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Xavier Massaneda

Publications and source records attributed to Xavier Massaneda.

At least 19 recordsLinked to original sources
arXiv2026

Separation properties of a hybrid point process with determinantal radii and uniform arguments

We recently characterized the almost sure separation of determinantal point processes $Λ_ϕ$ in the plane associated with Fock spaces $\mathcal F_ϕ$ with doubling weights $ϕ$. We also showed that, as expected, a more restrictive condition is required to characterize the almost surely separated Poisson processes with the same first intensity as $Λ_ϕ$. To gain further insight into this difference in behavior, we focus on radial weights $ϕ$ and introduce a hybrid process $Λ_ϕ^M=\{r_j e^{iθ_j}\}_{j}$, where the moduli $r_j$ are taken from $Λ_ϕ$, while the arguments $θ_j$ are chosen independently and uniformly in $[0,2π)$, independently of the radii. Our main result is that the hybrid process is almost surely separated under exactly the same condition as the corresponding Poisson process. Thus, despite preserving the joint radial distribution of the determinantal process, independent angular randomization destroys its improved separation properties.

math.CV↗
arXiv2025

Separated determinantal point processes and generalized Fock spaces

We study conditions so that the determinantal point process $Λ_ϕ$ associated to a generalized Fock space defined by a doubling subharmonic weight $ϕ$ is almost surely a separated sequence in $\mathbb C$. Under a natural assumption on $ϕ$, we provide a characterization of such processes. Additionally, we emphasize the role of intrinsic repulsion in determinantal processes by comparing $Λ_ϕ$ with the Poisson process of the same first intensity. As an application, we show that the determinantal process $Λ_α$ associated to the canonical weight $ϕ_α(z)=|z|^α$, $α>0$, is almost surely separated if and only if $α<4/3$. In contrast, the Poisson process $Λ_α^P$ having the same first intensity as $Λ_α$ is almost surely separated if and only if $α<1$.

math.CV↗
arXiv2024

Multiple sampling and interpolation in a space of polynomials

We study sampling and interpolation arrays with multiplicities for the spaces P_k of holomorphic polynomials of degree at most k. We find that the geometric conditions satisfied by these arrays are in accordance with the conditions satisfied by the sampling and interpolating sequences with unbounded multiplicities in the Fock space, which can be seen as a limiting case of the space P_k as k tends to infinity.

math.CV↗
arXiv2022

Inhomogeneous Poisson processes in the disk and interpolation

We investigate different geometrical properties of the inhomogeneous Poisson point process $Λ_μ$ associated to a positive, locally finite, $σ$-finite measure $μ$ on the unit disk. In particular, we characterize the processes $Λ_μ$ such that almost surely: 1) $Λ_μ$ is a Carleson-Newman sequence; 2) $Λ_μ$ is the union of a given number M of separated sequences. We use these results to discuss the measures $μ$ such that the associated process $Λ_μ$ is almost surely an interpolating sequence for the Hardy, Bloch or weighted Dirichlet spaces.

math.CV↗
arXiv2020

An enhanced uncertainty principle for the Vaserstein distance

We improve some recent results of Sagiv and Steinerberger that quantify the following uncertainty principle: for a function $f$ with mean zero, either the size of the zero set of the function or the cost of transporting the mass of the positive part of $f$ to its negative part must be big. We also provide a sharp upper estimate of the transport cost of the positive part of an eigenfunction of the Laplacian. This proves a conjecture of Steinerberger and provides a lower bound of the size of the nodal set of the eigenfunction.

math.CA↗
arXiv2019

From $ H^\infty$ to N. Pointwise properties and algebraic structure in the Nevanlinna class

This survey shows how, for the Nevanlinna class N of the unit disc, one can define and often characterize the analogues of well-known objects and properties related to the algebra of bounded analytic functions $ H^\infty$: interpolating sequences, Corona theorem, sets of determination, stable rank, as well as the more recent notions of Weak Embedding Property and threshold of invertibility for quotient algebras. The general rule we observe is that a given result for $H^\infty$ can be transposed to N by replacing uniform bounds by a suitable control by positive harmonic functions. We show several instances where this rule applies, as well as some exceptions. We also briefly discuss the situation for the related Smirnov class.

math.CV↗
arXiv2018

The Corona Property in Nevanlinna quotient algebras and Interpolating sequences

Let $I$ be an inner function in the unit disk $\mathbb D$ and let $\mathcal N$ denote the Nevanlinna class. We prove that under natural assumptions, Bezout equations in the quotient algebra $\mathcal N/I\mathcal N$ can be solved if and only if the zeros of $I$ form a finite union of Nevanlinna interpolating sequences. This is in contrast with the situation in the algebra of bounded analytic functions, where being a finite union of interpolating sequences is a sufficient but not necessary condition. An analogous result in the Smirnov class is proved as well as several equivalent descriptions of Blaschke products whose zeros form a finite union of interpolating sequences in the Nevanlinna class.

math.CA↗
arXiv2016

Finitely generated ideals in the Nevanlinna class

In this paper we investigate finitely generated ideals in the Nevanlinna class. We prove analogues to some known results for the algebra of bounded analytic functions $H^{\infty}$. We also show that, in contrast to the $H^{\infty}$-case, the stable rank of the Nevanlinna class is strictly bigger than 1.

math.CV↗
arXiv2014

Gaussian Analytic functions in the polydisk

We study hyperbolic Gaussian analytic functions in the unit polydisk of $\mathbb C^n$. Following the scheme previously used in the unit ball we first study the asymptotics of fluctuations of linear statistics as the directional intensities $L_j$, $j=1,\dots,n$ tend to $\infty$. Then we estimate the probability of large deviations of such linear statistics and use the estimate to prove a hole theorem. Our proofs are inspired by the methods of M. Sodin and B. Tsirelson for the one-dimensional case, and B. Shiffman and S. Zelditch for the study of the analogous problem for compact Kähler manifolds.

math.CV↗
arXiv2014

Gaussian Analytic functions in the unit ball

We study some properties of hyperbolic Gaussian analytic functions of intensity $L$ in the unit ball of $\mathbb C^n$. First we deal with the asymptotics of fluctuations of linear statistics as $L\to\infty$. Then we estimate the probability of large deviations (with respect to the expected value) of such linear statistics and use this estimate to prove a hole theorem.

math.CV↗
arXiv2014

Volume fluctuations of random analytic varieties in the unit ball

Given a Gaussian analytic function $f_L$ of intesity $L$ in the unit ball of $\mathbb C^n$, $n\geq 2$, consider its (random) zero variety $Z(f_L)$. We study the variance of the $(n-1)$-dimensional volume of $Z(f_L)$ inside a pseudo-hyperbolic ball of radius $r$. We first express this variance as an integral of a positive function in the unit disk. Then we study its asymptotic behaviour as $L\to\infty$ and as $r\to 1^{-}$. Both the results and the proofs generalise to the ball those given by Jeremiah Buckley for the unit disk.

math.CV↗
arXiv2013

Reverse Carleson measures in Hardy spaces

We give a necessary and sufficient condition for a measure $μ$ in the closed unit disk to be a reverse Carleson measure for Hardy spaces. This extends a previous result of Lefévre, Li, Queffélec and Rodr\'ıguez-Piazza \cite{LLQR}. We also provide a simple example showing that the analogue for the Paley-Wiener space does not hold. This example can be generalised to model spaces associated to one-component inner functions.

math.CV↗
arXiv2013

Non cyclic functions in the Hardy space of the bidisc with arbitrary decrease

We construct an example to show that no condition of slow decrease of the modulus of a function is sufficient to make it cyclic in the Hardy space of the bidisc. This is similar to what is well known in the case of the Hardy space of the disc, but in contrast to the case of the Bergman space of the disc.

math.CV↗
arXiv2012

Interpolation and peak functions for the Nevanlinna and Smirnov classes

It is known (implicit in [HMNT]) that when $Λ$ is an interpolating sequence for the Nevanlinna or the Smirnov class then there exist functions $f_λ$ in these spaces, with uniform control of their growth and attaining values 1 on $λ$ and 0 in all other $λ'\neqλ$. We provide an example showing that, contrary to what happens in other algebras of holomorphic functions, the existence of such functions does not imply that $Λ$ is an interpolating sequence.

math.CV↗
arXiv2012

Inhomogenous random zero sets

We construct random point processes in the complex plane that are asymptotically close to a given doubling measure. The processes we construct are the zero sets of random entire functions that are constructed through generalised Fock spaces. We offer two alternative constructions, one via bases for these spaces and another via frames, and we show that for both constructions the average distribution of the zero set is close to the given doubling measure, and that the variance is much less than the variance of the corresponding Poisson point process. We prove some asymptotic large deviation estimates for these processes, which in particular allow us to estimate the `hole probability', the probability that there are no zeroes in a given open bounded subset of the plane. We also show that the `smooth linear statistics' are asymptotically normal, under an additional regularity hypothesis on the measure. These generalise previous results by Sodin and Tsirelson for the Lebesgue measure.

math.CV↗
arXiv2010

Traces of functions in Fock spaces on lattices of critical density

Following a scheme of Levin we describe the values that functions in Fock spaces take on lattices of critical density in terms of both the size of the values and a cancelation condition that involves discrete versions of the Cauchy and Beurling-Ahlfors transforms.

math.CV↗