Skip to main content

Azita Mayeli

Followers

4

Following

1

Public Views

Interests

Uploads

Papers by Azita Mayeli

Tight Wavelet Frame Sets in Finite Vector Spaces

arXiv (Cornell University), Mar 20, 2017

Let q ≥ 2 be an integer, and F d q , d ≥ 1, be the vector space over the cyclic space F q. The pu... more Let q ≥ 2 be an integer, and F d q , d ≥ 1, be the vector space over the cyclic space F q. The purpose of this paper is twofold. First, we obtain sufficient conditions on E ⊂ F d q such that the inverse Fourier transform of 1 E generates a tight wavelet frame in L 2 (F d q). We call these sets (tight) wavelet frame sets. The conditions are given in terms of multiplicative and translational tilings, which is analogous with Theorem 1.1 ([20]) by Wang in the setting of finite fields. In the second part of the paper, we exhibit a constructive method for obtaining tight wavelet frame sets in F d q , d ≥ 2, q an odd prime and q ≡ 3 (mod 4).

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

arXiv (Cornell University), Jan 22, 2023

Let A be a countable and discrete subset of R d , d ≥ 2, of positive upper Beurling density. Let ... more Let A be a countable and discrete subset of R d , d ≥ 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 E(A) = {e 2πix•a } a∈A cannot serve as an orthogonal basis for L 2 (K) [12]. 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 φ ≥ 0 be a continuous monotonically non-increasing function on [0, ∞) such that the approximate orthogonality condition holds 1 2 j 2 j+1 2 j φ p (t)dt 1 p ≤ c j 2 −j d+1 2 and | χ K (a−a ′)| ≤ φ(ρ * (a−a ′)) ∀a = a, a, a ′ ∈ A, where ρ * is the Minkowski functional on K * , the dual body of K. Then, if lim sup j→∞ c j = 0, then the upper density of A is equal to 0, hence E(A) = {e 2πix•a } a∈A is not a frame for L 2 (K). The case p = ∞ was previously established by the authors of this paper in [9]. The point is that if E(A) is a frame for L 2 (K), then very few pairs of distinct exponentials e 2πix•a , e 2πix•a ′ from 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 R d , d ≥ 2. We also improve the L ∞ version of this result originally established in [9]. By using an extension of the combinatorial idea from [13], we prove that under the L ∞ hypothesis, A is finite if d = 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.

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

arXiv (Cornell University), Jan 23, 2023

Prolate spheroidal wave functions are an orthogonal family of bandlimited functions on R that hav... more Prolate spheroidal wave functions are an orthogonal family of bandlimited functions on 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 spatio-spectral limiting operator (SSLO) on L 2 (R d), which is an alternating product of projection operators associated to given spatial and frequency domains in 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 (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.

Commutative and noncommutative harmonic analysis and applications : AMS Special Session in Memory of Daryl Geller on Wavelet and Frame Theoretic Methods in Harmonic Analysis and Partial Differential Equations, September 22-23, 2012, Rochester Institute of Technology, Rochester, NY

American Mathematical Society eBooks, 2013

Riesz Wavelets, Tiling and Spectral Sets in LCA Groups

arXiv (Cornell University), Mar 20, 2017

This paper is devoted to the study of geometry properties of wavelet and Riesz wavelet sets on lo... more This paper is devoted to the study of geometry properties of wavelet and Riesz wavelet sets on locally compact abelian groups. The catalyst for our research is a result by Wang ([32], Theorem 1.1) in the Euclidean wavelet theory. Here, we extend the result to wavelet and Riesz wavelet collection of sets in infinite locally compact abelian groups.

Lattice sub-tilings and frames in LCA groups

arXiv (Cornell University), May 11, 2016

Given a lattice Λ in a locally compact abelian group G and a measurable subset Ω with finite and ... more Given a lattice Λ in a locally compact abelian group G and a measurable subset Ω with finite and positive measure, then the set of characters associated to the dual lattice form a frame for L 2 (Ω) if and only if the distinct translates by Λ of Ω have almost empty intersections. Some consequences of this results are the well-known Fuglede theorem for lattices, as well as a simple characterization for frames of modulates.

Identification of Besov spaces via Littlewood-Paley-Stein Type g-functions

arXiv (Cornell University), Mar 20, 2017

We use Littlewood-Paley-Stein type g-functions (also called generalized square functions) associa... more We use Littlewood-Paley-Stein type g-functions (also called generalized square functions) associated to symmetric diffusion semigroups to obtain a characterization of inhomogeneous abstract Besov spaces on the abstract Hilbert spaces. Then we apply our results for the abstract Besov spaces defined through the Poisson and Gauss-Weierstrass semigroups.

Gabor orthogonal bases and convexity

arXiv (Cornell University), Aug 21, 2017

Let g(x) = χ B (x) be the indicator function of a bounded convex set B in R d , d ≥ 2, with a smo... more Let g(x) = χ B (x) be the indicator function of a bounded convex set B in R d , d ≥ 2, with a smooth boundary and everywhere non-vanishing Gaussian curvature. Using a combinatorial approach we prove that if d = 1 mod 4, then there does not exist S ⊂ R 2d such that {g(x − a)e 2πix•b } (a,b)∈S is an orthonormal basis for L 2 (R d).

Heisenberg Frame Sets

arXiv (Cornell University), Mar 28, 2009

We develop a notion of Heisenberg frame sets in the dual of the Heisenberg group by means of inva... more We develop a notion of Heisenberg frame sets in the dual of the Heisenberg group by means of invariant multiplicity one subspaces. We prove a characterization of Heisenberg frame sets that is precisely analogous to the characterization of classical frame sets. In the process we obtain a necessary condition for general frame vectors of Weyl-Heisenberg systems.

Non-convex optimization via strongly convex majoirziation-minimization

arXiv (Cornell University), Jun 13, 2019

In this paper, we introduce a class of nonsmooth nonconvex least square optimization problem usin... more In this paper, we introduce a class of nonsmooth nonconvex least square optimization problem using convex analysis tools and we propose to use the iterative minimization-majorization (MM) algorithm on a convex set with initializer away from the origin to find an optimal point for the optimization problem. For this, first we use an approach to construct a class of convex majorizers which approximate the value of non-convex cost function on a convex set. The convergence of the iterative algorithm is guaranteed when the initial point $x^{(0)}$ is away from the origin and the iterative points $x^{(k)}$ are obtained in a ball centred at $x^{(k-1)}$ with small radius. The algorithm converges to a stationary point of cost function when the surregators are strongly convex. For the class of our optimization problems, the proposed penalizer of the cost function is the difference of $\ell_1$-norm and the Moreau envelope of a convex function, and it is a generalization of GMC non-separable penalty function previously introduced by Ivan Selesnick in \cite{IS17}.

On complete and incomplete exponential systems

arXiv (Cornell University), Jul 15, 2020

Given a bounded domain Ω ⊂ R d with positive measure and a finite set A = {a 1 , a 2 ,. .. , a d ... more Given a bounded domain Ω ⊂ R d with positive measure and a finite set A = {a 1 , a 2 ,. .. , a d }, we say that the set E(A) = {e 2πix•a j } a j ∈A is a complete exponential system if for every ξ ∈ R d , there exists 1 ≤ j ≤ d + 1 such that (0.1) Ω e −2πix•(a j −ξ) dx = 0; otherwise 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 2πix•a , e 2πix•a ′ are φ-approximately orthogonal if | χ Ω (a − a ′)| ≤ φ(|a − a ′ |), a = a ′ where φ : [0, ∞) → [0, ∞) 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 ([5]).

Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science

Applied and numerical harmonic analysis, 2017

The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Novel methods in harmonic analysis

Birkhäuser eBooks, 2017

Bracket map for Heisenberg group and the characterization of cyclic subspaces

arXiv (Cornell University), Mar 10, 2013

The bracket map was originally considered in [7] for locally compact abelian groups. In this work... more The bracket map was originally considered in [7] for locally compact abelian groups. In this work we extend the study of bracket maps to the noncommutative setting, providing characterizations of bases and frames for cyclic subspaces of the Heisenberg group. We also indicate how to generalize these results to a class of non-abelian nilpotent Lie groups whose irreducible representations are square integrable modulo the center. 1

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

SSRN Electronic Journal

On the Road to Tenure

Tight wavelet frame sets in finite vector spaces

Applied and Computational Harmonic Analysis, 2019

Let q ≥ 2 be an integer, and F d q , d ≥ 1, be the vector space over the cyclic space F q. The pu... more Let q ≥ 2 be an integer, and F d q , d ≥ 1, be the vector space over the cyclic space F q. The purpose of this paper is twofold. First, we obtain sufficient conditions on E ⊂ F d q such that the inverse Fourier transform of 1 E generates a tight wavelet frame in L 2 (F d q). We call these sets (tight) wavelet frame sets. The conditions are given in terms of multiplicative and translational tilings, which is analogous with Theorem 1.1 ([20]) by Wang in the setting of finite fields. In the second part of the paper, we exhibit a constructive method for obtaining tight wavelet frame sets in F d q , d ≥ 2, q an odd prime and q ≡ 3 (mod 4).

Homogeneous Besov Spaces on Stratiﬁed Lie Groups and Their Wavelet Characterization

We establish wavelet characterizations of homogeneous Besov spaces on stratified Lie groups, both... more We establish wavelet characterizations of homogeneous Besov spaces on stratified Lie groups, bothin terms of continuous and discrete wavelet systems. We first introduce a notion of homogeneous Besov space ̇Bsp,qin terms of a Littlewood-Paley-type decomposition, in analogy to the well-known characterization of the Euclidean case. Such decompositions can be defined via the spectral measure of a suitably chosen sub-Laplacian. We prove that the scale of Besov spaces is independent of the precise choice of Littlewood-Paley decomposition. In particular, different sub-Laplacians yield the same Besov spaces. We then turn to wavelet characterizations, first via continuous wavelet transforms which can be viewed as continuous-scale Littlewood-Paley decompositions, then via discretely indexed systems. We prove the existence of wavelet frames and associate datomic decomposition formulas for all homogeneous Besov spaces ̇Bsp,q with 1≤p, q \u3c∞ands R

The Fuglede Conjecture holds in Zp × Zp

In this paper we study subsets E of Z_p^d such that any function f : E → C can be written as a li... more In this paper we study subsets E of Z_p^d such that any function f : E → C can be written as a linear combination of characters orthogonal with respect to E. We shall refer to such sets as spectral. In this context, we prove the Fuglede Conjecture in Z_p^2 which says that E ⊂ Z_p^2 is spectral if and only if E tiles Z_p^2 by translation. Arithmetic properties of the finite field Fourier transform, elementary Galois theory and combinatorial geometric properties of direction sets play the key role in the proof.

Bracket map for the Heisenberg group and the characterization of cyclic subspaces

Applied and Computational Harmonic Analysis, 2014

The bracket map was originally considered in [7] for locally compact abelian groups. In this work... more The bracket map was originally considered in [7] for locally compact abelian groups. In this work we extend the study of bracket maps to the noncommutative setting, providing characterizations of bases and frames for cyclic subspaces of the Heisenberg group. We also indicate how to generalize these results to a class of non-abelian nilpotent Lie groups whose irreducible representations are square integrable modulo the center. 1