arXiv subjects

María Ángeles García-Ferrero

Publications and source records attributed to María Ángeles García-Ferrero.

7 recordsLinked to original sources

arXiv2026

Stability of the generalized Wehrl entropy and the local concentration of homogeneous polynomials

We study two notions of concentration for homogeneous polynomials of degree $N$ in $d+1$ complex variables on the unit sphere: a local notion measuring the fraction of the $L^2$-norm supported on a measurable subset; and a global notion given by the generalized Wehrl entropy. In both cases, the extremizers are known to be reproducing kernels, that is, monomials up to a unitary rotation, by results of Lieb--Solovej. We establish stability results for both inequalities in higher dimensions. For the local concentration, we show that for sets of sufficiently small measure, the almost-maximizers are quantitatively close to reproducing kernels, both in the polynomial and in the domain, extending previous resuls in one dimension. For the generalized Wehrl entropy, we prove that for any non-linear convex function $Φ$ and all sufficiently large degree $N$, the reproducing kernels are the unique minimizers up to stability, complementing recent results by Nicola-Riccardi-Tilli, which require a non-linearity condition near $1$ that exclude key examples such as the concentration functional. As a consequence, by passing to the large-$N$ limit, we recover stability results for both problems in the Bargmann-Fock space.

math.CA↗

arXiv2025

Stability of the concentration inequality on polynomials

In this paper, we study the stability of the concentration inequality for one-dimensional complex polynomials. We provide the stability of the local concentration inequality and a global version using a Wehrl-type entropy.

math.CA↗

arXiv2024

Reconstruction for the Calderón problem with Lipschitz conductivities

We determine the conductivity of the interior of a body using electrical measurements on its surface. We assume only that the conductivity is bounded below by a positive constant and that the conductivity and surface are Lipschitz continuous. To determine the conductivity we first solve an associated integral equation locally, finding solutions in $H^1(B)$, where $B$ is a ball that properly contains the body. A key ingredient is to equip this Sobolev space with an equivalent norm which depends on two auxiliary parameters that can be chosen to yield a contraction.

math.AP↗

arXiv2021

On the Calderón problem for nonlocal Schrödinger equations with homogeneous, directionally antilocal principal symbols

In this article we consider direct and inverse problems for $α$-stable, elliptic nonlocal operators whose kernels are possibly only supported on cones and which satisfy the structural condition of \emph{directional antilocality} as introduced in \cite{I86}. We consider the Dirichlet problem for these operators on the "domain of dependence of the operator" and in several, adapted function spaces. This formulation allows one to avoid natural "gauges" which would else have to be considered in the study of the associated inverse problems. Exploiting the directional antilocality of these operators we complement the investigation of the \emph{direct problem} with infinite data and single measurement uniqueness results for the associated \emph{inverse problems}. Here, due to the only directional antilocality, new geometric conditions arise on the measurement domains. We discuss both the setting of symmetric and a particular class of non-symmetric nonlocal elliptic operators, and contrast the corresponding results for the direct and inverse problems. In particular for only "one-sided operators" new phenomena emerge both in the direct and inverse problems: For instance, it is possible to study the problem in data spaces involving local and nonlocal data, the unique continuation property may not hold in general and further restrictions on the measurement set for the inverse problem arise.

math.AP↗

arXiv2021

Exceptional Legendre Polynomials and Confluent Darboux Transformations

Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for polynomial sequences that miss a finite number of "exceptional" degrees. In this paper we introduce a new construction of multi-parameter exceptional Legendre polynomials by considering the isospectral deformation of the classical Legendre operator. Using confluent Darboux transformations and a technique from inverse scattering theory, we obtain a fully explicit description of the operators and polynomials in question. The main novelty of the paper is the novel construction that allows for exceptional polynomial families with an arbitrary number of real parameters.

math.CA↗

arXiv2021

Runge Approximation and Stability Improvement for a Partial Data Calderón Problem for the Acoustic Helmholtz Equation

In this article, we discuss quantitative Runge approximation properties for the acoustic Helmholtz equation and prove stability improvement results in the high frequency limit for an associated partial data inverse problem modelled on \cite{AU04, KU19}. The results rely on quantitative unique continuation estimates in suitable function spaces with explicit frequency dependence. We contrast the frequency dependence of interior Runge approximation results from non-convex and convex sets.

math.AP↗

arXiv2020

On Two Methods for Quantitative Unique Continuation Results for Some Nonlocal Operators

In this article we present two mechanisms for deducing logarithmic quantitative unique continuation bounds for certain classes of integral operators. In our first method, expanding the corresponding integral kernels, we exploit the logarithmic stability of the moment problem. In our second method we rely on the presence of branch-cut singularities for certain Fourier multipliers. As an application we present quantitative Runge approximation results for the operator $ L_s(D) = \sum\limits_{j=1}^{n}(-\partial_{x_j}^2)^{s} + q$ with $s\in [\frac{1}{2},1)$ and $q\in L^{\infty}$ acting on functions on $\mathbb{R}^n$.

math.AP↗