Andrei Martínez-Finkelshtein

Andrei Martínez-Finkelshtein
Baylor University | BU · Department of Mathematics

Ph. D.

About

140
Publications
27,436
Reads
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2,248
Citations
Additional affiliations
February 2018 - present
Baylor University
Position
  • Professor
February 2018 - present
Baylor University
Position
  • Professor
January 2006 - present
Universidad de Almería
Education
February 1987 - February 1991
September 1981 - May 1986
University of Havana
Field of study

Publications

Publications (140)
Preprint
Full-text available
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 $P$...
Article
We study algebraic curves that are envelopes of families of polygons supported on the unit circle T. We address, in particular, a characterization of such curves of minimal class and show that all realizations of these curves are essentially equivalent and can be described in terms of orthogonal polynomials on the unit circle (OPUC), also known as...
Chapter
Given a natural number n ≥ 3 and two points a and b in the unit disk \(\mathbb {D}\) in the complex plane, it is known that there exists a unique elliptical disk having a and b as foci that can also be realized as the intersection of a collection of convex cyclic n-gons whose vertices fill the whole unit circle \(\mathbb {T}\). What is less clear i...
Article
Full-text available
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 (a.k.a. St...
Preprint
Full-text available
Given a natural number $n\geq3$ and two points $a$ and $b$ in the unit disk $\mathbb D$ in the complex plane, it is known that there exists a unique elliptical disk having $a$ and $b$ as foci that can also be realized as the intersection of a collection of convex cyclic $n$-gons whose vertices fill the whole unit circle $\mathbb T$. What is less cl...
Preprint
Full-text available
We study algebraic curves that are envelopes of families of polygons supported on the unit circle T. We address, in particular, a characterization of such curves of minimal class and show that all realizations of these curves are essentially equivalent and can be described in terms of orthogonal polynomials on the unit circle (OPUC), also known as...
Book
The main topics of this volume, dedicated to Lance Littlejohn, are operator and spectral theory, orthogonal polynomials, combinatorics, number theory, and the various interplays of these subjects. Although the event, originally scheduled as the Baylor Analysis Fest, had to be postponed due to the pandemic, scholars from around the globe have contri...
Article
Full-text available
In a recent paper (Martínez-Finkelshtein et al. in Proc Am Math Soc 147:2625–2640, 2019) some interesting results were obtained concerning complementary Romanovski–Routh polynomials, a class of orthogonal polynomials on the unit circle and extended regular Coulomb wave functions. The class of orthogonal polynomials here are generalization of the cl...
Article
Purpose: To evaluate in a sample of normal and keratoconic eyes a simple Bayesian network classifier for keratoconus identification that uses previously developed topographic indices, calculated directly from the digital analysis of the Placido ring images. Methods: A comparative study was performed on a total of 60 eyes from 60 patients (age 20...
Preprint
Full-text available
We consider a hermitian 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 is expressed in terms of the Cauchy transform of the asymptot...
Article
Full-text available
We consider the type I multiple orthogonal polynomials (MOPs) (A n,m ,B n,m ), deg⁡A n,m ≤n−1, deg⁡B n,m ≤m−1, and type II MOPs P n,m , deg⁡P n,m =n+m, satisfying non-hermitian orthogonality with respect to the weight e −z 3 on two unbounded contours γ 1 and γ 2 on C, with (in the case of type II MOPs) n conditions on γ 1 and m on γ 2 . Under the a...
Article
There has been considerable recent literature connecting Poncelet's theorem to ellipses, Blaschke products and numerical ranges, summarized, for example, in the recent book [16]. We show how those results can be understood using ideas from the theory of orthogonal polynomials on the unit circle (OPUC) and, in turn, can provide new insights to the t...
Preprint
There has been considerable recent literature connecting Poncelet's theorem to ellipses, Blaschke products and numerical ranges, summarized, for example, in the recent book [11]. We show how those results can be understood using ideas from the theory of orthogonal polynomials on the unit circle (OPUC) and, in turn, can provide new insights to the t...
Article
Full-text available
Close-form expression for the Strehl ratio calculated in the spatial frequency domain of the optical transfer function (SOTF) is considered for the case of an optical system that has circular symmetry. First, it is proved that the SOTF for the aberration-free diffraction limited optical system is equal to one. Further, a semi-analytic solution for...
Article
Full-text available
Close-form expression for the Strehl ratio calculated in the spatial frequency domain of the optical transfer function (SOTF) is considered for the case of time-varying dynamic optical system that has circular symmetry. Specifically, closed-form expressions for the temporally averaged SOTF are considered, which can be easily evaluated numerically (...
Preprint
Full-text available
We consider properties and applications of a sequence of polynomials known as complementary Romanovski-Routh polynomials (CRR polynomials for short). These polynomials, which follows from the Romanovski-Routh polynomials or complexified Jacobi polynomials, are known to be useful objects in the studies of the one-dimensional Schr\"{o}dinger equation...
Preprint
Full-text available
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 $$ n,m \to \infty, \quad \frac{n}{n+m}\to \alpha \in (0, 1) $$ we find the detailed asymptotic...
Preprint
Full-text available
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 $$ n,m \to \infty, \quad \frac{n}{n+m}\to \alpha \in (0, 1) $$ we find the detailed asymptotic...
Article
Given a nontrivial Borel measure μ on the unit circle T{double-struck}, the corresponding reproducing (or Christoffel-Darboux) kernels with one of the variables fixed at z = 1 constitute a family of so-called para-orthogonal polynomials, whose zeros belong to T. With a proper normalization they satisfy a three-term recurrence relation determined by...
Article
Full-text available
Purpose To develop an objective refraction formula based on the ocular wavefront error (WFE) expressed in terms of Zernike coefficients and pupil radius, which would be an accurate predictor of subjective spherical equivalent (SE) for different pupil sizes. Methods A sphere is fitted to the ocular wavefront at the center and at a variable distance...
Conference Paper
Full-text available
Purpose : To obtain fast and accurate measurements of the equivalent sphere (M) and amplitude of accommodation (AA; subjective SAA, objective OAA), taking into account the potential changes in refraction due to the changes of pupil size. Methods : An empirical wavefront refraction metric (MTR) was formulated using objective (OR) and subjective ref...
Article
Full-text available
Given a nontrivial positive measure $\mu$ on the unit circle, the associated Christoffel-Darboux kernels are $K_n(z, w;\mu) = \sum_{k=0}^{n}\overline{\varphi_{k}(w;\mu)}\,\varphi_{k}(z;\mu)$, $n \geq 0$, where $\varphi_{k}(\cdot; \mu)$ are the orthonormal polynomials with respect to the measure $\mu$. Let the positive measure $\nu$ on the unit circ...
Article
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.
Article
Full-text available
Motivated by the study of the asymptotic behavior of Jacobi polynomials $\left( P_{n}^{(nA,nB)}\right) _{n}$ with $A\in \mathbb C$ and $B>0$ we establish the global structure of trajectories of the related rational quadratic differential on $\mathbb C$. As a consequence, the asymptotic zero distribution (limit of the root-counting measures of $\lef...
Article
The radial expectation values of the probability density of a quantum system in position and momentum spaces allow one to describe numerous physical quantities of the system as well as to find generalized Heisenberg-like uncertainty relations and to bound entropic uncertainty measures. It is known that the position and momentum expectation values o...
Article
Full-text available
A pattern of interpolation nodes on the disk is studied, for which the interpolation problem is theoretically unisolvent, and which renders a minimal numerical condition for the collocation matrix when the standard basis of Zernike polynomials is used. It is shown that these nodes have an excellent performance also from several alternative points o...
Research
Full-text available
Sometimes the analysis of the stress state, crack initiation and some others phenomenon in heterogeneous media is complicated. In order to facilitate the ulterior study of such problems, we apply the asymptotic homogenization method (AHM) to a general laminated shell composite and an equivalent homogeneous elastic problem with effective properties...
Chapter
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) := Q n,0 f 0 +Q n,1 f 1 +Q n,2 f 2 +· · ·+Q n,s f s (z) = O 1 z sn+s , z → ∞, with the degree of all Q n,k ≤ n. We describe an approach for finding the asymptotic zero distribution of...
Article
Full-text available
The complex or non-hermitian orthogonal polynomials with analytic weights are ubiquitous in several areas such as approximation theory, random matrix models, theoretical physics and in numerical analysis, to mention a few. Due to the freedom in the choice of the integration contour for such polynomials, the location of their zeros is a priori not c...
Article
Full-text available
Saddle points of a vector logarithmic energy with a vector polynomial external field on the plane constitute the vector critical measures, a notion that finds a natural motivation in several branches of analysis. We study in depth the case of measures µ = (µ 1 , µ 2 , µ 3) when the mutual interaction comprises both attracting and repelling forces....
Article
Full-text available
We implement an efficient method of computation of two dimensional Fourier-type integrals based on approximation of the integrand by Gaussian radial basis functions, which constitute a standard tool in approximation theory. As a result, we obtain a rapidly converging series expansion for the integrals, allowing for their accurate calculation. We ap...
Article
Full-text available
Type I Hermite--Pad\'e 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 a...
Article
Given a sequence of orthonormal polynomials on $\Bbb R$,$\{p_n\}_{n\geq 0}$, with $p_n$ of degree $n$, we define the discrete probability distribution $\Psi_n(x) = \left(\Psi_{n,1}(x) , \dots \Psi_{n,n}(x) \right) $, with $\Psi_{n,j}(x) = \big(\sum_{j=0}^{n-1} p_j^2(x)\big)^{-1} p_{j-1}^2(x)$, $j=1, \dots, n$. In this paper, we study the asymptotic...
Article
Full-text available
We study the asymptotic properties of a class of multiple orthogonal polynomials with respect to a Nikishin system generated by two measures $(\sigma_1, \sigma_2)$ with unbounded supports (${supp}(\sigma_1) \subset \mathbb{R}_+$, ${supp}(\sigma_2) \subset \mathbb{R}_-$), and such that the second measure $\sigma_2$ is discrete. The weak asymptotics...
Article
Full-text available
Calculating through-focus characteristics of the human eye from a single objective measurement of wavefront aberration can be accomplished through a range of methods that are inherently computationally cumbersome. A simple yet accurate and computationally efficient method is developed, which combines the philosophy of the extended Nijboer–Zernike a...
Article
Full-text available
We consider multiple orthogonal polynomials with respect to Nikishin systems generated by two measures $(\sigma_1, \sigma_2)$ with unbounded supports ($\mbox{supp} \, \sigma_1 \subseteq \mathbb{R}_+$, $\mbox{supp} \, \sigma_2 \subseteq \,\mathbb{R}_-$) and $\sigma_2$ is discrete. A Nikishin type equilibrium problem in the presence of an external fi...
Article
Full-text available
The variation of equilibrium energy is analyzed for three different functionals that naturally arise in solving a number of problems in the theory of constructive rational approximation of multivalued analytic functions. The variational approach is based on the relationship between the variation of the equilibrium energy and the equilibrium measure...
Article
In this paper we study the asymptotics (as $n\to \infty$) of the sequences of Laguerre polynomials with varying complex parameters $\alpha$ depending on the degree $n$. More precisely, we assume that $\alpha_n = n A_n, $ and $ \lim_n A_n=A \in \mathbb{C}$. This study has been carried out previously only for $\alpha_n\in \mathbb{R}$, but complex val...
Article
We investigate multiple Charlier polynomials and in particular we will use the (nearest neighbor) recurrence relation to find the asymptotic behavior of the ratio of two multiple Charlier polynomials. This result is then used to obtain the asymptotic ...
Article
Full-text available
The paper is devoted to a study of phase transitions in the Hermitian random matrix models with a polynomial potential. In an alternative equivalent language, we study families of equilibrium measures on the real line in a polynomial external field. The total mass of the measure is considered as the main parameter, which may be interpreted also eit...
Article
Purpose: To assess in a sample of normal, keratoconic, and keratoconus (KC) suspect eyes the performance of a set of new topographic indices computed directly from the digitized images of the Placido rings. Methods: This comparative study was composed of a total of 124 eyes of 106 patients from the ophthalmic clinics Vissum Alicante and Vissum A...
Article
Full-text available
This paper studies a variation of the equilibrium energy for a certain fairly general functional which appears naturally in the solution of many rational approximation problems of multi-valued analytic functions. The main result of this work states that for the energy functional under consideration and a certain class of admissible compact sets, re...
Article
Full-text available
This is a survey of results constituting the foundations of the modern convergence theory of Padé approximants.Bibliography: 204 titles.
Article
Full-text available
In 1986 J. Nuttall published in Constructive Approximation the paper "Asymptotics of generalized Jacobi polynomials", where with his usual insight he studied the behavior of the denominators ("generalized Jacobi polynomials") and the remainders of the Pade approximants to a special class of algebraic functions with 3 branch points. 25 years later w...
Article
To construct a set of indices that measure the irregularity of the anterior corneal surface, computed directly from the image of the Placido disks reflected on the cornea. Besides the high sensitivity and specificity, this approach allows bypassing the surface or curvature reconstruction step that is currently performed by the software of any comme...
Article
Full-text available
We give a Riemann-Hilbert approach to the theory of matrix orthogonal polynomials. We will focus on the algebraic aspects of the problem, obtaining difference and differential relations satisfied by the corresponding orthogonal polynomials. We will show that in the matrix case there is some extra freedom that allows us to obtain a family of ladder...
Article
Full-text available
We consider the orthogonal polynomials on [−1,1] with respect to the weight $w_c(x)=h(x)(1-x)^{\alpha}(1+x)^{\beta} \varXi _{c}(x),\quad\alpha,\beta>-1,$w_c(x)=h(x)(1-x)^{\alpha}(1+x)^{\beta} \varXi _{c}(x),\quad\alpha,\beta>-1, where h is real analytic and strictly positive on [−1,1] and Ξ c is a step-like function: Ξ c (x)=1 for x∈[−1,0) and...
Article
Full-text available
To introduce an iterative, multiscale procedure that allows for better reconstruction of the shape of the anterior surface of the cornea from altimetric data collected by a corneal topographer. The report describes, first, an adaptive, multiscale mathematical algorithm for the parsimonious fit of the corneal surface data that adapts the number of f...
Article
We show that for many families of OPUC, one has ‖φn′‖2/n→1, a condition we call normal behavior. We prove that this implies |αn|→0 and that it holds if ∑n=0∞|αn|∞. We also prove it is true for many sparse sequences. On the other hand, it is often destroyed by the insertion of a mass point.
Article
Full-text available
We consider the double scaling limit for a model of n non-intersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = 1 at x = 0. After appropriate rescaling, the paths fill a region in the tx–plane as n → ∞ that intersects the h...
Article
Full-text available
In this paper we describe an adaptive and multi-scale algorithm for the parsimonious fit of the corneal surface data that allows to adapt the number of functions used in the reconstruction to the conditions of each cornea. The method implements also a dynamical selection of the parameters and the management of noise. It can be used for the real-tim...
Article
We show that for many families of OPUC, one has $||\varphi'_n||_2/n -> 1$, a condition we call normal behavior. We prove that this implies $|\alpha_n| -> 0$ and that it holds if the sequence $\alpha_n$ is in $\ell^1$. We also prove it is true for many sparse sequences. On the other hand, it is often destroyed by the insertion of a mass point. Comme...
Chapter
The asymptotics of entropic integrals of Laguerre and Gegenbauer polynomials is used to calculate the Shannon information entropy of Rydberg atoms (i.e. giant atoms of hydrogenic type), which provides a bulky spreading measure of their charge density much more appropriate than the standard deviation or Heisenberg measure. These systems are stepping...
Article
Full-text available
The radial position (< r(alpha)>, alpha is an element of R) and momentum (< p(beta)>, beta is an element of (-1, 3)) expectation values of the D-dimensional Rydberg hydrogenic states (i.e. states where the electron has a large hyperquantum number n) are rigorously determined by means of powerful tools of the modern approximation theory relative to...
Article
This is a brief account on some results and methods of the asymptotic theory dealing with the entropy of orthogonal polynomials for large degree. This study is motivated primarily by quantum mechanics, where the wave functions and the densities of the states of solvable quantum-mechanical systems are expressed by means of orthogonal polynomials. Mo...