principle for solutions of the Schr{\"o}dinger equation on the Heisenberg group

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-Berthier-Benedicks's Uncertainty Principle that shows that if $u_0$ and $u_1=u(s=1)$ have both small support then $u=0$. For the second result, we add some potential to the equation and we obtain a dynamical version of the Paley-Wiener Theorem in the spirit of the result of Kenig, Ponce, Vega \cite{KPV}. Both results are obtained by suitably transfering results from the Euclidean setting.We also establish some limitations to Dynamical Uncertainty Principles.

math.CA↗
arXiv2025

Uncertainty Principle, annihilating pairs and Fourier restriction

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: \[\|f\|_{L^2(G)} \leq C(S, \Sigma) \bigl( \|f\|_{L^2(G \setminus S)} + \|\widehat{f}\|_{L^2(\widehat{G} \setminus \Sigma)} \bigr),\] where $\widehat{f}$ denotes the Fourier transform of $f$. This variant of the uncertainty principle is particularly useful in applications such as signal processing and control theory.The purpose of this paper is to show that such estimates can be strengthened when $S$ or $\Sigma$ satisfies a restriction theorem and to provide an estimate for the constant $C(S, \Sigma)$. This result serves as a quantitative counterpart to a recent finding by the first and last author. In the setting of finite groups, the results also extend those of Matolcsi-Sz\"ucs and Donoho-Stark.

math.CA↗
arXiv2024

On L1-norms for non-harmonic trigonometric polynomials with sparse frequencies

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 $N\geq 1$, $$ A\sum_{k=0}^N\frac{|a_k|}{1+k}\leq\frac{1}{T}\int_{-T/2}^{T/2} \left|\sum_{k=0}^N a_k e^{2i\pi\lambda_k t}\right|\,\mbox{d}t$$ further, if $\sum_{k\in\mathbb{Z}}\dfrac{1}{1+|\lambda_k|}<+\infty$,$$ B\max_{|k|\leq N}|a_k|\leq\frac{1}{T}\int_{-T/2}^{T/2} \left|\sum_{k=-N}^N a_k e^{2i\pi\lambda_k t}\right|\,\mbox{d}t $$ where $A,B$ are constants that depend on $T$ and $\Lambda$ only. The first inequality was obtained by Nazarov for $T>1$ and the second one by Ingham for $T\geq 1$ under the condition that $\lambda_{k+1}-\lambda_k\geq 1$. The main novelty is that if those gaps go to infinity, then $T$ can be taken arbitrarily small. The result is new even when the $\lambda_k$'s are integers where it extends a result of McGehee, Pigno and Smith. The results are then applied to observability of Schr\"odinger equations with moving sensors.

math.CA↗
arXiv2023

From Ingham to Nazarov's inequality: a survey on some trigonometric inequalities

The aim of this paper is to give an overview of some inequalities about $L^p$-norms ($p= 1$ or $p= 2$) of harmonic (periodic) and non-harmonic trigonometric polynomials. Among the material covered, we mention Ingham's Inequality about 2 norms of non-harmonic trigonometric polynomials, the proof of the Littlewood conjecture by Mc Gehee, Pigno and Smith on the lower bound of the 1 norm of harmonic trigonometric polynomials as well as its counterpart in the non-harmonic case due to Nazarov. For the latter one, we give a quantitative estimate that completes our recent result with an estimate of 1-norms over small intervals. We also give some stronger lower bounds when the frequencies satisfy some more restrictive conditions (lacunary Fourier series, "multi-step arithmetic sequences"). Most proofs are close to existing ones and some open questions are mentioned at the end.

math.CA↗
arXiv2023

Gabor phase retrieval via semidefinite programming

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 $$\mathcal{G} f(x,\omega) := 2^{\frac14} \int_{\mathbb{R}} f(t) e^{-\pi (t-x)^2} e^{-2\pi i y t}\,\mbox{d}t,\quad (x,y)\in\mathbb{R}^2.$$More precisely, given sampling positions $\Omega\subseteq \mathbb{R}^2$ the task is to reconstruct $f$ (up to global phase) from measurements $\{|\mathcal{G} f(\omega)|: \,\omega\in\Omega\}$. This non-linear inverse problem is known to suffer from severe ill-posedness. As for any other phase retrieval problem, constructive recovery is a notoriously delicate affair due to the lack of convexity. One of the fundamental insights in this line of research is that the connectivity of the measurements is both necessary and sufficient for reconstruction of phase information to be theoretically possible. In this article we propose a reconstruction algorithm which is based on solving two convex problems and, as such, amenable to numerical analysis. We show, empirically as well as analytically, that the scheme accurately reconstructs from noisy data within the connected regime.Moreover, to emphasize the practicability of the algorithm we argue that both convex problems can actually be reformulated as semi-definite programs for which efficient solvers are readily available. The approach is based on ideas from complex analysis, Gabor frame theory as well as matrix completion.

math.CA↗
arXiv2023

Null-controllability of the Generalized Baouendi-Grushin heat like equations

In this article, we prove null-controllability results for the heat equation associated tofractional Baouendi-Grushin operators $$\partial_t u+\bigl(-\Delta_x-V(x)\Delta_y\bigr)^s u= \mathbb{1}_\Omega h$$ where $V$ is a potential that satisfies some power growth conditions and the set $\Omega$is thick in some sense. This extends previously known results for potentials $V(x)=|x|^{2k}$.To do so, we study Zhu-Zhuge's spectral inequality for Schr{\"o}dinger operators with power growth potentials, and give a precised quantitative form of it.

math.OC↗
arXiv2023

Oversampling and Donoho-Logan type theorems in model spaces

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 the Donoho--Logan Large Sieve Theorem which is a quantitative estimate of the embedding of the Paley--Wiener space into an $L^2(\R,\mu)$ space.

math.CA↗
arXiv2023

The Littlewood problem and non-harmonic Fourier series

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 covers a resultby Hudson and Leckband when the length of the interval goes to infinity.

math.CA↗
arXiv2022

Uniqueness of phase retrieval from three measurements

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 operators obey further restrictions. Of particular interest here are so-called {\em coded diffraction paterns} where the operators are of the form $T_j\varphi=\mathcal{F}m_j\varphi$, $\mathcal{F}$ the Fourier transform and $m_j\in L^\infty(R)$ are "masks". Here we explicitely construct three real-valued masks $m_1,m_2,m_3\in L^\infty(R)$ so that the associated coded diffraction patterns do phase retrieval. This implies that the three self-adjoint operators $T_j\varphi=\mathcal{F}[m_j\mathcal{F}^{-1}\varphi]$ also do phase retrieval. The proof uses complex analysis.We then show that some natural analogues of these operators in the finite dimensional setting do not always lead to the same uniqueness result due to an undersampling effect.

math.CA↗
arXiv2021

On the effect of zero-flipping on the stability of the phase retrieval problem in the Paley-Wiener class

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 $\mathfrak{F}_a f$ the resulting function. The operator $\mathfrak{F}_a$ is defined even if $a$ is not a genuine zero of $f$, that is if we make an error on the location of the zero. Our main goal is to investigate the effect of $\mathfrak{F}_a$. We show that $\mathfrak{F}_af$ is no longer bandlimited but is still wide-banded. We then investigate the effect of $\mathfrak{F}_a$ on the stability of phase retrieval by estimating the quantity $\inf_{|c|=1}\|cf-\mathfrak{F}_af\|_2$. We show that this quantity is in general not well-suited to investigate stability, and so we introduce the quantity $\inf_{|c|=1}\|c\mathfrak{F}_bf-\mathfrak{F}_af\|_2$. We show that this quantity is dominated by the distance between $a$ and $b$.

math.CA↗