Global Asymptotics of the Discrete Chebyshev Polynomials

In this paper, we study the asymptotics of the discrete Chebyshev polynomials tn (z, N) as the degree grows to infinity. Global asymptotic formulas are obtained as n \rightarrow \infty, when the ratio of the parameters n/N = c is a constant in the interval (0, 1). Our method is based on a modified version of the Riemann-Hilbert approach first introduced by Deift and Zhou

On the use of Hahn's asymptotic formula and stabilized recurrence for a fast, simple, and stable Chebyshev--Jacobi transform

We describe a fast, simple, and stable transform of Chebyshev expansion coefficients to Jacobi expansion coefficients and its inverse based on the numerical evaluation of Jacobi expansions at the Chebyshev--Lobatto points. This is achieved via a decomposition of Hahn's interior asymptotic formula into a small sum of diagonally scaled discrete sine and cosine transforms and the use of stable recurrence relations. It is known that the Clenshaw--Smith algorithm is not uniformly stable on the entire interval of orthogonality. Therefore, Reinsch's modification is extended for Jacobi polynomials and employed near the endpoints to improve numerical stability

Minkowski's Question Mark Measure

Minkowski's question mark function is the distribution function of a singular continuous measure: we study this measure from the point of view of logarithmic potential theory and orthogonal polynomials. We conjecture that it is regular, in the sense of Ullman--Stahl--Totik and moreover it belongs to a Nevai class: we provide numerical evidence of the validity of these conjectures. In addition, we study the zeros of its orthogonal polynomials and the associated Christoffel functions, for which asymptotic formulae are derived. Rigorous results and numerical techniques are based upon Iterated Function Systems composed of Mobius maps.Comment: 31 pages, 17 figures 2-nd revision: added section 7.2, upper and lower bounds to the Hausdorff dimension of the measur

Orthogonal polynomials of equilibrium measures supported on Cantor sets

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 Green's functions and conformal mappings of interest in constructive function theory.Comment: 28 pages, 15 figure

Numerical computation of the isospectral torus of finite gap sets and of IFS Cantor sets

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.Comment: Contribution to Ed Saff's 70th birthday celebrative volum

Global Asymptotics of the Hahn Polynomials

In this paper, we study the asymptotics of the Hahn polynomials Q_n(x; {\alpha}, {\beta}, N) as the degree n grows to infinity, when the parameters {\alpha} and {\beta} are fixed and the ratio of n/N = c is a constant in the interval (0, 1). Uniform asymptotic formulas in terms of Airy functions and elementary functions are obtained for z in three overlapping regions, which together cover the whole complex plane. Our method is based on a modified version of the Riemann-Hilbert approach introduced by Deift and Zhou.Comment: 44 pages, 7 figure

Distributing many points on spheres: minimal energy and designs

This survey discusses recent developments in the context of spherical designs and minimal energy point configurations on spheres. The recent solution of the long standing problem of the existence of spherical $t$-designs on $\mathbb{S}^d$ with $\mathcal{O}(t^d)$ number of points by A. Bondarenko, D. Radchenko, and M. Viazovska attracted new interest to this subject. Secondly, D. P. Hardin and E. B. Saff proved that point sets minimising the discrete Riesz energy on $\mathbb{S}^d$ in the hypersingular case are asymptotically uniformly distributed. Both results are of great relevance to the problem of describing the quality of point distributions on $\mathbb{S}^d$, as well as finding point sets, which exhibit good distribution behaviour with respect to various quality measures.Comment: 50 pages, 1 table; revision; references added and update

Tridiagonal Models for Dyson Brownian Motion

In this paper, we consider tridiagonal matrices the eigenvalues of which evolve according to $\beta$-Dyson Brownian motion. This is the stochastic gradient flow on $\mathbb{R}^n$ given by, for all $1 \leq i \leq n,$ $$d\lambda_{i,t} = \sqrt{\frac{2}{\beta}}dZ_{i,t} - \biggl( \frac{V'(\lambda_i)}{2} - \sum_{j: j \neq i} \frac{1}{\lambda_i - \lambda_j} \biggr)\,dt$$ where $V$ is a constraining potential and $\left\{ Z_{i,t} \right\}_1^n$ are independent standard Brownian motions. This flow is stationary with respect to the distribution $$\rho^{\beta}_N(\lambda) = \frac{1}{Z^{\beta}_N} e^{-\frac{\beta}{2} \left( -\sum_{1 \leq i \neq j \leq N} \log|\lambda_i - \lambda_j| + \sum_{i=1}^N V(\lambda_i) \right) }.$$ The particular choice of $V(t)=2t^2$ leads to an eigenvalue distribution constrained to lie roughly in $(-\sqrt{n},\sqrt{n}).$ We study evolution of the entries of one choice of tridiagonal flow for this $V$ in the $n\to \infty$ limit. On the way to describing the evolution of the tridiagonal matrices we give the derivative of the Lanczos tridiagonalization algorithm under perturbation

Global dynamics of a Yang-Mills field on an asymptotically hyperbolic space

We consider a spherically symmetric (purely magnetic) SU(2) Yang-Mills field propagating on an ultrastatic spacetime with two asymptotically hyperbolic regions connected by a throat of radius $\alpha$. Static solutions in this model are shown to exhibit an interesting bifurcation pattern in the parameter $\alpha$. We relate this pattern to the Morse index of the static solution with maximal energy. Using a hyperboloidal approach to the initial value problem, we describe the relaxation to the ground state solution for generic initial data and unstable static solutions for initial data of codimension one, two, and three.Comment: 23 pages, 11 figure

Asymptotic properties of Jacobi matrices for a family of fractal measures

We study the properties and asymptotics of the Jacobi matrices associated with equilibrium measures of the weakly equilibrium Cantor sets. These family of Cantor sets were defined and different aspects of orthogonal polynomials on them were studied recently. Our main aim is numerically examine some conjectures concerning orthogonal polynomials which do not directly follow from previous results. We also compare our results with more general conjectures made for recurrence coefficients associated with fractal measures supported on $\mathbb{R}$