arXiv subjects

Andrei Martínez-Finkelshtein

Publications and source records attributed to Andrei Martínez-Finkelshtein.

8 recordsLinked to original sources

arXiv2026

Compactly supported radial basis functions as probability density functions

Compactly Supported Radial Basis Functions (CS-RBFs) are a fundamental tool in multivariate approximation theory. However, their use in statistics and probability modeling remains underexplored, having been used mainly to express covariance functions in Gaussian processes or as kernel functions. This work explores CS-RBFs as a novel parametric family of probability density functions, focusing in particular on Wendland $\mathscr{C}^2$ kernels. The primary contribution of this work is the derivation of analytical expressions for various statistical properties, such as moments and the cumulative distribution function, of CS-RBFs as univariate and conditional densities. The approach comprises two alternative scenarios: when the CS-RBF support lies entirely within the variable's domain (untruncated support) and when part of it is outside (truncated support). Mixture models employing CS-RBFs are also analyzed, and their main properties are detailed. Furthermore, we introduce an incremental learning algorithm for density estimation with CS-RBF mixture models, in which centers are determined using k-means and weights and shape parameters are optimized by stochastic gradient descent. Experiments on synthetic and real-world datasets show that CS-RBF densities provide competitive results in terms of likelihood and model complexity in comparison with Gaussian mixture models. In addition, these CS-RBF densities allow the exact computation of key distributional properties in univariate and conditional settings.

math.ST↗

arXiv2025

Sampling patterns for Zernike-like bases in non-standard geometries

Zernike polynomials are widely used in optics and ophthalmology due to their direct connection to classical optical aberrations. While orthogonal on the unit disk, their application to discrete data or non-circular domains--such as ellipses, annuli, and hexagons--presents challenges in terms of numerical stability and accuracy. In this work, we extend Zernike-like orthogonal functions to these non-standard geometries using diffeomorphic mappings and construct sampling patterns that preserve favorable numerical conditioning. We provide theoretical bounds for the condition numbers of the resulting collocation matrices and validate them through extensive numerical experiments. As a practical application, we demonstrate accurate wavefront interpolation and reconstruction in segmented mirror telescopes composed of hexagonal facets. Our results show that appropriately transferred sampling configurations, especially Optimal Concentric Sampling and Lebesgue points, allow stable high-order interpolation and effective wavefront modeling in complex optical systems. Moreover, the Optimal Concentric Samplings can be computed with an explicit expression, which is a significant advantage in practice.

math.NA↗

arXiv2022

Electrostatic partners and zeros of orthogonal and multiple orthogonal polynomials

For a given polynomial $P$ with simple zeros, and a given semiclassical weight $w$, we present a construction that yields a linear second-order differential equation (ODE), and in consequence, an electrostatic model for zeros of $P$. The coefficients of this ODE are written in terms of a dual polynomial that we call the electrostatic partner of $P$. This construction is absolutely general and can be carried out for any polynomial with simple zeros and any semiclassical weight on the complex plane. An additional assumption of quasi-orthogonality of $P$ with respect to $w$ allows us to give more precise bounds on the degree of the electrostatic partner. In the case of orthogonal and quasi-orthogonal polynomials, we recover some of the known results and generalize others. Additionally, for the Hermite--Padé or multiple orthogonal polynomials of type II, this approach yields a system of linear second-order differential equations, from which we derive an electrostatic interpretation of their zeros in terms of a vector equilibrium. More detailed results are obtained in the special cases of Angelesco, Nikishin, and generalized Nikishin systems. We also discuss the discrete-to-continuous transition of these models in the asymptotic regime, as the number of zeros tends to infinity, into the known vector equilibrium problems. Finally, we discuss how the system of obtained second-order ODEs yields a third-order differential equation for these polynomials, well described in the literature. We finish the paper by presenting several illustrative examples.

math.CA↗

arXiv2020

Spectral curves, variational problems and the hermitian matrix model with external source

We consider the hermitian random matrix model with external source and general polynomial potential, when the source has two distinct eigenvalues but is otherwise arbitrary. All such models studied so far have a common feature: an associated cubic equation (spectral curve), one of whose solutions can be expressed in terms of the Cauchy transform of the limiting eigenvalue distribution. This is our starting point: we show that to any such spectral curve (not necessarily given by a random matrix ensemble) it corresponds a unique vector-valued measure with three components on the complex plane, characterized as a solution of a variational problem stated in terms of their logarithmic energy. We describe all possible geometries of the supports of these measures; in particular the third component, if non-trivial, lives on a contour on the plane. This general result is applied to the random matrix model with external source, under an additional assumption of uniform boundedness of the zeros of a sequence of average characteristic polynomials in the large matrix limit. It is shown that any limiting zero distribution for such a sequence can be written in terms of a solution of a spectral curve, and thus admits the variational description obtained in the first part of the paper. As a consequence of our analysis, we obtain that the density of this limiting measure can have only a handful of local behaviors: sine, Airy and their higher order type behavior, Pearcey or yet the fifth power of the cubic (but no higher order cubics can appear). We also compare our findings with the most general results available in the literature, showing that once an additional symmetry is imposed, our vector critical measure contains enough information to recover the solutions to the constrained equilibrium problem that was known to describe the limiting eigenvalue distribution in this symmetric situation.

math-ph↗

arXiv2018

Critical measures for vector energy: asymptotics of non-diagonal multiple orthogonal polynomials for a cubic weight

We consider the type I multiple orthogonal polynomials (MOPs) $(A_{n,m}, B_{n,m})$ and type II MOPs $P_{n,m}$, satisfying non-hermitian orthogonality with respect to the weight $e^{-z^3}$ on two unbounded contours on $\mathbb C$. Under the assumption that $$ n,m \to \infty, \quad \frac{n}{n+m}\to α\in (0, 1) $$ we find the detailed asymptotics of the MOPs, and describe the phase transitions of this limit behavior as a function of $α$. This description is given in terms of vector critical measures, which are saddle points of an energy functional comprising both attracting and repelling forces. These critical measures are characterized by a cubic equation (spectral curve), and their components $μ_j$ live on trajectories of a canonical quadratic differential $\varpi$ on the Riemann surface of this equation, which was object of study in our previous paper [Adv. Math. 302 (2016), 1137--1232]. The asymptotic zero distribution of the polynomials $A_{n,m}$ and $P_{n,m}$ are given by appropriate combinations of the components of the vector critical measure. However, in the case of the zeros of $B_{n,m}$ the behavior is totally different, and can be described in terms of the balayage of $μ_2 - μ_3$ onto certain curves on the plane. These curves are constructed with the aid of $\varpi$, and their topology has three very distinct characters, depending on the value of $α$, and are obtained from the critical graph of $\varpi$. Once the trajectories and vector critical measures are studied, the main asymptotic technical tool is the analysis of a $3\times 3$ Riemann-Hilbert problem characterizing the MOPs. We illustrate our findings with results of several numerical experiments, and formulate some conjectures and empirical observations based on these experiments.

math.CV↗

arXiv2017

What is $\ldots\ $ a multiple orthogonal polynomial?

This is an extended version of our note in the Notices of the American Mathematical Society 63 (2016), no. 9, in which we explain what multiple orthogonal polynomials are and where they appear in various applications.

math.CA↗

arXiv2017

Christoffel formula for kernel polynomials on the unit circle

Given a nontrivial positive measure $μ$ on the unit circle, the associated Christoffel-Darboux kernels are $K_n(z, w;μ) = \sum_{k=0}^{n}\overline{φ_{k}(w;μ)}\,φ_{k}(z;μ)$, $n \geq 0$, where $φ_{k}(\cdot; μ)$ are the orthonormal polynomials with respect to the measure $μ$. Let the positive measure $ν$ on the unit circle be given by $d ν(z) = |G_{2m}(z)|\, d μ(z)$, where $G_{2m}$ is a conjugate reciprocal polynomial of exact degree $2m$. We establish a determinantal formula expressing $\{K_n(z,w;ν)\}_{n \geq 0}$ directly in terms of $\{K_n(z,w;μ)\}_{n \geq 0}$. Furthermore, we consider the special case of $w=1$; it is known that appropriately normalized polynomials $K_n(z,1;μ) $ satisfy a recurrence relation whose coefficients are given in terms of two sets of real parameters $ \{c_n(μ)\}_{n=1}^{\infty}$ and $ \{g_{n}(μ)\}_{n=1}^{\infty}$, with $0<g_n<1 $ for $n\geq 1$. The double sequence $\{(c_n(μ), g_{n}(μ))\}_{n=1}^{\infty}$ characterizes the measure $μ$. A natural question about the relation between the parameters $c_n(μ)$, $g_n(μ)$, associated with $μ$, and the sequences $c_n(ν)$, $g_n(ν)$, corresponding to $ν$, is also addressed. Finally, examples are considered, such as the Geronimus weight (a measure supported on an arc of the unit circle), a class of measures given by basic hypergeometric functions, and a class of measures with hypergeometric orthogonal polynomials.

math.CA↗

arXiv2015

Asymptotics of type I Hermite-Padé polynomials for semiclassical functions

Type I Hermite--Padé polynomials for a set of functions $f_0, f_1, ..., f_s$ at infinity, $Q_{n,0}$, $Q_{n,1}$, ..., $Q_{n,s}$, is defined by the asymptotic condition $$ R_n(z):=\bigl(Q_{n,0}f_0+Q_{n,1}f_1+Q_{n,2}f_2+...+Q_{n,s}f_s\bigr)(z) =\mathcal O (\frac1{z^{s n+s}}), \quad z\to\infty, $$ with the degree of all $Q_{n,k}\leq n$. We describe an approach for finding the asymptotic zero distribution of these polynomials as $n\to \infty$ under the assumption that all $f_j$'s are semiclassical, i.e. their logarithmic derivatives are rational functions. In this situation $R_n$ and $Q_{n,k}f_k$ satisfy the same differential equation with polynomials coefficients. We discuss in more detail the case when $f_k$'s are powers of the same function $f$ ($f_k=f^k$); for illustration, the simplest non trivial situation of $s=2$ and $f$ having two branch points is analyzed in depth. Under these conditions, the ratio or comparative asymptotics of these polynomials is also discussed. From methodological considerations and in order to make the situation clearer, we start our exposition with the better known case of Padé approximants (when $s=1$).

math.CA↗