Andrei Martínez-Finkelshtein

Publications

• Heine, Hilbert, Pade, Riemann, and Stieltjes: a John Nuttall's work 25 years later

  Andrei Martinez-Finkelshtein, Evgenii A. Rakhmanov, Sergey P. Suetin

  11/2011;

  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 speci... [more] 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 we try to look at this problem from a modern perspective. On one hand, the generalized Jacobi polynomials constitute an instance of the so-called Heine-Stieltjes polynomials, i.e. they are solutions of linear ODE with polynomial coefficients. On the other, they satisfy complex orthogonality relations, and thus are suitable for the Riemann-Hilbert asymptotic analysis. Along with the names mentioned in the title, this paper features also a special appearance by Riemann surfaces, quadratic differentials, compact sets of minimal capacity, special functions and other characters.
• 1.53

  Impact points

  Placido-based indices of corneal irregularity.

  Darío Ramos-López, Andrei Martínez-Finkelshtein, Gracia M Castro-Luna, David Piñero, Jorge L Alió

  Optometry and vision science : official publication of the American Academy of Optometry. 07/2011; 88(10):1220-31.

  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 i... [more] 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 commercial Placido topographer. Several basic indices are proposed to detect irregularities on the anterior surface of the cornea, via analyzing some geometric and mathematical properties of the mires. These individual primary indices are built directly from the displacement of the digitized images of the rings reflected on the cornea. In addition, compound metrics are proposed (such as the generalized linear model or the classification trees) by combining some of the primary indices to improve their efficiency. The computed metrics were developed and tested for the CSO topography system (CSO, Firenze, Italy), but the methodology proposed here extends easily to any other commercial Placido disks topographer. The primary indices allow discriminating, with excellent accuracy, between normal eyes and eyes with keratoconic corneas. Sensitivity and specificity of the primary indices is analyzed by using the receiver operating characteristic (ROC) curve methodology. Some combined indices are presented, which raise the efficiency to optimal. All the primary indices proposed in this work exhibit very good performance in discriminating between normal and irregular corneas. The accuracy of the combined indices is optimal within the test group (perfect classification), allowing their use in clinical practice as corneal markers of a disease. All these indices are fast to compute and can be easily implemented in any corneal topography system.

• Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization

  F. Alberto Grünbaum, Manuel D. de la Iglesia, Andrei Martinez-Finkelshtein

  06/2011;

  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 fr... [more] 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 operators, some of them of 0-th order, something that is not possible in the scalar case. The combination of the ladder operators will lead to a family of second-order differential equations satisfied by the orthogonal polynomials, some of them of 0-th and first order, something also impossible in the scalar setting. This shows that the differential properties in the matrix case are much more complicated than in the scalar situation. We will study several examples given in the last years as well as others not considered so far.
• 3.43

  Impact points

  Adaptive cornea modeling from keratometric data.

  Andrei Martínez-Finkelshtein, Darío Ramos López, Gracia M Castro, Jorge L Alió

  Investigative ophthalmology & visual science. 03/2011; 52(8):4963-70.

  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 th... [more] 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 functions used in the reconstruction to the conditions of each cornea. The method also implements a dynamic selection of the parameters and the management of noise. Then, several numerical experiments are performed, comparing it with the results obtained by the standard Zernike-based procedure. The numerical experiments showed that the algorithm exhibits steady exponential error decay, independent of the level of aberration of the cornea. The complexity of each anisotropic Gaussian-basis function in the functional representation is the same, but the parameters vary to fit the current scale. This scale is determined only by the residual errors and not by the number of the iteration. Finally, the position and clustering of the centers, as well as the size of the shape parameters, provides additional spatial information about the regions of higher irregularity. The methodology can be used for the real-time reconstruction of both altimetric data and corneal power maps from the data collected by keratoscopes, such as the Placido ring-based topographers, that will be decisive in early detection of corneal diseases such as keratoconus.

• Editorial.

  Dolores Barrios Rolanía, Francisco Marcellán, Andrei Martínez-Finkelshtein

  Journal of Approximation Theory. 01/2011; 163:1-2.

• Non-intersecting squared Bessel paths: critical time and double scaling limit

  A. B. J. Kuijlaars, A. Martinez-Finkelshtein, F. Wielonsky

  11/2010;

  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 r... [more] 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 $n\to \infty$ that intersects the hard edge at $x=0$ at a critical time $t=t^{*}$. In a previous paper (arXiv:0712.1333), the scaling limits for the positions of the paths at time $t\neq t^{*}$ were shown to be the usual scaling limits from random matrix theory. Here, we describe the limit as $n\to \infty$ of the correlation kernel at critical time $t^{*}$ and in the double scaling regime. We derive an integral representation for the limit kernel which bears some connections with the Pearcey kernel. The analysis is based on the study of a $3\times 3$ matrix valued Riemann-Hilbert problem by the Deift-Zhou steepest descent method. The main ingredient is the construction of a local parametrix at the origin, out of the solutions of a particular third-order linear differential equation, and its matching with a global parametrix. Comment: 53 pages, 15 figures

• An adaptive algorithm for the cornea modeling from keratometric data

  Andrei Martinez-Finkelshtein, Dario Ramos-Lopez, Gracia M. Castro-Luna, Jorge L. Alio

  09/2010;

  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 ma... [more] 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-time reconstruction of both altimetric data and corneal power maps from the data collected by keratoscopes, such as the Placido rings based topographers, decisive for an early detection of corneal diseases such as keratoconus. Numerical experiments show that the algorithm exhibits a steady exponential error decay, independently of the level of aberration of the cornea. The complexity of each anisotropic gaussian basis functions in the functional representation is the same, but their parameters vary to fit the current scale. This scale is determined only by the residual errors and not by the number of the iteration. Finally, the position and clustering of their centers, as well as the size of the shape parameters, provides an additional spatial information about the regions of higher irregularity. These results are compared with the standard approximation procedures based on the Zernike polynomials expansions. Comment: 19 pages, 9 figures

• Asymptotics of the L^2 Norm of Derivatives of OPUC

  Andrei Martinez-Finkelshtein, Barry Simon

  06/2010;

  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, i... [more] 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. Comment: 36 pages, no figures. Minor corrections, to appear in the Journal of Approximation Theory

• Asymptotics of orthogonal polynomial's entropy.

  A. I. Aptekarev, Jesús Sánchez-Dehesa, Andrei Martínez-Finkelshtein

  J. Computational Applied Mathematics. 01/2010; 233:1355-1365.
• 3.43

  Impact points

  COMPARATIVE ANALYSIS OF SOME MODAL RECONSTRUCTION METHODS OF THE SHAPE OF THE CORNEA FROM CORNEAL ELEVATION DATA.

  Andrei Martinez-Finkelshtein, Antonia M Delgado, Gracia M Castro, Alejandro Zarzo, Jorge L. Alio

  Investigative ophthalmology & visual science. 08/2009;

  Purpose. A comparative study of the ability of some modal schemes to reproduce corneal shapes of varying complexity is performed, using both standard radial polynomials and the radial basis functions (RBF). Our claim is that the correct approach in the case of highly irregular corneas should combine... [more] Purpose. A comparative study of the ability of some modal schemes to reproduce corneal shapes of varying complexity is performed, using both standard radial polynomials and the radial basis functions (RBF). Our claim is that the correct approach in the case of highly irregular corneas should combine several bases. Methods. Standard approaches of reconstruction by Zernike and other types of radial polynomials are compared with the discrete least squares fit (LSF) by the RBF in three theoretical surfaces, synthetically generated by computer algorithms in the lack of measurement noise. For the reconstruction by polynomials the maximal radial order 6 was chosen, which corresponds to the first 28 Zernike polynomials or the first 49 Bhatia-Wolf polynomials. The fit with the RBF has been carried out using a regular grid of centers. Results. The quality of fit was assessed by computing for each surface the mean square errors (MSE) of the reconstruction by LSF, measured at the same nodes where the heights were collected. Another criterion of the fitting quality used was the accuracy in recovery of the Zernike coefficients, especially in the case of incomplete data. Conclusions. The Zernike (and especially, the Bhatia-Wolf) polynomials constitute a reliable reconstruction method of a non-severely aberrated surface with a small surface regularity index (SRI). However, they fail to capture small deformations of the anterior surface of a synthetic cornea. The most promising is a combined approach that balances the robustness of the Zernike fit with the localization of the RBF.

• On a conjecture of A. Magnus concerning the asymptotic behavior of the recurrence coefficients of the generalized Jacobi polynomials

  A. Foulquie Moreno, A. Martinez-Finkelshtein, V. L. Sousa

  05/2009;

  In 1995 Magnus posed a conjecture about the asymptotics of the recurrence coefficients of orthogonal polynomials with respect to the weights on [-1,1] of the form $$ (1-x)^\alpha (1+x)^\beta |x_0 - x|^\gamma \times a jump at x_0, $$ with $\alpha, \beta, \gamma>-1$ and $x_0 \in (-1,1)$. We show ri... [more] In 1995 Magnus posed a conjecture about the asymptotics of the recurrence coefficients of orthogonal polynomials with respect to the weights on [-1,1] of the form $$ (1-x)^\alpha (1+x)^\beta |x_0 - x|^\gamma \times a jump at x_0, $$ with $\alpha, \beta, \gamma>-1$ and $x_0 \in (-1,1)$. We show rigorously that Magnus' conjecture is correct even in a more general situation, when the weight above has an extra factor, which is analytic in a neighborhood of [-1,1] and positive on the interval. The proof is based on the steepest descendent method of Deift and Zhou applied to the non-commutative Riemann-Hilbert problem characterizing the orthogonal polynomials. A feature of this situation is that the local analysis at $x_0$ has to be carried out in terms of confluent hypergeometric functions. Comment: 29 pages, 4 figures

• Asymptotics of orthogonal polynomials for a weight with a jump on [-1,1]

  A. Foulquie Moreno, A. Martinez-Finkelshtein, V. L. Sousa

  04/2009;

  We consider the orthogonal polynomials on $[-1,1]$ with respect to the weight $$ w_c(x)=h(x)(1-x)^{\alpha}(1+x)^{\beta} \Xi_{c}(x), \quad \alpha, \beta >-1, $$ where $h$ is real analytic and strictly positive on $[-1, 1]$, and $\Xi_{c}$ is a step-like function: $\Xi_{c}(x)=1$ for $x\in [-1, 0)$ a... [more] We consider the orthogonal polynomials on $[-1,1]$ with respect to the weight $$ w_c(x)=h(x)(1-x)^{\alpha}(1+x)^{\beta} \Xi_{c}(x), \quad \alpha, \beta >-1, $$ where $h$ is real analytic and strictly positive on $[-1, 1]$, and $\Xi_{c}$ is a step-like function: $\Xi_{c}(x)=1$ for $x\in [-1, 0)$ and $\Xi_{c}(x)=c^2$, $c>0$, for $x\in [0, 1]$. We obtain strong uniform asymptotics of the monic orthogonal polynomials in $\mathbb{C}$, as well as first terms of the asymptotic expansion of the main parameters (leading coefficients of the orthonormal polynomials and the recurrence coefficients) as $n\to \infty$. In particular, we prove for $w_c$ a conjecture of A. Magnus regarding the asymptotics of the recurrence coefficients. The main focus is on the local analysis at the origin. We study the asymptotics of the Christoffel-Darboux kernel in a neighborhood of the jump and show that the zeros of the orthogonal polynomials no longer exhibit the clock behavior. For the asymptotic analysis we use the steepest descendent method of Deift and Zhou applied to the non-commutative Riemann-Hilbert problems characterizing the orthogonal polynomials. The local analysis at $x=0$ is carried out in terms of the confluent hypergeometric functions. Incidentally, we establish some properties of these functions that may have an independent interest. Comment: 48 pages, 5 figures. New structure, further results on confluent hypergeometric functions, relation with de Brange space of analytic functions

• On asymptotic behavior of Heine-Stieltjes and Van Vleck polynomials

  A. Martinez-Finkelshtein, E. A. Rakhmanov

  04/2009;

  We investigate the strong asymptotics of Heine-Stieltjes polynomials - polynomial solutions of a second order differential equations with complex polynomial coefficients. The solution is given in terms of critical measures (saddle points of the weighted logarithmic energy on the plane), that are tig... [more] We investigate the strong asymptotics of Heine-Stieltjes polynomials - polynomial solutions of a second order differential equations with complex polynomial coefficients. The solution is given in terms of critical measures (saddle points of the weighted logarithmic energy on the plane), that are tightly related to quadratic differentials with closed trajectories on the plane. The paper is a continuation of the research initiated in [arXiv:0902.0193]. However, the starting point here is the WKB method, which allows to obtain the strong asymptotics.

• Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials

  A. Martinez-Finkelshtein, E. A. Rakhmanov

  02/2009;

  We investigate the asymptotic zero distribution of Heine-Stieltjes polynomials - polynomial solutions of a second order differential equations with complex polynomial coefficients. In the case when all zeros of the leading coefficients are all real, zeros of the Heine-Stieltjes polynomials were inte... [more] We investigate the asymptotic zero distribution of Heine-Stieltjes polynomials - polynomial solutions of a second order differential equations with complex polynomial coefficients. In the case when all zeros of the leading coefficients are all real, zeros of the Heine-Stieltjes polynomials were interpreted by Stieltjes as discrete distributions minimizing an energy functional. In a general complex situation one deals instead with a critical point of the energy. We introduce the notion of discrete and continuous critical measures (saddle points of the weighted logarithmic energy on the plane), and prove that a weak-* limit of a sequence of discrete critical measures is a continuous critical measure. Thus, the limit zero distributions of the Heine-Stieltjes polynomials are given by continuous critical measures. We give a detailed description of such measures, showing their connections with quadratic differentials. In doing that, we obtain some results on the global structure of rational quadratic differentials on the Riemann sphere that have an independent interest. Comment: 70 pages, 14 figures. Minor corrections, to appear in Comm. Math. Physics

• Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights

  A. B. J. Kuijlaars, A. Martinez-Finkelshtein, F. Wielonsky

  12/2007;

  We study 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 $t = T$ at $x = 0$. In the limit $n \to \infty$, after appropriate rescaling, the paths fil... [more] We study 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 $t = T$ at $x = 0$. In the limit $n \to \infty$, after appropriate rescaling, the paths fill out a region in the $tx$-plane that we describe explicitly. In particular, the paths initially stay away from the hard edge at $x = 0$, but at a certain critical time $t^*$ the smallest paths hit the hard edge and from then on are stuck to it. For $t \neq t^*$ we obtain the usual scaling limits from random matrix theory, namely the sine, Airy, and Bessel kernels. A key fact is that the positions of the paths at any time $t$ constitute a multiple orthogonal polynomial ensemble, corresponding to a system of two modified Bessel-type weights. As a consequence, there is a $3 \times 3$ matrix valued Riemann-Hilbert problem characterizing this model, that we analyze in the large $n$ limit using the Deift-Zhou steepest descent method. There are some novel ingredients in the Riemann-Hilbert analysis that are of independent interest. Comment: 59 pages, 11 figures

• Discrete entropies of orthogonal polynomials

  A. I. Aptekarev, J. S. Dehesa, A. Martinez-Finkelshtein, R. Yañez

  11/2007;

  Let $p_n$ be the $n$-th orthonormal polynomial on the real line, whose zeros are $\lambda_j^{(n)}$, $j=1, ..., n$. Then for each $j=1, ..., n$, $$ \vec \Psi_j^2 = (\Psi_{1j}^2, ..., \Psi_{nj}^2) $$ with $$ \Psi_{ij}^2= p_{i-1}^2 (\lambda_j^{(n)}) (\sum_{k=0}^{n-1} p_k^2(\lambda_j^{(n)}))^{-1}, \quad... [more] Let $p_n$ be the $n$-th orthonormal polynomial on the real line, whose zeros are $\lambda_j^{(n)}$, $j=1, ..., n$. Then for each $j=1, ..., n$, $$ \vec \Psi_j^2 = (\Psi_{1j}^2, ..., \Psi_{nj}^2) $$ with $$ \Psi_{ij}^2= p_{i-1}^2 (\lambda_j^{(n)}) (\sum_{k=0}^{n-1} p_k^2(\lambda_j^{(n)}))^{-1}, \quad i=1, >..., n, $$ defines a discrete probability distribution. The Shannon entropy of the sequence $\{p_n\}$ is consequently defined as $$ \mathcal S_{n,j} = -\sum_{i=1}^n \Psi_{ij}^{2} \log (\Psi_{ij}^{2}) . $$ In the case of Chebyshev polynomials of the first and second kinds an explicit and closed formula for $\mathcal S_{n,j}$ is obtained, revealing interesting connections with the number theory. Besides, several results of numerical computations exemplifying the behavior of $\mathcal S_{n,j}$ for other families are also presented.

• Discrete entropies of orthogonal polynomials

  A. I. Aptekarev, Jesús Sánchez-Dehesa, Andrei Martínez-Finkelshtein, R. J. Yáñez

  CoRR. 01/2007; abs/0710.2134.

• Asymptotics of orthogonal polynomials with respect to an analytic weight with algebraic singularities on the circle

  A. Martinez-Finkelshtein, K. T. -R. McLaughlin, E. B. Saff

  06/2006;

  Strong asymptotics of polynomials orthogonal on the unit circle with respect to a weight of the form $$ W(z) = w(z) \prod_{k=1}^m |z-a_k|^{2\beta_k}, \quad |z|=1, \quad |a_k|=1, \quad \beta_k>-1/2, \quad k=1, ..., m, $$ where $w(z)>0$ for $|z|=1$ and can be extended as a holomorphic and non-va... [more] Strong asymptotics of polynomials orthogonal on the unit circle with respect to a weight of the form $$ W(z) = w(z) \prod_{k=1}^m |z-a_k|^{2\beta_k}, \quad |z|=1, \quad |a_k|=1, \quad \beta_k>-1/2, \quad k=1, ..., m, $$ where $w(z)>0$ for $|z|=1$ and can be extended as a holomorphic and non-vanishing function to an annulus containing the unit circle. The formulas obtained are valid uniformly in the whole complex plane. As a consequence, we obtain some results about the distribution of zeros of these polynomials, the behavior of their leading and Verblunsky coefficients, as well as give an alternative proof of the Fisher-Hartwig conjecture about the asymptotics of Toeplitz determinants for such type of weights. The main technique is the steepest descent analysis of Deift and Zhou, based on the matrix Riemann-Hilbert characterization proposed by Fokas, Its and Kitaev.

• Electrostatic models for zeros of polynomials: old, new, and some open problems

  F. Marcellan, A. Martinez-Finkelshtein, P. Martinez-Gonzalez

  01/2006;

  We give a survey concerning both very classical and recent results on the electrostatic interpretation of the zeros of some well-known families of polynomials, and the interplay between these models and the asymptotic distribution of their zeros when the degree of the polynomials tends to infinity. ... [more] We give a survey concerning both very classical and recent results on the electrostatic interpretation of the zeros of some well-known families of polynomials, and the interplay between these models and the asymptotic distribution of their zeros when the degree of the polynomials tends to infinity. The leading role is played by the differential equation satisfied by these polynomials. Some new developments, applications and open problems are presented.

• On Extensions of a Theorem of Baxter

  J. S. Geronimo, A. Martinez-Finkelshtein

  09/2005;

  We combine the Riemann-Hilbert approach with the techniques of Banach algebras to obtain an extension of Baxter's Theorem for polynomials orthogonal on the unit circle. This is accomplished by using the link between the negative Fourier coefficients of the scattering function and the coefficient... [more] We combine the Riemann-Hilbert approach with the techniques of Banach algebras to obtain an extension of Baxter's Theorem for polynomials orthogonal on the unit circle. This is accomplished by using the link between the negative Fourier coefficients of the scattering function and the coefficients in the recurrence formula satisfied by these polynomials.