Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOrthogonal polynomials of equilibrium measures supported on Cantor sets
108
0
0.0
(
0
)
Authors
Giorgio Mantica
Ask ChatGPT about the research
No Arabic abstract
We study the orthogonal polynomials associated with the equilibrium measure, in logarithmic potential theory, living on the attractor of an Iterated Function System. We construct sequences of discrete measures, that converge weakly to the equilibrium measure, and we compute their Jacobi matrices via standard procedures, suitably enhanced for the scope. Numerical estimates of the convergence rate to the limit Jacobi matrix are provided, that show stability and efficiency of the whole procedure. As a secondary result, we also compute Jacobi matrices of equilibrium measures on finite sets of intervals, and of balanced measures of Iterated Function Systems. These algorithms can reach large orders: we study the asymptotic behavior of the orthogonal polynomials and we show that they can be used to efficiently compute Greens functions and conformal mappings of interest in constructive function theory.
rate research
Read More
Let $G$ be a finite union of disjoint and bounded Jordan domains in the complex plane, let $mathcal{K}$ be a compact subset of $G$ and consider the set $G^star$ obtained from $G$ by removing $mathcal{K}$; i.e., $G^star:=Gsetminus mathcal{K}$. We refer to $G$ as an archipelago and $G^star$ as an archipelago with lakes. Denote by ${p_n(G,z)}_{n=0}^infty$ and ${p_n(G^star,z)}_{n=0}^infty$, the sequences of the Bergman polynomials associated with $G$ and $G^star$, respectively; that is, the orthonormal polynomials with respect to the area measure on $G$ and $G^star$. The purpose of the paper is to show that $p_n(G,z)$ and $p_n(G^star,z)$ have comparable asymptotic properties, thereby demonstrating that the asymptotic properties of the Bergman polynomials for $G^star$ are determined by the boundary of $G$. As a consequence we can analyze certain asymptotic properties of $p_n(G^star,z)$ by using the corresponding results for $p_n(G,z)$, which were obtained in a recent work by B. Gustafsson, M. Putinar, and two of the present authors. The results lead to a reconstruction algorithm for recovering the shape of an archipelago with lakes from a partial set of its complex moments.
We describe a numerical technique to compute the equilibrium measure, in logarithmic potential theory, living on the attractor of Iterated Function Systems composed of one-dimensional affine maps. This measure is obtained as the limit of a sequence of equilibrium measures on finite unions of intervals. Although these latter are known analytically, their computation requires the evaluation of a number of integrals and the solution of a non-linear set of equations. We unveil the potential numerical dangers hiding in these problems and we propose detailed solutions to all of them. Convergence of the procedure is illustrated in specific examples and is gauged by computing the electrostatic potential.
We describe a numerical procedure to compute the so-called isospectral torus of finite gap sets, that is, the set of Jacobi matrices whose essential spectrum is composed of finitely many intervals. We also study numerically the convergence of specific Jacobi matrices to their isospectral limit. We then extend the analyis to the definition and computation of an isospectral torus for Cantor sets in the family of Iterated Function Systems. This analysis is developed with the ultimate goal of attacking numerically the conjecture that the Jacobi matrices of I.F.S. measures supported on Cantor sets are asymptotically almost-periodic.
We analyze different measures for the backward error of a set of numerical approximations for the roots of a polynomial. We focus mainly on the element-wise mixed backward error introduced by Mastronardi and Van Dooren, and the tropical backward error introduced by Tisseur and Van Barel. We show that these measures are equivalent under suitable assumptions. We also show relations between these measures and the classical element-wise and norm-wise backward error measures.
We present a simple construction for a tridiagonal matrix $T$ that commutes with the hopping matrix for the entanglement Hamiltonian ${cal H}$ of open finite free-Fermion chains associated with families of discrete orthogonal polynomials. It is based on the notion of algebraic Heun operator attached to bispectral problems, and the parallel between entanglement studies and the theory of time and band limiting. As examples, we consider Fermionic chains related to the Chebychev, Krawtchouk and dual Hahn polynomials. For the former case, which corresponds to a homogeneous chain, the outcome of our construction coincides with a recent result of Eisler and Peschel; the latter cases yield commuting operators for particular inhomogeneous chains. Since $T$ is tridiagonal and non-degenerate, it can be readily diagonalized numerically, which in turn can be used to calculate the spectrum of ${cal H}$, and therefore the entanglement entropy.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
