arXiv subjects

Alfredo Deaño

Publications and source records attributed to Alfredo Deaño.

25 records · Page 2Linked to original sources

arXiv2015

Construction and implementation of asymptotic expansions for Jacobi--type orthogonal polynomials

We are interested in the asymptotic behavior of orthogonal polynomials of the generalized Jacobi type as their degree $n$ goes to $\infty$. These are defined on the interval $[-1,1]$ with weight function $w(x)=(1-x)^α(1+x)^βh(x)$, $α,β>-1$ and $h(x)$ a real, analytic and strictly positive function on $[-1,1]$. This information is available in the work of Kuijlaars, McLaughlin, Van Assche and Vanlessen, where the authors use the Riemann--Hilbert formulation and the Deift--Zhou non-linear steepest descent method. We show that computing higher-order terms can be simplified, leading to their efficient construction. The resulting asymptotic expansions in every region of the complex plane are implemented both symbolically and numerically, and the code is made publicly available. The main advantage of these expansions is that they lead to increasing accuracy for increasing degree of the polynomials, at a computational cost that is actually independent of the degree. In contrast, the typical use of the recurrence relation for orthogonal polynomials in computations leads to a cost that is at least linear in the degree. Furthermore, the expansions may be used to compute Gaussian quadrature rules in $\mathcal{O}(n)$ operations, rather than $\mathcal{O}(n^2)$ based on the recurrence relation.

cs.MS↗

arXiv2015

Asymptotics of orthogonal polynomials generated by a Geronimus perturbation of the Laguerre measure

This paper deals with monic orthogonal polynomials generated by a Geronimus canonical spectral transformation of the Laguerre classical measure: \[ \frac{1}{x-c}x^{α}e^{-x}dx+Nδ(x-c), \] for $x\in[0,\infty)$, $α>-1$, a free parameter $N\in \mathbb{R}_{+}$ and a shift $c<0$. We analyze the asymptotic behavior (both strong and relative to classical Laguerre polynomials) of these orthogonal polynomials as $n$ tends to infinity.

math.CA↗

arXiv2014

Large degree asymptotics of orthogonal polynomials with respect to an oscillatory weight on a bounded interval

We consider polynomials $p_n^ω(x)$ that are orthogonal with respect to the oscillatory weight $w(x)=e^{iωx}$ on $[-1,1]$, where $ω>0$ is a real parameter. A first analysis of $p_n^ω(x)$ for large values of $ω$ was carried out in connection with complex Gaussian quadrature rules with uniform good properties in $ω$. In this contribution we study the existence, asymptotic behavior and asymptotic distribution of the roots of $p_n^ω(x)$ in the complex plane as $n\to\infty$. The parameter $ω$ grows with $n$ linearly. The tools used are logarithmic potential theory and the $S$-property, together with the Riemann--Hilbert formulation and the Deift--Zhou steepest descent method.

math.CA↗

arXiv2014

Asymptotic behavior and zero distribution of polynomials orthogonal with respect to Bessel functions

We consider polynomials $P_n$ orthogonal with respect to the weight $J_ν$ on $[0,\infty)$, where $J_ν$ is the Bessel function of order $ν$. Asheim and Huybrechs considered these polynomials in connection with complex Gaussian quadrature for oscillatory integrals. They observed that the zeros are complex and accumulate as $n \to \infty$ near the vertical line $\textrm{Re}\, z = \frac{νπ}{2}$. We prove this fact for the case $0 \leq ν\leq 1/2$ from strong asymptotic formulas that we derive for the polynomials $P_n$ in the complex plane. Our main tool is the Riemann-Hilbert problem for orthogonal polynomials, suitably modified to cover the present situation, and the Deift-Zhou steepest descent method. A major part of the work is devoted to the construction of a local parametrix at the origin, for which we give an existence proof that only works for $ν\leq 1/2$.

math.CA↗

arXiv2013

Strong and ratio asymptotics for Laguerre polynomials revisited

In this paper we consider the strong asymptotic behavior of Laguerre polynomials in the complex plane. The leading behavior is well known from Perron and Mehler-Heine formulas, but higher order coefficients, which are important in the context of Krall-Laguerre or Laguerre-Sobolev-type orthogonal polynomials, are notoriously difficult to compute. In this paper, we propose the use of an alternative expansion, due to Buchholz, in terms of Bessel functions of the first kind. The coefficients in this expansion can be obtained in a straightforward way using symbolic computation. As an application, we derive extra terms in the asymptotic expansion of ratios of Laguerre polynomials in $C\[0,\infty)$.

math.CA↗

arXiv2012

A Gaussian quadrature rule for oscillatory integrals on a bounded interval

We investigate a Gaussian quadrature rule and the corresponding orthogonal polynomials for the oscillatory weight function $e^{iωx}$ on the interval $[-1,1]$. We show that such a rule attains high asymptotic order, in the sense that the quadrature error quickly decreases as a function of the frequency $ω$. However, accuracy is maintained for all values of $ω$ and in particular the rule elegantly reduces to the classical Gauss-Legendre rule as $ω\to 0$. The construction of such rules is briefly discussed, and though not all orthogonal polynomials exist, it is demonstrated numerically that rules with an even number of points are always well defined. We show that these rules are optimal both in terms of asymptotic order as well as in terms of polynomial order.

math.NA↗

arXiv2011

Topological expansion in the cubic random matrix model

In this paper we return to the classical work by Brézin, Itzykson, Parisi and Zuber, in which, among other things, the authors explicitly calculated the coefficients of the topological expansion in the cubic random matrix model in genus 0. Our main goal will be to rigorously prove the results of Brézin, Itzykson, Parisi and Zuber and to obtain an explicit formula for the coefficients of the topological expansion in genus 1. We will also prove some formulae and asymptotic results for the coefficients of the topological expansion in higher genera.

math-ph↗