# Andrei Martínez-FinkelshteinBaylor University | BU · Department of Mathematics

Andrei Martínez-Finkelshtein

Ph. D.

## About

140

Publications

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2,248

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Introduction

Additional affiliations

February 2018 - present

February 2018 - present

January 2006 - present

Education

February 1987 - February 1991

September 1981 - May 1986

## Publications

Publications (140)

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 $P$...

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...

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...

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...

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...

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...

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...

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...

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...

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...

We consider the type I multiple orthogonal polynomials (MOPs) (A n,m ,B n,m ), degA n,m ≤n−1, degB n,m ≤m−1, and type II MOPs P n,m , degP 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...

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...

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...

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...

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 (...

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...

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...

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...

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...

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...

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...

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...

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.

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...

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...

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...

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...

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...

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...

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....

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...

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...

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...

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...

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...

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...

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...

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...

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 ...

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...

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...

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...

This is a survey of results constituting the foundations of the modern convergence theory of Padé approximants.Bibliography: 204 titles.

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...

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...

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...

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...

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...

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.

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...

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...

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...

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...

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...

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...