arXiv subjects

Jorge Antezana

Publications and source records attributed to Jorge Antezana.

At least 19 recordsLinked to original sources

arXiv2026

Variations on two Cabrelli's works

In this paper we present two different problems within the framework of shift-invariant theory. First, we develop a triangular form for shift-preserving operators acting on finitely generated shift-invariant spaces. In case of the normal operators, we recover a diagonal decomposition. The results show, in particular, that any finitely generated shift-invariant space can be decomposed into an orthogonal sum of principal shift-invariant spaces, with additional invariance properties under a shift-preserving operator. Second, we provide a new characterization of the multi-tiling sets $Ω\subset\mathbb{R}^d$ of positive measure for which $L^2(Ω)$ admits a structured Riesz basis of exponentials that is formulated in the ambient space $\mathbb{T}^{k\times k}$. In addition, we show a simpler sufficient condition which generalizes the admissibility property, that is also necessary for 2-tiling sets.

math.FA↗

arXiv2025

Characterization of multipliers on vector-valued Hardy spaces

This work characterizes the multipliers on vector-valued Hardy spaces over the infinite polydisk and the infinite polytorus, as well as in the context of Dirichlet series. Unlike the scalar-valued setting, where these frameworks are completely analogous reformulations of one another, there are significant differences in the vector-valued context. We prove that while the space of multipliers on the infinite polydisk is $H_\infty(\mathbb{D}^\infty_2, B(X))$, the situation on the infinite polytorus is distinct; assuming $X$ is separable, the multiplier space can be identified as $H_\infty^{sot}(\mathbb{T}^\infty, B(X))$, consisting of essentially bounded SOT-measurable functions. These spaces coincide when $X$ possesses the analytic Radon-Nikodym property. Finally, we extend these results to the associated Hardy spaces of Dirichlet series, $\mathcal{H}_p^+(X)$ and $\mathcal{H}_p(X)$, providing characterizations for their respective multiplier spaces.

math.FA↗

arXiv2024

Weak Type Boundedness of the Hardy Littlewood Maximal Operator on Weighted Lorentz Spaces

The main goal of this paper is to provide a complete characterization of the weak-type boundedness of the Hardy-Littlewood maximal operator, $M$, on weighted Lorentz spaces $Λ^p_u(w)$, whenever $p>1$. This solves a problem left open in \cite{crs:crs}. Moreover, with this result, we complete the program of unifying the study of the boundedness of $M$ on weighted Lebesgue spaces and classical Lorentz spaces, which was initiated in the aforementioned monograph.

math.CA↗

arXiv2024

Revisiting Yano and Zygmund extrapolation theory

We prove a pointwise estimate for the decreasing rearrangement of $Tf$, where $T$ is any sublinear operator satisfying the weak-type boundedness $$ T:L^{p,1}(μ) \to L^{p,\infty}(ν), \quad \forall p: 1<p_0 < p\leq p_1<\infty, $$ with norm controlled by $Cφ\left(\left[{p_0^{-1}} - p^{-1}\right]^{-1}\right)$ and $φ$ satisfies some admissibility conditions. The pointwise estimate is: \begin{equation*} \begin{split} (Tf)^*_ν(t) &\lesssim \frac 1{p_0 - 1}\left(\frac 1{t^\frac 1{p_0}}\int_0^t φ\left(1 - \log \frac rt\right)f^*_μ(r)\frac{dr}{r^{1 - \frac 1{p_0}}} + \frac 1{t^\frac 1{p_1}}\int_t^\infty f^*_μ(r)\frac{dr}{r^{1 - \frac 1{p_1}}}\right). \end{split} \end{equation*} In particular, this estimate allows to obtain extensions of Yano and Zygmund extrapolation results.

math.CA↗

arXiv2024

From weak type weighted inequality to pointwise estimate for the decreasing rearrangement

We shall prove pointwise estimates for the decreasing rearrangement of $Tf$, where $T$ covers a wide range of interesting operators in Harmonic Analysis such as operators satisfying a Fefferman-Stein inequality, the Bochner-Riesz operator, rough operators, sparse operators, Fourier multipliers, etc. In particular, our main estimate is of the form $$ (Tf)^*(t) \leq C\left( \frac 1t\int_0^tf^*(s)\,ds + \int_t^\infty \left( 1 + \log\frac st \right)^{- 1}φ\left(1 + \log\frac st \right) f^*(s)\,\frac{ds}s \right), $$ where $φ$ is determined by the Muckenhoupt $A_p$-weight norm behaviour of the operator.

math.CA↗

arXiv2024

Lorentz-Shimogaki and Boyd theorems for weighted Lorentz spaces

We prove the Lorentz-Shimogaki and Boyd theorems for the spaces $Λ^p_u(w)$. As a consequence, we give the complete characterization of the strong boundedness of $H$ on these spaces in terms of some geometric conditions on the weights $u$ and $w$, whenever $p>1$. For these values of $p$, we also give the complete solution of the weak-type boundedness of the Hardy-Littlewood operator on $Λ^p_u(w)$.

math.CA↗

arXiv2023

Random periodic sampling patterns for shift-invariant spaces

We consider multi-variate signals spanned by the integer shifts of a set of generating functions with distinct frequency profiles and the problem of reconstructing them from samples taken on a random periodic set. We show that such a sampling strategy succeeds with high probability provided that the density of the sampling pattern exceeds the number of frequency profiles by a logarithmic factor. The signal model includes bandlimited functions with multi-band spectra. While in this well-studied setting delicate constructions provide sampling strategies that meet the information theoretic benchmark of Shannon and Landau, the sampling pattern that we consider provides, at the price of a logarithmic oversampling factor, a simple alternative that is accompanied by favorable a priori stability margins (snug frames). More generally, we also treat bandlimited functions with arbitrary compact spectra, and different measures of its complexity and approximation rates by integer tiles. At the technical level, we elaborate on recent work on relevant sampling, with the key difference that the reconstruction guarantees that we provide hold uniformly for all signals, rather than for a subset of well-concentrated ones. This is achieved by methods of concentration of measure formulated on the Zak domain.

math.FA↗

arXiv2023

Weighted maximal inequalities on hyperbolic spaces

In this work we develop a weight theory in the setting of hyperbolic spaces. Our starting point is a variant of the well-known endpoint Fefferman-Stein inequality for the centered Hardy-Littlewood maximal function. This inequality generalizes, in the hyperbolic setting, the weak $(1,1)$ estimates obtained by Strömberg in "Weak type L1 estimates for maximal functions on noncompact symmetric spaces", Ann. of Math. 114 (1981), where Strömberg answered a question posed by Stein and Wainger in "Problems in harmonic analysis related to curvature", Bull. Amer. Math. Soc. 84 (1978). Our approach is based on a combination of geometrical arguments and the techniques used in the discrete setting of regular trees by Naor and Tao in "Random martingales and localization of maximal inequalities", J. Funct. Anal. 259 (2010). This variant of the Fefferman-Stein inequality paves the road to weighted estimates for the maximal function for $p>1$. On the one hand, we show that the classical $A_p$ conditions are not the right ones in this setting. On the other hand, we provide sharp sufficient conditions for weighted weak and strong type $(p,p)$ boundedness of the centered maximal function, when $p>1$. The sharpness is in the sense that, given $p>1$, we can construct a weight satisfying our sufficient condition for that $p$, and so it satisfies the weak type $(p,p)$ inequality, but the strong type $(p,p)$ inequality fails. In particular, the weak type $(q,q)$ fails as well for every $q < p$.

math.CA↗

arXiv2021

Splitting the Riesz basis condition for systems of dilated functions]{Splitting the Riesz basis condition for systems of dilated functions through Dirichlet series

Inspired by the work of Hedenmalm, Lindqvist and Seip, we consider different properties of dilations systems of a fixed function $φ\in L^2(0,1)$. More precisely, we study when the system $\{φ(nx)\}_n$ is a Bessel sequence, a Riesz sequence, or it satisfies the lower frame bound. We are able to characterize these properties in terms of multipliers of the Hardy space $\mathcal{H}^2$ of Dirichtet series and, also, in terms of Hardy spaces on the infinite polytorus. We also address the multivariate case.

math.FA↗

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↗

arXiv2019

Model subspaces techniques to study Fourier expansions in L^2 spaces associated to singular measures

Let $μ$ be a probability measure on $\mathbb{T}$ that is singular with respect to the Haar measure. In this paper we study Fourier expansions in $L^2(\mathbb{T},μ)$ using techniques from the theory of model subspaces of the Hardy space. Since the sequence of monomials $\{z^n\}_{n\in \mathbb{N}}$ is effective in $L^2(\mathbb{T},μ)$, it has a Parseval frame associated via the Kaczmarz algorithm. Our first main goal is to identify the aforementioned frame with boundary values of the frame $P_φ(z^n)$ for the model subspace $\mathcal{H}(φ)= H^2 \ominus φH^2$, where $P_φ$ is the orthogonal projection from the Hardy space $H^2$ onto $\mathcal{H}(φ)$. The study of Fourier expansions in $L^2(\mathbb{T},μ)$ also leads to consider positive kernels in the Hardy space. Our second main goal is to study the set of measures $μ$ which reproduce a kernel contained in a model subspace. We completely characterize this set when the kernel is the reproducing kernel of a model subspace, and we study the consequences of this characterization.

math.CA↗

arXiv2019

Minimal curves in U(n) and Gl(n)+ with respect to the spectral and the trace norms

Consider the Lie group of n x n complex unitary matrices U(n) endowed with the bi-invariant Finsler metric given by the spectral norm, ||X||_U = ||U*X||_{sp} = ||X||_{sp} for any X tangent to a unitary operator U. Given two points in U(n), in general there exists infinitely many curves of minimal length. The aim of this paper is to provide a complete description of such curves. As a consequence of this description, we conclude that there is a unique curve of minimal length between U and V if and only if the spectrum of U*V is contained in a set of the form \{e^{i θ}, e^{-i θ}\} for some θ\in [0, \infty). Similar studies are done for the Grassmann manifolds. Now consider the cone of n x n positive invertible matrices Gl(n)+ endowed with the bi-invariant Finsler metric given by the trace norm, ||X||_{1, A} = ||A^{-1/2}XA^{-1/2}||_1 for any X tangent to A \in Gl(n)+. In this context, given two points A,B \in Gl(n)+ there exists infinitely many curves of minimal length. In order to provide a complete description of such curves, we provide a characterization of the minimal curves joining two Hermitian matrices X, Y \in H(n). As a consequence of the last description, we provide a way to construct minimal paths in the group of unitary matrices U(n) endowed with the bi-invariant Finsler metric ||X||_{1, U} = ||U*X||_{1} = ||X||_{1} for any X tangent to U \in U(n). We also study the set of intermediate points in all the previous contexts. Between two given unitary matrices U and V we prove that this set is geodesically convex provided ||U - V||_{sp} < 1. In Gl(n)+ this set is geodesically convex for every unitarily invariant norm.

math.FA↗

arXiv2018

Ergodic theorem in Hadamard spaces in terms of inductive means

Let (G,+) be a compact, abelian, and metrizable topological group. In this group we take $g\in G$ such that the corresponding automorphism t_g is ergodic. The main result of this paper is a new ergodic theorem for functions in L^1(G,M), where M is a Hadamard space. The novelty of our result is that we use inductive means to average the elements of the orbit {t_g^n(h)}_{n\in N}. The advantage of inductive means is that they can be explicitly computed in many important examples. The proof of the ergodic theorem is done firstly for continuous functions, and then it is extended to L^1 functions. The extension is based in a new construction of mollifiers in Hadamard spaces. This construction has the advantage that it only uses the metric structure and the existence of barycenters, and do not require the existence of an underlying vector space. For this reason, it can be used in any Hadamard space, in contrast with those results that need to use the tangent space or some chart to define the mollifier.

math.MG↗

arXiv2017

Tiling functions and Gabor orthonormal basis

We study the existence of Gabor orthonormal bases with window the characteristic function of the set W=[0,a] U [b+a, b+1] of measure 1, with a, b>0. By the symmetries of the problem, we can restrict our attention to the case a<=1/2. We prove that either if a<1/2 or (a=1/2 and b>= 1/2) there exist such Gabor orthonormal bases, with window the characteristic function of the set W, if and only if W tiles the line. Furthermore, in both cases, we completely describe the structure of the set of time-frequency shifts associated to these bases

math.CA↗

arXiv2016

Approximation by partial isometries and symmetric approximation of finite frames

We solve the problem of best approximation by partial isometries of given rank to an arbitrary rectangular matrix, when the distance is measured in any unitarily invariant norm. In the case where the norm is strictly convex, we parametrize all the solutions. In particular, this allow us to give a simple necessary and sufficient condition for uniqueness. We then apply these results to solve the global problem of approximation by partial isometries, and to extend the notion of symmetric approximation of frames introduced in M. Frank, V. Paulsen, T. Tiballi, Symmetric Approximation of frames and bases in Hilbert Spaces, Trans. Amer. Math. Soc. 354 (2002), 777-793. In addition, we characterize symmetric approximations of frames belonging to a prescribed subspace.

math.FA↗

arXiv2015

Multi-tiling sets, Riesz bases, and sampling near the critical density in LCA groups

We prove the existence of sampling sets and interpolation sets near the critical density, in Paley Wiener spaces of a locally compact abelian (LCA) group G . This solves a problem left by Gröchenig, Kutyniok, and Seip in the article: `Landau's density conditions for LCA groups ' (J. of Funct. Anal. 255 (2008) 1831-1850). To achieve this result, we prove the existence of universal Riesz bases of characters for L2(Omega), provided that the relatively compact subset Omega of the dual group of G satisfies a multi-tiling condition. This last result generalizes Fuglede's theorem, and extends to LCA groups setting recent constructions of Riesz bases of exponentials in bounded sets of Rd.

math.CA↗

arXiv2015

Zeros of random functions generated with de Branges kernels

We study the point process given by the set of real zeros of random sums of orthonormal bases of reproducing kernels of de Branges spaces. Examples of these kernels are the cardinal sine, Airy and Bessel kernels. We find an explicit formula for the first intensity function in terms of the phase of the Hermite-Biehler function. We prove that the first intensity of the point process completely characterizes the underlying de Branges space. This result is a real version of the so called Calabi rigidity for GAFs proved by M. Sodin.

math.CA↗

arXiv2011

Gap probabilities for the cardinal sine

We study the zero set of random analytic functions generated by a sum of the cardinal sine functions that form an orthogonal basis for the Paley-Wiener space. As a model case, we consider real-valued Gaussian coefficients. It is shown that the asymptotic probability that there is no zero in a bounded interval decays exponentially as a function of the length.

math.CV↗