arXiv subjects

Felipe Marceca

Publications and source records attributed to Felipe Marceca.

11 recordsLinked to original sources

arXiv2026

Spectral deviation of concentration operators on reproducing kernel Hilbert spaces

We study the eigenvalue profile of concentration operators (multiplication by an indicator function followed by projection) acting on reproducing kernel Hilbert spaces. The spectral profile of such operators provides a useful notion of local degrees of freedom. We formalize this idea by estimating the number of eigenvalues that lie away from 0 and 1, commonly referred to as the plunge region. Our main motivation is to treat discrete and continuous settings simultaneously and uniformly, and to be able to argue that approximations arising from discretization schemes reflect, in a non-asymptotic sense, the spectral profile of their continuous counterparts. As a case in point, we show that Gabor multipliers computed on sufficiently fine grids obey spectral deviation estimates similar to those available for the short-time Fourier transform (STFT) with bounds that are uniform in the discretization step. Concretely, this means that the theoretical localization properties of the STFT are observable in practice.

math.SP↗

arXiv2025

Sampling properties of the zeroes of the Gaussian entire function

We study sampling properties of the zero set of the Gaussian entire function on Fock spaces. Firstly, we relax Seip and Wallst\'en's density and separation conditions for sampling sets on Fock spaces to obtain weighted inequalities for sets that are not necessarily sampling. On the probabilistic front, we estimate the number of zeroes of the Gaussian entire functions that are close to each other. We use these to prove random sampling inequalities for polynomials of degree at most $d$ using ${d}+o(d)$ points, and show that, with high probability, the sampling constants grow slower than $d^\varepsilon$ for any $\varepsilon>0$. In particular, we recover a result from Lyons and Zhai in the case of the Gaussian entire function, where it is shown that the zeroes are (almost surely) a uniqueness set for the Fock space.

math.PR↗

arXiv2024

Hypercontractivity and strips of convergence in Hardy spaces of general Dirichlet series

For a general Dirichlet series $\sum a_n e^{-\lambda_n s}$ with frequency $\lambda=(\lambda_n)_n$, we study how horizontal translation (i.e. convolution with a Poisson kernel) improves its integrability properties. We characterize hypercontractive frequencies in terms of their additive structure answering some questions posed by Bayart. We also provide sharp bounds for the strips $S_p(\lambda)$ that encode the minimum translation necessary for series in the Hardy space $\mathcal{H}_p(\lambda)$ to have absolutely convergent coefficients.

math.FA↗

arXiv2023

Eigenvalue estimates for Fourier concentration operators on two domains

We study concentration operators associated with either the discrete or the continuous Fourier transform, that is, operators that incorporate a spatial cut-off and a subsequent frequency cut-off to the Fourier inversion formula. Their spectral profiles describe the number of prominent degrees of freedom in problems where functions are assumed to be supported on a certain domain and their Fourier transforms are known or measured on a second domain. We derive eigenvalue estimates that quantify the extent to which Fourier concentration operators deviate from orthogonal projectors, by bounding the number of eigenvalues that are away from 0 and 1 in terms of the geometry of the spatial and frequency domains, and a factor that grows at most poly-logarithmically on the inverse of the spectral margin. The estimates are non-asymptotic in the sense that they are applicable to concrete domains and spectral thresholds, and almost match asymptotic benchmarks. Our work covers for the first time non-convex and non-symmetric spatial and frequency concentration domains, as demanded by numerous applications that exploit the expected approximate low dimensionality of the modeled phenomena. The proofs build on Israel's work on one dimensional intervals [arXiv: 1502.04404v1]. The new ingredients are the use of redundant wave-packet expansions and a dyadic decomposition argument to obtain Schatten norm estimates for Hankel operators.

math.FA↗

arXiv2022

Improved discrepancy for the planar Coulomb gas at low temperatures

We study the planar Coulomb gas in the regime where the inverse temperature $\beta_n$ grows at least logarithmically with respect to the number of particles $n$ (freezing regime, $\beta_n\gtrsim \log n$). We show that, almost surely for large $n$, the discrepancy between the number of particles in any microscopic region and their expected value (given with adequate precision by the equilibrium measure) is, up to log factors, of the order of the perimeter of the observation window. The estimates are valid throughout the whole droplet (the region where the particles accumulate), and are particularly interesting near the boundary, while in the bulk they offer technical improvements over known results. Our work builds on recent results on equidistribution at low temperatures and improves on them by providing refined spectral asymptotics for certain Toeplitz operators on the range of the erfc-kernel (sometimes called Faddeeva or plasma dispersion kernel).

math.PR↗

arXiv2022

Gaussian beta ensembles: the perfect freezing transition and its characterization in terms of Beurling-Landau densities

The Gaussian $\beta$-ensemble is a real $n$-point configuration $\{x_j\}_1^n$ picked randomly with respect to the Boltzmann factor $e^{-\frac\beta 2H_n}$, $H_n=\sum_{i\ne j}\log\frac 1{|x_i-x_j|}+n\sum_{i=1}^n\tfrac 12x_i^2.$ The point process $\{x_j\}_1^n$ tends to follow the semicircle law $\sigma(x)=\tfrac 1{2\pi}\sqrt{(4-x^2)_+}$ in certain average senses. A Fekete configuration (minimizer of $H_n$) is spread out in a much more uniform way in the interval $[-2,2]$ with respect to the regularization $\sigma_n(x)=\max\{\sigma(x),n^{-\frac 1 3}\}$ of the semicircle law. In particular, Fekete configurations are "equidistributed" with respect to $\sigma_n(x)$, in a certain technical sense of Beurling-Landau densities. We consider the problem of characterizing sequences $\beta_n$ of inverse temperatures, which guarantee almost sure equidistribution as $n\to\infty$. We find that a necessary and sufficient condition is that $\beta_n$ grows at least logarithmically in $n$: $$\beta_n\gtrsim \log n.$$ We call this growth rate the perfect freezing regime. We give several further results on the distribution of particles when $\beta_n\gtrsim\log n$, for example on minimal spacing, discrepancies, and sampling and interpolation for weighted polynomials. The condition $\beta_n\gtrsim\log n$ was introduced by some of the authors in the context of two-dimensional Coulomb gas ensembles, where it is shown to be sufficient for equidistribution. Although the technical implementation requires some considerable modifications, the strategy from dimension two adapts well to prove sufficiency also for one-dimensional Gaussian ensembles. On a technical level, we use estimates for weighted polynomials due to Levin, Lubinsky, Gustavsson and others. The other direction (necessity) involves estimates due to Ledoux and Rider on the distribution of particles which fall near or outside the boundary.

math.PR↗

arXiv2021

Spectral deviation of concentration operators for the short-time Fourier transform

Time-frequency concentration operators restrict the integral analysis-synthesis formula for the short-time Fourier transform to a given compact domain. We estimate how much the corresponding eigenvalue counting function deviates from the Lebesgue measure of the time-frequency domain. For window functions in the Gelfand-Shilov class, the bounds almost match known asymptotics, with the advantage of being effective for concrete domains and spectral thresholds. As such our estimates allow for applications where the spectral threshold depends on the geometry of the time-frequency concentration domain. We also consider window functions that decay only polynomially in time and frequency.

math.SP↗

arXiv2020

Decoupling inequalities with exponential constants

Decoupling inequalities disentangle complex dependence structures of random objects so that they can be analyzed by means of standard tools from the theory of independent random variables. We study decoupling inequalities for vector-valued homogeneous polynomials evaluated at random variables. We focus on providing geometric conditions ensuring decoupling inequalities with good constants depending only exponentially on the degree of the polynomial. Assuming the Banach space has finite cotype we achieve this for classical decoupling inequalities that compare the polynomials with their associated multilinear operators. Under stronger geometric assumptions on the involved Banach spaces, we also obtain decoupling inequalities between random polynomials and fully independent random sums of their coefficients. Finally, we present decoupling inequalities where in the multilinear operator just two independent copies of the random vector are involved (one repeated $m-1$ times).

math.FA↗

arXiv2019

Hausdorff-Young type inequalities for vector-valued Dirichlet series

We study Hausdorff-Young type inequalities for vector-valued Dirichlet series which allow to compare the norm of a Dirichlet series in the Hardy space $\mathcal{H}_{p} (X)$ with the $q$-norm of its coefficients. In order to obtain inequalities completely analogous to the scalar case, a Banach space must satisfy the restrictive notion of Fourier type/cotype. We show that variants of these inequalities hold for the much broader range of spaces enjoying type/cotype. We also consider Hausdorff-Young type inequalities for functions defined on the infinite torus $\mathbb{T}^{\infty}$ or the boolean cube $\{-1,1\}^{\infty}$.

math.FA↗

arXiv2018

Random unconditional convergence of vector-valued Dirichlet series

We study random unconditionality of Dirichlet series in vector-valued Hardy spaces $\mathcal H_p(X)$. It is shown that a Banach space $X$ has type 2 (respectively, cotype 2) if and only if for every choice $(x_n)_n\subset X$ it follows that $(x_n n^{-s})_n$ is Random unconditionally convergent (respectively, divergent) in $\mathcal H_2(X)$. The analogous question on $\mathcal H_p(X)$ spaces for $p\neq2$ is also explored. We also provide explicit examples exhibiting the differences between the unconditionality of $(x_n n^{-s})_n$ in $\mathcal H_p(X)$ and that of $(x_n z^n)_n$ in $H_p(X)$.

math.FA↗

arXiv2018

Some remarks on non-symmetric polarization

Let $P:\mathbb{C}^n\rightarrow \mathbb{C}$ be an $m$-homogeneous polynomial given by \[P(x)= \sum_{1\leq j_1\leq \ldots \leq j_m \leq n} c_{j_1 \ldots j_m} x_{j_1}\ldots x_{j_m}.\] Defant and Schl\"uters defined a non-symmetric associated $m$-form $L_P: \left(\mathbb{C}^n \right)^m\rightarrow \mathbb{C}$ by \[L_P \left(x^{(1)},\ldots,x^{(m)} \right)= \sum_{1\leq j_1\leq \ldots \leq j_m \leq n} c_{j_1 \ldots j_m} x_{j_1}^{(1)}\ldots x_{j_m}^{(m)}.\] They estimated the norm of $L_P$ on $(\mathbb{C}^n, \| \cdot\|)^m$ by the norm of $P$ on $(\mathbb{C}^n, \| \cdot\|)$ times a $(c\log n)^{m^2}$ factor for every 1-unconditional norm $\|\cdot\|$ on $\mathbb{C}^n$. A symmetrization procedure based on a card-shuffling algorithm which (together with Defant and Schl\"uters' argument) brings the constant term down to $(c m \log n)^{m-1}$ is provided. Regarding the lower bound, it is shown that the optimal constant is bigger than $(c \log n)^{m/2}$ when $n\gg m$. Finally, the case of $\ell_p$-norms $\|\cdot \|_p$ with $1\leq p <2$ is addressed.

math.FA↗