arXiv subjects

Azita Mayeli

Publications and source records attributed to Azita Mayeli.

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.

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arXiv2026

Localized frames on Euclidean balls

We construct explicit wave packet frames adapted to Euclidean balls and use them to obtain quantitative eigenvalue estimates for spatio--spectral limiting operators. Let $d\geq 2$, let $B_d(R)\subset \R^d$ be the Euclidean ball of radius $R$, and let $S\subset \R^d$ be a measurable set such that $\partial S$ has finite $(d-\eta)$-upper Minkowski content for $0 < \eta \leq 1$. We construct a unit-norm frame for $L^2(B_d(R))$, with frame bounds depending only on the dimension $d$, whose elements are adapted to the radial and angular geometry of the ball. We prove quantitative Fourier localization estimates for this frame: Relative to $S$, the frame decomposes into packets concentrated in $S$, packets concentrated in $\R^d\setminus S$, and an exceptional family whose cardinality is bounded explicitly in terms of $R$, and the Minkowski content of $\partial S$. As an application, we derive an upper bound for the plunge region of the spatio-spectral limiting operator associated to the sets $B_d(R)$ and $S$.

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arXiv2026

Trace bounds for limiting operators on rough domains

This work concerns a quantitative form of Landau's eigenvalue theorem for spatio-spectral limiting operators. We isolate a simple mechanism that converts the problem of estimating the distribution of eigenvalues of a limiting operator into the problem of bounding the trace of the difference between the operator and its square. This mechanism allows us to analyze limiting operators for domains with fractal boundaries. When the boundaries have finite perimeter, we recover the expected optimal dependence on the scaling parameter.

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arXiv2026

Wave Packets and Eigenvalue Estimates for Limiting Operators on the Disk

We study two-dimensional spatio-spectral limiting operators \[ T_R := P_{D(R)} B_S P_{D(R)} : L^2(\mathbb{R}^2) \rightarrow L^2(\mathbb{R}^2), \] where $D(R)$ is a disk of radius $R>1$, $S\subset\mathbb{R}^2$ is a domain with well-shaped boundary, $P_{D(R)}$ is the orthogonal projection on the subspace of functions supported on $D(R)$, and $B_S$ is the orthogonal projection on the subspace of functions whose Fourier transform is supported on $S$. We construct a disk-adapted wave-packet frame for $L^2(D(R))$ with frame bounds uniform in $R$ using Gevrey-$s$ cutoffs ($s>1$) to obtain near-exponential Fourier localization. Exploiting these localization estimates, we bound the size of the eigenvalue plunge-region for $T_R$ and prove that for each $s>1$ and each $\varepsilon\in(0,1/2)$, \[ \#\{k : \lambda_k(T_R)\in(\varepsilon,1-\varepsilon)\} = O\!\left(R (\log(R/\varepsilon))^{1+2s}\right), \] with constants depending on $s$ and the geometric parameters of $S$. This bound improves existing plunge-region estimates in the classical setting where both domains are disks, when $\varepsilon$ scales like $R^{-\nu}$ for a fixed $\nu > 0$. By an affine transformation, the same result holds if $D(R)$ is a scaled ellipse.

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arXiv2026

Wave packet systems and connections to spectral analysis of limiting operators

We discuss the design of ``wave packet systems'' that admit strong concentration properties in phase space. We make a connection between this problem and topics in signal processing related to the spectral behavior of spatial and frequency-limiting operators. The results have engineering applications in medical imaging, geophysics, and astronomy.

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arXiv2025

Eigenvalue distribution analysis of multidimensional prolate matrices

We extend classical time-frequency limiting analysis, historically applied to one-dimensional finite signals, to the multidimensional discrete setting. This extension is relevant for images, videos, and other multidimensional signals, as it enables a rigorous study of joint time-frequency localization in higher dimensions. To achieve this, we define multidimensional time-limiting and frequency-limiting matrices tailored to signals on a Cartesian grid and construct a multi-indexed prolate matrix. We prove that the spectrum of this matrix exhibits an eigenvalue concentration phenomenon: the bulk of eigenvalues cluster near 1 or 0 with a narrow transition band separating these regions. Moreover, we derive quantitative bounds on the width of the transition band in terms of the time-bandwidth product and prescribed accuracy. Concretely, our contributions are twofold: (i) we extend existing one-dimensional results to higher-dimensional Cartesian discrete signals; and (ii) we develop a multidimensional non-asymptotic eigenvalue-distribution analysis for prolate matrices. The advances are summarized in Theorem 1.1. Numerical experiments in one- and two-dimensional settings confirm the predicted eigenvalue concentration and illustrate potential applications in fast computation for image analysis, multidimensional spectral estimation, and related signal-processing tasks.

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arXiv2025

Fourier minimization and imputation of time series

One of the most common procedures in modern data analytics is filling in missing values in times series. For a variety of reasons, the data provided by clients to obtain a forecast, or other forms of data analysis, may have missing values, and those values need to be filled in before the data set can be properly analyzed. Many freely available forecasting software packages, such as the sktime library, have built-in mechanisms for filling in missing values. The purpose of this paper is to adapt the classical $L^1$ minimization method for signal recovery to the filling of missing values in times. The theoretical justifications of these methods leverage results by Bourgain (\cite{Bourgain89}), Talagrand (\cite{Talagrand98}), the second and the third listed authors (\cite{IM24}), and the result by the second listed author, Kashin, Limonova and the third listed author (\cite{IKLM24}). Brief numerical tests for these algorithms are given but more extensive will be discussed in a companion paper.

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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.

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arXiv2024

Fourier Uncertainty Principles on Riemannian Manifolds

The purpose of this paper is to develop a Fourier uncertainty principle on compact Riemannian manifolds and contrast the underlying ideas with those arising in the setting of locally compact abelian groups. The key obstacle is the growth of eigenfunctions, and connections to Bourgain's celebrated $\Lambda_q$ theorem are discussed in this context.

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arXiv2024

On the eigenvalue distribution of spatio-spectral limiting operators in higher dimensions, II

Let $F$, $S$ be bounded measurable sets in $\mathbb{R}^d$. Let $P_F : L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d) $ be the orthogonal projection on the subspace of functions with compact support on $F$, and let $B_S : L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$ be the orthogonal projection on the subspace of functions with Fourier transforms having compact support on $S$. In this paper, we derive improved distributional estimates on the eigenvalue sequence $1 \geq \lambda_1(F,S) \geq \lambda_2(F,S) \geq \cdots > 0$ of the \emph{spatio-spectral limiting operator} $B_S P_F B_S : L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$. The significance of such estimates lies in their diverse applications in medical imaging, signal processing, geophysics and astronomy. Our proof is based on the decomposition techniques developed in \cite{MaRoSp23}. The novelty of our approach is in the use of a two-stage dyadic decomposition with respect to both the spatial and frequency domains, and the application of the results in \cite{ArieAzita23} on the eigenvalues of spatio-spectral limiting operators associated to cubical domains.

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arXiv2023

Uncertainty Principles on Finite Abelian Groups, Restriction Theory, and Applications to Sparse Signal Recovery

Let $G$ be a finite abelian group. Let $f: G \to {\mathbb C}$ be a signal (i.e. function). The classical uncertainty principle asserts that the product of the size of the support of $f$ and its Fourier transform $\hat f$, $\text{supp}(f)$ and $\text{supp}(\hat f)$ respectively, must satisfy the condition: $$|\text{supp}(f)| \cdot |\text{supp}(\hat f)| \geq |G|.$$ In the first part of this paper, we improve the uncertainty principle for signals with Fourier transform supported on generic sets. This improvement is achieved by employing {\it the restriction theory} and {\it the Salem set} mechanism from harmonic analysis. Then we investigate some applications of uncertainty principles that were developed in the first part of this paper, to the problem of unique recovery of finite sparse signals in the absence of some frequencies. Donoho and Stark (\cite{DS89}) showed that a signal of length $N$ can be recovered exactly, even if some of the frequencies are unobserved, provided that the product of the size of the number of non-zero entries of the signal and the number of missing frequencies is not too large, leveraging the classical uncertainty principle for vectors. Our results broaden the scope for a natural class of signals in higher-dimensional spaces. In the case when the signal is binary, we provide a very simple exact recovery mechanism through the DRA algorithm.

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arXiv2023

Spectral Tile Direction in the Group $\mathbb{Z}_{p^2} \times \mathbb{Z}_{q^2} \times \mathbb{Z}_r$

Let $p$, $q$, and $r$ be distinct primes such that $p^2q^2<r$. We prove that every spectral set in the cyclic group $\mathbb{Z}_{p^2q^2r}$ is a tile. Since the reverse direction is already known, this shows that $\mathbb{Z}_{p^2q^2r}$ is a Fuglede group under this condition. The proof is based on divisibility properties of mask polynomials and on the structure of spectral sets in finite cyclic groups.

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arXiv2023

On the Eigenvalue Distribution of Spatio-Spectral Limiting Operators in Higher Dimensions

Prolate spheroidal wave functions are an orthogonal family of bandlimited functions on $\mathbb{R}$ that have the highest concentration within a specific time interval. They are also identified as the eigenfunctions of a time-frequency limiting operator (TFLO), and the associated eigenvalues belong to the interval $[0, 1]$. Previous work has studied the asymptotic distribution and clustering behavior of the TFLO eigenvalues. In this paper, we extend these results to multiple dimensions. We prove estimates on the eigenvalues of a \emph{spatio-spectral limiting operator} (SSLO) on $L^2(\mathbb{R}^d)$, which is an alternating product of projection operators associated to given spatial and frequency domains in $\mathbb{R}^d$. If one of the domains is a hypercube, and the other domain is a convex body satisfying a symmetry condition, we derive quantitative bounds on the distribution of the SSLO eigenvalues in the interval $[0,1]$. To prove our results, we design an orthonormal system of wave packets in $L^2(\mathbb{R}^d)$ that are highly concentrated in the spatial and frequency domains. We show that these wave packets are ``approximate eigenfunctions'' of a spatio-spectral limiting operator. To construct the wave packets, we use a variant of the Coifman-Meyer local sine basis for $L^2[0,1]$, and we lift the basis to higher dimensions using a tensor product.

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arXiv2023

Approximate orthogonality, Bourgain's pinned distance theorem and exponential frames

Let $A$ be a countable and discrete subset of ${\Bbb R}^d$, $d \ge 2$, of positive upper Beurling density. Let $K$ denote a bounded symmetric convex set with a smooth boundary and everywhere non-vanishing Gaussian curvature. It is known that ${\mathcal E}(A)=\{e^{2 \pi i x \cdot a}\}_{a \in A}$ cannot serve as an orthogonal basis for $L^2(K)$ \cite{IKT01}. In this paper, we prove that even approximate average orthogonality is an obstacle to the existence of an exponential frame in the following sense. Let $A$ be as above and $\phi \ge 0$ be a continuous monotonically nonincreasing function on $[0, \infty)$ such that the approximate orthogonality condition holds \begin{align}\notag {\left( \frac{1}{2^j} \int_{2^j}^{2^{j+1}} \phi^p(t) dt \right)}^{1/p} \leq c_j 2^{-j\frac{d+1}{2}} \quad \text{and} \quad |\widehat{\chi}_K(a-a')| \leq \phi(\rho^*(a-a')) \ \forall a \not=a , a,a' \in A, \end{align} where $\rho^*$ is the Minkowski functional on $K^*$, the dual body of $K$. Then, if $$\limsup_{j \to \infty} c_j=0,$$ then the upper density of $A$ is equal to $0$, hence ${\mathcal E}(A)$ is not a frame for $L^2(K)$. The case $p=\infty$ was previously established by the authors of this paper in \cite{IM2020}. The point is that if ${\mathcal E}(A)$ is a frame for $L^2(K)$, then very few pairs of distinct exponentials $e^{2 \pi i x.a}, e^{2 \pi i x.a'}$ from ${\mathcal E}(A)$ come anywhere near being orthogonal. Our proof uses a generalization of Bourgain's result on pinned distances determined by sets of positive Lebesgue upper density in ${\Bbb R}^d$, $d \ge 2$. We also improve the $L^{\infty}$ version of this result originally established in \cite{IM2020}. By using an extension of the combinatorial idea from \cite{IR03}, we prove that under the $L^{\infty}$ hypothesis, $A$ is finite if $d \not=1 \mod 4$. If $d=1$ mod $4, A$ may be infinite, but if it is, then it must be a subset of a line.

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arXiv2022

A characterization of Gabor Riesz bases with separable time-frequency shifts

A Gabor system generated by a window function $g\in L^2(\mathbb{R}^d)$ and a separable set $Λ\times Γ\subset \mathbb{R}^{2d}$ is the collection of time-frequency shifts of $g$ given by $\mathcal G(g, Λ\times Γ) = \left\{ e^{2πi ξ\cdot t}g(t-x)\right\}_{ (x,ξ)\in Λ\times Γ}$. One of the fundamental problems in Gabor analysis is to characterize all windows and time-frequency sets that generate a Gabor frame or Gabor orthonormal basis. The case of Gabor orthonormal bases generated by characteristic functions $g=χ_Ω$ has been solved by Han and Wang. In this paper, we build on these results and obtain a full characterization of Riesz Gabor systems of the form $\mathcal G(χ_Ω, Λ\times Γ)$ when $Ω$ is a tiling of $\mathbb{R}^d$ with respect to $Λ$. Furthermore, for a certain class of lattices $Λ\times Γ$, we prove that a necessary condition for the characteristic function of a multi-tiling set to serve as a window function for a Riesz Gabor basis is that the set must be a tiling set. To prove this, we develop new results on the zeros of the Zak transform and connect these results to Gabor frames.

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arXiv2021

Frame spectral pairs and exponential bases

Given a domain $Ω\subset\Bbb R^d$ with positive and finite Lebesgue measure and a discrete set $Λ\subset \Bbb R^d$, we say that $(Ω, Λ)$ is a {\it frame spectral pair} if the set of exponential functions $\mathcal E(Λ):=\{e^{2πi λ\cdot x}: λ\in Λ\}$ is a frame for $L^2(Ω)$. Special cases of frames include Riesz bases and orthogonal bases. In the finite setting $\Bbb Z_N^d$, $d, N\geq 1$, a frame spectral pair can be similarly defined. %(Here, $\Bbb Z_N$ is the cyclic abelian group of order.) We show how to construct and obtain new classes of frame spectral pairs in $\Bbb R^d$ by "adding" frame spectral pairs in $\Bbb R^{d}$ and $\Bbb Z_N^d$. Our construction unifies the well-known examples of exponential frames for the union of cubes with equal volumes. We also remark on the link between the spectral property of a domain and sampling theory.

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arXiv2021

Riesz bases of exponentials and multi-tiling in finite abelian groups

Motivated by the open problem of exhibiting a subset of Euclidean space which has no exponential Riesz basis, we focus on exponential Riesz bases in finite abelian groups. We point out that that every subset of a finite abelian group has such a basis, removing interest in the existence question in this context. We then define tightness quantities for subsets to measure the conditioning of Riesz bases; for normalized tightness quantities, a value of one corresponds to an orthogonal basis, and a value of infinity corresponds to nonexistence of a basis. As an application, we obtain new weak evidence in favor of the open problem by giving a sequence of subsets of finite abelian groups whose tightness quantities go to infinity in the limit. We also prove that the Cartesian product of a set with a finite abelian group has the same tightness quantities as the original set. Lastly, under an additional hypothesis, explicit bounds are given for tightness quantities in terms of a subset's lowest multi-tiling level by a subgroup and its geometric configuration. This establishes a quantitative link between discrete geometry and harmonic analysis in this setting.

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arXiv2020

On complete and incomplete exponential systems

Given a bounded domain $Ω\subset {\Bbb R}^d$ with positive measure and a finite set $A=\{a^1, a^2, \dots, a^d\}$, we say that the set ${\mathcal E}(A)={\{e^{2 πi x \cdot a^j}\}}_{a^j \in A}$ is a complete exponential system if for every $ξ\in {\Bbb R}^d$, there exists $1 \leq j \leq d+1$ such that \begin{equation} \label{completedef} \int_Ω e^{-2 πi x \cdot (a^j-ξ)} dx \not=0; \end{equation} otherwise ${\mathcal E}(A)$ is called an incomplete exponential system. In this paper, we essentially classify complete and incomplete exponential systems when $Ω=B_d$, the unit ball, and when $Ω=Q_d$, the unit cube. Given a bounded domain $Ω$, we say that $e^{2 πi x \cdot a}, e^{2 πi x \cdot a'}$ are $ϕ$-approximately orthogonal if $$|\widehatχ_Ω(a-a')| \leq ϕ(|a-a'|), \ a\neq a'$$ where $ϕ: [0, \infty) \to [0, \infty)$ is a bounded measurable function that tends to $0$ at infinity. We prove that $L^2(B_d)$ does not possess a $ϕ$-approximate orthogonal basis of exponentials for a wide range of functions $ϕ$. The proof involves connections with the theory of distances in sets of positive Lebesgue upper density originally developed by Furstenberg, Katznelson and Weiss (\cite{FKW90}).

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