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Luís Daniel Abreu

Publications and source records attributed to Luís Daniel Abreu.

At least 19 recordsLinked to original sources
arXiv2026

A computable wandering and tracelike vector for modular orbits in the Bergman space

We construct a function $Φ$ such that the orbit under the representation of PSL(2,Z) is an orthonormal basis for the Bergman space with weight $α=12$. Moreover, we show that $Φ$ is effectively computable as a holomorphic function on the upper half-plane (in the precise sense of computable analysis), by providing an effective procedure. This constructs a wandering and tracelike vector for PSL(2,Z), whose abstract existence was proved by Sir Vaughan Jones in his last paper, where the corresponding construction was left as a problem. The function is built using an orthonormalization and modularization method, and it displays modular reminiscencies, despite not being modular itself. provides a computable implementing vector for the abstract anti-isomorphism between the von Neumann algebra $M_{12}(Γ)$ and its commutant, which is generated, in Rădulescu's sense, by cusp-form Toeplitz operators, while Voiculescu's results provide a random matrix model for $M_{12}(Γ)$.

math.OA↗
arXiv2026

Inverse problems for the zeros of the Wigner function

We study the nodal (zero) set of the Wigner transform, obtaining a rigidity theorem for the eigenstates of one-dimensional harmonic oscillator. For every $k \in \mathbb{N}_0$, and $f\in L^{2}(\mathbb{R})$, equality between the nodal sets of $Wf$ and $Wh_{k}$ determines $f$, up to a constant phase and an explicitly defined nodal-preserving pseudo-displacement. The proof requires a result with independent interest, where it is shown that a Wigner distribution has bounded nodal set if and only if its underlying state is, up to time-frequency and metaplectic symmetries, a finite Hermite expansion. The demonstration of the results combines new analytic, geometric and arithmetic tools: a property of classical convolution, a geometric study of phase space displacements and a new divisibility theorem for generalized Laguerre polynomials. As a by-product, we obtain several results concerning the structure and admissible geometry of bounded nodal sets of Wigner distributions.

quant-ph↗
arXiv2026

Optimal concentration in the Paley-Wiener space

Let $Ω\subset \mathbb{R}$ be a bounded interval and let $PW(Ω)$ be the corresponding Paley--Wiener space. For a measurable set $E\subset \mathbb{R}$ of finite measure, consider the largest possible fraction of the $L^{2}$-mass of a function in $PW(Ω)$ that can lie in $E$. We prove that this concentration is no larger than the concentration attained on an interval of measure $\lvert E\rvert $. Thus, \emph{intervals optimize concentration in the Paley-Wiener space of band-limited functions.} The proof, based on an universality-type limit of the reproducing kernel of analytic trigonometric polynomials on the circle, has two steps. First, we establish an \emph{optimal concentration theorem for analytic trigonometric polynomials on the circle}. Second, the universality-type limit transfers the result from the circle to the real line, by controlling the expansion of circles whose projection kernels are midpoint Riemann sums for the Paley--Wiener sinc kernel.

math.CA↗
arXiv2026

Zero correlations and averaged fields of orthonormal Gaussian functions

We consider the family of point processes $\{\mathcal{Z}_{f_{n}}\}_{n=0}^{\infty}$ of zeros of Gaussian random functions $\{f_{n}(z,\overline{z})\}_{n=0}^{\infty} $, arising from the Gaussian Entire Function \[ f_{0}(z):=\sum_{k=0}^{\infty} ζ_{k} \frac{z^{k}}{\sqrt{k!}}, \quad ζ_{k} \sim N_{\mathbb{C}}(0,1)\text{ i.i.d.} \] by iteration of the Landau raising operator, and orthonormal at each point in expectation in the sense that \[ \mathbb{E}\left[ e^{-\left\vert z\right\vert^{2}}f_{n}(z,\overline{z})\overline{f_{n^{\prime }}(z,\overline{z})}\right] ={δ}_{nn'}. \] We first show that the normalized pair correlations $g_{n,n+k}(z,w)$ of the pairs $(\mathcal{Z}_{f_{n}},\mathcal{Z}_{f_{n+k}})$ exhibit \emph{a pattern reminiscent of the classical interlacing of zeros of orthogonal polynomials}: when $w\rightarrow z$, $g_{n,n+k}$ displays repulsion for $k=1$, attraction for $k=2$, and no short-range second-order correlation for $k \ge 3$. We complement this with the convergence of real-valued averaged fields on compacts $K \subset \mathbb{C}$, \[ \lim_{N \to \infty} \frac{1}{N}\sum_{n=0}^{N-1}\left\vert f_{n}(z,\overline{z})e^{-\frac{\left\vert z\right\vert^{2}}{2}} \right\vert^{2} \rightarrow 1 \quad \text{ almost surely in $C(K)$}, \] and a functional central limit theorem for the corresponding scaled fluctuations, which converge to the Gaussian process \[\mathcal{G}(z) = \frac{1}{\sqrtπ} \int_{\mathbb{C}} \mathbf{1}_{B(z,1)}(u) dW_{\mathbb{R}}(u), \] where $W_{\mathbb{R}}$ denotes real white noise on $\mathbb{C}$ and $B(z,1)$ is the unit disk centered at $z$. The results are motivated by problems in signal processing. Due to an identification with white noise spectrograms, they confirm conjectures of Flandrin and Bayram-Baraniuk and provide a rationale for the efficiency of high resolution time-frequency algorithms, namely \emph{ConceFT}, by Daubechies, Wang and Wu.

math.PR↗
arXiv2025

On quantum large sieve inequalities and operator recovery from incomplete information

We obtain large sieve type inequalities for the Rayleigh quotient of the restriction of phase space representations of higher rank operators, via an operator analogue of the short-time Fourier transform (STFT). The resulting bounds are referred to as `quantum large sieve inequalities'. On the shoulders of Donoho and Stark, we demonstrate that these inequalities guarantee the recovery of an operator whose phase-space information is missing or unobservable over a 'measure-sparse' region $Ω$, by solving an $L^{1}$-minimization program. This is an operator version of what is commonly known as `Logan's phenomenon'. Moreover, our results can be viewed as a deterministic, continuous variable version, on phase space, of `low-rank' matrix recovery, which itself can be regarded as an operator version of (finite-rank) compressive sensing. Our results depend on an abstract large sieve principle for operators with integrable STFT and on a non-commutative analogue of the local reproducing formula in rotationally invariant domains (first stated by Seip for the Fock space of entire functions). As an application, we obtain concentration estimates for Cohen's class distributions and the Husimi function. We motivate the paper by comparing with Nicola and Tilli's Faber-Krahn inequality for the STFT, illustrating that norm bounds on a domain $Ω$, obtained by large sieve methods, introduce a trade-off between sparsity and concentration properties of $Ω$: If $Ω$ is sparse, the large sieve bound may significantly improve known operator norm bounds, while if $Ω$ is concentrated, it produces worse bounds.

math.FA↗
arXiv2024

Donoho-Logan large sieve principles for the wavelet transform

In this paper we formulate Donoho and Logan's large sieve principle for the wavelet transform on the Hardy space, adapting the concept of maximum Nyquist density to the hyperbolic geometry of the underlying space. The results provide deterministic guarantees for $L_{1}$-minimization methods and hold for a class of mother wavelets which constitute an orthonormal basis of the Hardy space and can be associated with higher hyperbolic Landau levels. Explicit calculations of the basis functions reveal a connection with the Zernike polynomials. We prove a novel local reproducing formula for the spaces in consideration and use it to derive concentration estimates of the large sieve type for the corresponding wavelet transforms. We conclude with a discussion of optimality of localization in the analytic case by building on recent results of Kulikov, Ramos and Tilli based on the groundbreaking methods of Nicola and Tilli. This leads to a sharp uncertainty principle and a local Lieb inequality for the wavelet transform.

math.FA↗
arXiv2023

Entanglement entropy and hyperuniformity of Ginibre and Weyl-Heisenberg ensembles

We show that, for a class of planar determinantal point processes (DPP) X, the entanglement entropy of X on a compact region grows exactly at the rate of the variance fluctuation in that region. Therefore, such DPPs satisfy an area law if they are of Class I hyperuniformity, while the area law is violated if they are of Class II hyperuniformity. As a result, the entanglement entropy of Weyl-Heisenberg ensembles (a family of DPPs containing the Ginibre ensemble and Ginibre-type ensembles in higher Landau levels), satisfies an area law, as a consequence of its hyperuniformity.

math-ph↗
arXiv2023

The average singular value of a complex random matrix decreases with dimension

We obtain a recurrence relation in $d$ for the average singular value $% α(d)$ of a complex valued $d\times d$\ matrix $\frac{1}{\sqrt{d}}X$ with random i.i.d., N( 0,1) entries, and use it to show that $α(d)$ decreases monotonically with $d$ to the limit given by the Marchenko-Pastur distribution.\ The monotonicity of $α(d)$ has been recently conjectured by Bandeira, Kennedy and Singer in their study of the Little Grothendieck problem over the unitary group $\mathcal{U}_{d}$ \cite{BKS}, a combinatorial optimization problem. The result implies sharp global estimates for $α(d)$, new bounds for the expected minimum and maximum singular values, and a lower bound for the ratio of the expected maximum and the expected minimum singular value. The proof is based on a connection with the theory of Turán determinants of orthogonal polynomials. We also discuss some applications to the problem that originally motivated the conjecture.

math.PR↗
arXiv2022

Comment on 'Comment on 'Affine density, von Neumann dimension and a problem of Perelomov"

In the Letter 'Comment on 'Affine density, von Neumann dimension and a problem of Perelomov', arXiv.2211.04879, by Prof. J. L. Romero, it is claimed that the main theorem of Ref2 := [Adv. Math. 407, Article ID 108564, 22 p. (2022)] is included in the prior research survey Ref1 := [Expo. Math., 40(2), 265-301, 2022]. We provide some background, and demonstrate with hard facts that this claim is false, as well as other claims in the Letter which, in most cases, either result from wrong mathematical reasoning or objectively differ from what is written in the published material. The author might be interested in publicly retracting and apologising for his comments.

math.FA↗
arXiv2019

Harmonic analysis in phase space and finite Weyl-Heisenberg ensembles

Weyl-Heisenberg ensembles are translation-invariant determinantal point processes on $\mathbb{R}^{2d}$ associated with the Schrödinger representation of the Heisenberg group, and include as examples the Ginibre ensemble and the polyanalytic ensembles, which model the higher Landau levels in physics. We introduce finite versions of the Weyl-Heisenberg ensembles and show that they behave analogously to the finite Ginibre ensembles. More specifically, guided by the observation that the Ginibre ensemble with $N$ points is asymptotically close to the restriction of the infinite Ginibre ensemble to the disk of area $N$, we define finite WH ensembles as adequate finite approximations of the restriction of infinite WH ensembles to a given domain $Ω$. We provide a precise rate for the convergence of the corresponding one-point intensities to the indicator function of $Ω$, as $Ω$ is dilated and the process is rescaled proportionally (thermodynamic regime). The construction and analysis rely neither on explicit formulas nor on the asymptotics for orthogonal polynomials, but rather on phase-space methods. Second, we apply our construction to study the pure finite Ginibre-type polyanalytic ensembles, which model finite particle systems in a single Landau level, and are defined in terms of complex Hermite polynomials. On a technical level, we show that finite WH ensembles provide an approximate model for finite polyanalytic Ginibre ensembles, and we quantify the corresponding deviation. By means of this asymptotic description, we derive estimates for the rate of convergence of the one-point intensity of polyanalytic Ginibre ensembles in the thermodynamic limit.

math-ph↗
arXiv2017

MSE estimates for multitaper spectral estimation and off-grid compressive sensing

We obtain estimates for the Mean Squared Error (MSE) for the multitaper spectral estimator and certain compressive acquisition methods for multi-band signals. We confirm a fact discovered by Thomson [Spectrum estimation and harmonic analysis, Proc. IEEE, 1982]: assuming bandwidth $W$ and $N$ time domain observations, the average of the square of the first $K=2NW$ Slepian functions approaches, as $K$ grows, an ideal band-pass kernel for the interval $[-W,W]$. We provide an analytic proof of this fact and measure the corresponding rate of convergence in the $L^{1}$ norm. This validates a heuristic approximation used to control the MSE of the multitaper estimator. The estimates have also consequences for the method of compressive acquisition of multi-band signals introduced by Davenport and Wakin, giving MSE approximation bounds for the dictionary formed by modulation of the critical number of prolates.

cs.IT↗
arXiv2017

Sharp rates of convergence for accumulated spectrograms

We investigate an inverse problem in time-frequency localization: the approximation of the symbol of a time-frequency localization operator from partial spectral information by the method of accumulated spectrograms (the sum of the spectrograms corresponding to large eigenvalues). We derive a sharp bound for the rate of convergence of the accumulated spectrogram, improving on recent results.

math.CA↗
arXiv2017

The Weyl-Heisenberg ensemble: hyperuniformity and higher Landau levels

Weyl-Heisenberg ensembles are a class of determinantal point processes associated with the Schrödinger representation of the Heisenberg group. Hyperuniformity characterizes a state of matter for which (scaled) density fluctuations diminish towards zero at the largest length scales. We will prove that Weyl-Heisenberg ensembles are hyperuniform. Weyl-Heisenberg ensembles include as a special case a multi-layer extension of the Ginibre ensemble modeling the distribution of electrons in higher Landau levels, which has recently been object of study in the realm of the Ginibre-type ensembles associated with polyanalytic functions. In addition, the family of Weyl-Heisenberg ensembles includes new structurally anisotropic processes, where point-statistics depend on the different spatial directions, and thus provide a first means to study directional hyperuniformity.

cond-mat.stat-mech↗
arXiv2016

The bias-variance trade-off in Thomson's multitaper estimator

At the heart of non-parametric spectral estimation, lies the dilemma known as the bias-variance trade-off: low biased estimators tend to have high variance and low variance estimators tend to have high bias. In 1982, Thomson introduced a multitaper method where this trade-off is made explicit by choosing a target bias resolution and obtaining a corresponding variance reduction. The method became the standard in many applications. Its favorable bias-variance trade-off is due to an empirical fact, conjectured by Thomson based on numerical evidence: assuming bandwidth W and N time domain observations, the average of the square of the first $K=\left\lfloor 2NW\right\rfloor$ Slepian functions approaches, as K grows, an ideal band-pass kernel for the interval [-W,W]. We provide an analytic proof of this fact and quantify the approximation error in the L1 norm; the approximation error is then used to control the bias of the multitaper estimator resulting from spectral leakage. This leads to new performance bounds for the method, explicit in terms of the bandwidth W and the number N of time domain observations. Our method is flexible and can be extended to higher dimensions and different geometries.

math.CA↗
arXiv2015

Pseudo prolate spheroidal functions

Let $D_{T}$ and $B_{Ω}$ denote the operators which cut the time content outside $T$ and the frequency content outside $Ω$, respectively. The prolate spheroidal functions are the eigenfunctions of the operator $P_{T,Ω}=D_{T}B_{Ω}D_{T}$. With the aim of formulating in precise mathematical terms the notion of Nyquist rate, Landau and Pollack have shown that, asymptotically, the number of such functions with eigenvalue close to one is $\approx \frac{\left\vert T\right\vert \left\vert Ω\right\vert }{2π}$. We have recently revisited this problem with a new approach: instead of counting the number of eigenfunctions with eigenvalue close to one, we count the maximum number of orthogonal $ε$-pseudoeigenfunctions with $ε$-pseudoeigenvalue one. Precisely, we count how many orthogonal functions have a maximum of energy $ε$ outside the domain $T\times Ω$, in the sense that $\left\Vert P_{T,Ω}f-f\right\Vert ^{2}\leq ε$. We have recently discovered that the sharp asymptotic number is $\approx (1-ε)^{-1}\frac{\left\vert T\right\vert \left\vert Ω\right\vert }{2π}$. The proof involves an explicit construction of the pseudoeigenfunctions of $P_{T,Ω}$. When $T$ and $Ω$ are intervals we call them pseudo prolate spheroidal functions. In this paper we explain how they are constructed.

math.FA↗
arXiv2014

Measures of localization and quantitative Nyquist densities

We obtain a refinement of the degrees of freedom estimate of Landau and Pollak. More precisely, we estimate, in terms of $ε$, the increase in the degrees of freedom resulting upon allowing the functions to contain a certain prescribed amount of energy $ε$ outside a region delimited by a set $T$ in time and a set $Ω$ in frequency. In this situation, the lower asymptotic Nyquist density $\vert T\vert \vert Ω\vert /2π$ is increased to $(1+ε)\vert T\vert \vert Ω\vert /2π$. At the technical level, we prove a pseudospectra version of the classical spectral dimension result of Landau and Pollak, in the multivariate setting of Landau. Analogous results are obtained for Gabor localization operators in a compact region of the time-frequency plane.

math.FA↗
arXiv2014

On accumulated spectrograms

We study the eigenvalues and eigenfunctions of the time-frequency localization operator $H_Ω$ on a domain $Ω$ of the time-frequency plane. The eigenfunctions are the appropriate prolate spheroidal functions for an arbitrary domain $Ω$. Indeed, in analogy to the classical theory of Landau-Slepian-Pollak, the number of eigenvalues of $H_Ω$ in $[1-δ, 1]$ is equal to the measure of $Ω$ up to an error term depending on the perimeter of the boundary of $Ω$. Our main results show that the spectrograms of the eigenfunctions corresponding to the large eigenvalues (which we call the accumulated spectrogram) form an approximate partition ofunity of the given domain $Ω$. We derive both asymptotic, non-asymptotic, and weak $L^2$ error estimates for the accumulated spectrogram. As a consequence the domain $Ω$ can be approximated solely from the spectrograms of eigenfunctions without information about their phase.

math.CA↗