arXiv subjects

Joaquim Ortega-Cerdà

Publications and source records attributed to Joaquim Ortega-Cerdà.

At least 19 recordsLinked to original sources

arXiv2026

The zeros of point evaluation extremals in Paley-Wiener spaces

For $1\le p < \infty$, $φ_p$ is the function of minimal $L^p$ norm among all functions $f$ of exponential type at most $π$ for which $f(0)=1$. We start from the ansatz that the positive zeros $τ_n$ of $φ_p$ are close to $n+1/p-1/2$ and use a weighted convex correction to first place them in an $\ell^2$-neighborhood of this sequence. A bootstrap argument then improves this to a {Poincaré asymptotic} expansion of $τ_n-(n+\frac1p-\frac12)$ in terms of odd powers of $(n+\frac1p-\frac12)^{-1}$. We also get precise information about the Fourier transform $\widehatφ_p$, a result that in particular implies that for all integers $n\ge 2$, $\widehatφ_{2n}$ is an integrable function which, when convolved with itself $2n-2$ times, is constant on the interval $[-π,π]$. In particular, we identify the extremizer in an autoconvolution problem posed by B. Green in 2001.

math.CA↗

arXiv2026

Contractivity of Rudin's diagonal restriction operator for $H^p$ of the polydisc

We establish that Rudin's diagonal restriction operator is contractive from the Hardy space $H^p$ of the bidisc to the Bergman space $A^p$ of the disc for $2\leq p<\infty$, and characterize all cases of equality. We extend the result to polydiscs and obtain applications to polynomial point evaluation among others. The proof relies on an explicit formula, deduced from certain Green's formulas, for the defect between the $p$th powers of the two $L^p$ norms.

math.FA↗

arXiv2026

There are no Riesz bases of exponentials in balls and triangles

We prove that $L^2(Ω)$ admits no Riesz basis of exponentials when $Ω$ is a ball in $\R^n$, $n\geq2$, or a nondegenerate triangle in $\R^2$. The ball result extends to every nonempty bounded convex open set with $C^2$ boundary in $\R^n$, $n\geq2$. Starting from a hypothetical Riesz basis, we use Beurling weak limits of translates of its frequency set to construct a stationary system. On the associated Hilbert space, we define spectral cutoff projections and a comparison family of orthogonal projections obtained from the synthesis operator. The distance between corresponding projections is uniformly bounded by a constant strictly smaller than one. The comparison family is continuous in operator norm, whereas the boundary geometry of $Ω$ produces two strong limits of the cutoff projections with strictly nested ranges. Both limiting projections would then be at distance less than one from the same comparison projection, which is impossible.

math.CA↗

arXiv2026

A set of points with optimal $L^2$ spherical cap discrepancy

We introduce an explicit deterministic collection of $N$ spherical points, $\cP_N\subset\mathbb S^2$, that we call the {\em deterministic Diamond points}. For every $0<α<2$ we prove that there exists a constant $C_α>0$ such that \[ 0\leq \frac{2^{α+1}}{α+2}N^2-\sum_{x,y\in\cP_N}|x-y|^α\le C_αN^{1-α/2}. \] This result shows that $\cP_N\subset\mathbb S^2$ has a Riesz energy deficit of optimal order in this range of the parameter. In particular, for $α=1$, Stolarsky's invariance principle implies that $\cP_N\subset\mathbb S^2$ has $L^2$ spherical cap discrepancy of asymptotically optimal order \[ D_{L^2}^C(\mathcal P_N)\asymp N^{-3/4}, \] making it, to our knowledge, the first explicit configuration proved to have this property. More generally, our result implies that the point set $\cP_N$ has Sobolev $H^{s}(\Sph)$ worst-case error of optimal order for all $1<s<2$.

math.CA↗

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↗

arXiv2026

The Hörmander--Bernhardsson extremal function

We characterize the function $φ$ of minimal $L^1$ norm among all functions $f$ of exponential type at most $π$ for which $f(0)=1$. This function, studied by Hörmander and Bernhardsson in 1993, has only real zeros $\pm τ_n$, $n=1,2, \ldots$. Starting from the fact that $n+\frac12-τ_n$ is an $\ell^2$ sequence, established in an earlier paper of ours, we identify $φ$ in the following way. We factor $φ(z)$ as $Φ(z)Φ(-z)$, where $Φ(z)= \prod_{n=1}^\infty(1+(-1)^n\frac{z}{τ_n})$ and show that $Φ$ satisfies a certain second order linear differential equation along with a functional equation either of which characterizes $Φ$. We use these facts to establish an odd power series expansion of $n+\frac12-τ_n$ in terms of $(n+\frac12)^{-1}$ and a power series expansion of the Fourier transform of $φ$, as suggested by the numerical work of Hörmander and Bernhardsson. The dual characterization of $Φ$ arises from a commutation relation that holds more generally for a two-parameter family of differential operators, a fact that is used to perform high precision numerical computations.

math.CA↗

arXiv2025

The Hörmander--Bernhardsson extremal function: A preliminary study

We study the function $φ_1$ of minimal $L^1$ norm among all functions $f$ of exponential type at most $π$ for which $f(0)=1$. This function, first studied by Hörmander and Bernhardsson in 1993, has only real zeros $\pm τ_n$, $n=1,2, \ldots$, and the sequence $(τ_n-n-\frac12)$ has $\ell^2$ norm bounded by $0.13$. The zeros $τ_n$ can be computed by means of a fixed point iteration.

math.FA↗

arXiv2025

A Bombieri-type inequality and equidistribution of points

In recent work, Etayo introduces a new Bombieri-type inequality for monic polynomials. Here we reinterpret this new inequality as a more general integral inequality involving the Green function for the sphere. This rather geometric interpretation allows for generalizations of the basic inequality, involving fractional zeros while also opening up the possibility to extend the setting to general compact Riemann surfaces. We derive a sharp form of these generalized Bombieri-type inequalities for the case of the sphere and the torus. These inequalities involve a quantity we call the packing number, which in turn is inspired by the geometric zero packing problems considered by Hedenmalm in the context of the asymptotic variance of the Bergman projection of a bounded function. As for the torus, we introduce analogs of polynomials (pseudopolynomials) based on the classical Weierstrass $σ$ function, and we explain how such pseudopolynomials fit in with the extended geometric Bombieri-type inequality. The sharpness of the packing number bound on the torus involves the construction of a lattice configuration on the torus for any given integer number of points. The corresponding bound for the sphere instead relies on the existence of well-conditioned polynomials in the sense of Shub and Smale.

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

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↗

arXiv2024

Linear statistics of determinantal point processes and norm representations

We study the asymptotic behavior of the fluctuations of smooth and rough linear statistics for determinantal point processes on the sphere and on the Euclidean space. The main tool is the generalization of some norm representation results for functions in Sobolev spaces and in the space of functions of bounded variation.

math.CA↗

arXiv2023

Point evaluation in Paley--Wiener spaces

We study the norm of point evaluation at the origin in the Paley--Wiener space $PW^p$ for $0 < p < \infty$, i. e., we search for the smallest positive constant $C$, called $\mathscr{C}_p$, such that the inequality $|f(0)|^p \leq C \|f\|_p^p$ holds for every $f$ in $PW^p$. We present evidence and prove several results supporting the following monotonicity conjecture: The function $p\mapsto \mathscr{C}_p/p$ is strictly decreasing on the half-line $(0,\infty)$. Our main result implies that $\mathscr{C}_p p/2$ for $1 \leq p < 2$. We also estimate the asymptotic behavior of $\mathscr{C}_p$ as $p \to \infty$ and as $p \to 0^+$. Our approach is based on expressing $\mathscr{C}_p$ as the solution of an extremal problem. Extremal functions exist for all $0<p<\infty$; they are real entire functions with only real zeros, and the extremal functions are known to be unique for $1\leq p < \infty$. Following work of Hörmander and Bernhardsson, we rely on certain orthogonality relations associated with the zeros of extremal functions, along with certain integral formulas representing respectively extremal functions and general functions at the origin. We also use precise numerical estimates for the largest eigenvalue of the Landau--Pollak--Slepian operator of time--frequency concentration. A number of qualitative and quantitative results on the distribution of the zeros of extremal functions are established. In the range $1<p<\infty$, the orthogonality relations associated with the zeros of the extremal function are linked to a de Branges space. We state a number of conjectures and further open problems pertaining to $\mathscr{C}_p$ and the extremal functions.

math.CA↗

arXiv2023

A monotonicity theorem for subharmonic functions on manifolds

We provide a sharp monotonicity theorem about the distribution of subharmonic functions on manifolds, which can be regarded as a new, measure theoretic form of the uncertainty principle. As an illustration of the scope of this result, we deduce contractivity estimates for analytic functions on the Riemann sphere, the complex plane and the Poincaré disc, with a complete description of the extremal functions, hence providing a unified and illuminating perspective of a number of results and conjectures on this subject, in particular on the Wehrl entropy conjecture by Lieb and Solovej. In this connection, we completely prove that conjecture for SU(2), by showing that the corresponding extremals are only the coherent states. Also, we show that the above (global) estimates admit a local counterpart and in all cases we characterize also the extremal subsets, among those of fixed assigned measure.

math.CA↗

arXiv2021

Idempotent Fourier multipliers acting contractively on $H^p$ spaces

We describe the idempotent Fourier multipliers that act contractively on $H^p$ spaces of the $d$-dimensional torus $\mathbb{T}^d$ for $d\geq 1$ and $1\leq p \leq \infty$. When $p$ is not an even integer, such multipliers are just restrictions of contractive idempotent multipliers on $L^p$ spaces, which in turn can be described by suitably combining results of Rudin and Andô. When $p=2(n+1)$, with $n$ a positive integer, contractivity depends in an interesting geometric way on $n$, $d$, and the dimension of the set of frequencies associated with the multiplier. Our results allow us to construct a linear operator that is densely defined on $H^p(\mathbb{T}^\infty)$ for every $1 \leq p \leq \infty$ and that extends to a bounded operator if and only if $p=2,4,\ldots,2(n+1)$.

math.FA↗

arXiv2021

Hilbert points in Hardy spaces

A Hilbert point in $H^p(\mathbb{T}^d)$, for $d\geq1$ and $1\leq p \leq \infty$, is a nontrivial function $φ$ in $H^p(\mathbb{T}^d)$ such that $\| φ\|_{H^p(\mathbb{T}^d)} \leq \|φ+ f\|_{H^p(\mathbb{T}^d)}$ whenever $f$ is in $H^p(\mathbb{T}^d)$ and orthogonal to $φ$ in the usual $L^2$ sense. When $p\neq 2$, $φ$ is a Hilbert point in $H^p(\mathbb{T})$ if and only if $φ$ is a nonzero multiple of an inner function. An inner function on $\mathbb{T}^d$ is a Hilbert point in any of the spaces $H^p(\mathbb{T}^d)$, but there are other Hilbert points as well when $d\geq 2$. We investigate the case of $1$-homogeneous polynomials in depth and obtain as a byproduct a new proof of the sharp Khintchin inequality for Steinhaus variables in the range $2 4$ but only numerically for $1\leq p<4$.

math.FA↗

arXiv2021

Marcinkiewicz-Zygmund Inequalities for Polynomials in Fock Space

We study the relation between Marcinkiewicz-Zygmund families for polynomials in a weighted $L^2$-space and sampling theorems for entire functions in the Fock space and the dual relation between uniform interpolating families for polynomials and interpolating sequences. As a consequence we obtain a description of signal subspaces spanned by Hermite functions by means of Gabor frames.

math.CV↗

arXiv2021

Interpolation by multivariate polynomials in convex domains

Let $Ω$ be a convex open set in $\mathbb R^n$ and let $Λ_k$ be a finite subset of $Ω$. We find necessary geometric conditions for $Λ_k$ to be interpolating for the space of multivariate polynomials of degree at most $k$. Our results are asymptotic in $k$. The density conditions obtained match precisely the necessary geometric conditions that sampling sets are known to satisfy, and they are expressed in terms of the equilibrium potential of the convex set. Moreover, we prove that in the particular case of the unit ball, for $k$ large enough, there is no family of orthogonal reproducing kernels in the space of polynomials of degree at most $k$.

math.CA↗

arXiv2020

A converse to the Schwarz lemma for planar harmonic maps

A sharp version of a recent inequality of Kovalev and Yang on the ratio of the $(H^1)^\ast$ and $H^4$ norms for certain polynomials is obtained. The inequality is applied to establish a sharp and tractable sufficient condition for the Wirtinger derivatives at the origin for harmonic self-maps of the unit disc which fix the origin.

math.CV↗