Asymptotic Behavior Of Orthogonal Polynomials Research Articles

Orthogonal rational functions with real poles, root asymptotics, and GMP matrices

There is a vast theory of the asymptotic behavior of orthogonal polynomials with respect to a measure on R \mathbb {R} and its applications to Jacobi matrices. That theory has an obvious affine invariance and a very special role for ∞ \infty . We extend aspects of this theory in the setting of rational functions with poles on R ¯ = R ∪ { ∞ } \overline {\mathbb {R}} = \mathbb {R} \cup \{\infty \} , obtaining a formulation which allows multiple poles and proving an invariance with respect to R ¯ \overline {\mathbb {R}} -preserving Möbius transformations. We obtain a characterization of Stahl–Totik regularity of a GMP matrix in terms of its matrix elements; as an application, we give a proof of a conjecture of Simon – a Cesàro–Nevai property of regular Jacobi matrices on finite gap sets.

Spectral analysis of Jacobi operators and asymptotic behavior of orthogonal polynomials

We find and discuss asymptotic formulas for orthonormal polynomials [Formula: see text] with recurrence coefficients [Formula: see text]. Our main goal is to consider the case where off-diagonal elements [Formula: see text] as [Formula: see text]. Formulas obtained are essentially different for relatively small and large diagonal elements [Formula: see text]. Our analysis is intimately linked with spectral theory of Jacobi operators [Formula: see text] with coefficients [Formula: see text] and a study of the corresponding second order difference equations. We introduce the Jost solutions [Formula: see text], [Formula: see text], of such equations by a condition for [Formula: see text] and suggest an Ansatz for them playing the role of the semiclassical Liouville–Green Ansatz for solutions of the Schrödinger equation. This allows us to study the spectral structure of Jacobi operators and their eigenfunctions [Formula: see text] by traditional methods of spectral theory developed for differential equations. In particular, we express all coefficients in asymptotic formulas for [Formula: see text] as [Formula: see text] in terms of the Wronskian of the solutions [Formula: see text] and [Formula: see text]. The formulas obtained for [Formula: see text] generalize the asymptotic formulas for the classical Hermite polynomials where [Formula: see text] and [Formula: see text]. The spectral structure of Jacobi operators [Formula: see text] depends crucially on a rate of growth of the off-diagonal elements [Formula: see text] as [Formula: see text]. If the Carleman condition is satisfied, which, roughly speaking, means that [Formula: see text], and the diagonal elements [Formula: see text] are small compared to [Formula: see text], then [Formula: see text] has the absolutely continuous spectrum covering the whole real axis. We obtain an expression for the corresponding spectral measure in terms of the boundary values [Formula: see text] of the Jost solutions. On the contrary, if the Carleman condition is violated, then the spectrum of [Formula: see text] is discrete. We also review the case of stabilizing recurrence coefficients when [Formula: see text] tend to a positive constant and [Formula: see text] as [Formula: see text]. It turns out that the cases of stabilizing and increasing recurrence coefficients can be treated in an essentially same way.

Entropy function and orthogonal polynomials

We give a simple proof of a classical theorem by A. Máté, P. Nevai, and V. Totik on asymptotic behavior of orthogonal polynomials on the unit circle. It is based on a new real-variable approach involving an entropy estimate for the orthogonality measure. Our second result is an extension of a theorem by G. Freud on averaged convergence of Fourier series. We also discuss some related open problems in the theory of orthogonal polynomials on the unit circle.

Asymptotic behavior of orthogonal polynomials. Singular critical case

Our goal is to find an asymptotic behavior as n→∞ of the orthogonal polynomials Pn(z) defined by Jacobi recurrence coefficients an (off-diagonal terms) and bn (diagonal terms). We consider the case an→∞, bn→∞ in such a way that ∑an−1<∞ (that is, the Carleman condition is violated) and γn:=2−1bn(anan−1)−1∕2→γ as n→∞. In the case |γ|≠1 asymptotic formulas for Pn(z) are known; they depend crucially on the sign of |γ|−1. We study the critical case |γ|=1. The formulas obtained are qualitatively different in the cases |γn|→1−0 and |γn|→1+0. Another goal of the paper is to advocate an approach to a study of asymptotic behavior of Pn(z) based on a close analogy of the Jacobi difference equations and differential equations of Schrödinger type.

Semiclassical asymptotic behavior of orthogonal polynomials

Our goal is to find asymptotic formulas for orthonormal polynomials $$P_{n}(z)$$ with the recurrence coefficients slowly stabilizing as $$n\rightarrow \infty $$ . To that end, we develop scattering theory of Jacobi operators with long-range coefficients and study the corresponding second-order difference equation. We introduce the Jost solutions $$f_{n}(z)$$ of this equation by a condition for $$n\rightarrow \infty $$ and suggest an Ansatz for them playing the role of the semiclassical Liouville–Green Ansatz for the corresponding solutions of the Schrodinger equation. This allows us to study Jacobi operators and their eigenfunctions $$P_{n}(z)$$ by traditional methods of spectral theory developed for differential equations. In particular, we express all coefficients in asymptotic formulas for $$P_{n}(z)$$ as $$\rightarrow \infty $$ in terms of the Wronskian of the solutions $$\{P_{n} (z)\}$$ and $$\{f_{n} (z)\}$$ .

Asymptotic behavior of orthogonal polynomials without the Carleman condition

Our goal is to find an asymptotic behavior as n→∞ of orthogonal polynomials Pn(z) defined by the Jacobi recurrence coefficients an,bn. We suppose that the off-diagonal coefficients an grow so rapidly that the series ∑an−1 converges, that is, the Carleman condition is violated. With respect to diagonal coefficients bn we assume that −bn(anan−1)−1/2→2β∞ for some β∞≠±1. The asymptotic formulas obtained for Pn(z) are quite different from the case ∑an−1=∞ when the Carleman condition is satisfied. In particular, if ∑an−1<∞, then the phase factors in these formulas do not depend on the spectral parameter z∈C. The asymptotic formulas obtained in the cases |β∞|<1 and |β∞|>1 are also qualitatively different from each other. As an application of these results, we find necessary and sufficient conditions for the essential self-adjointness of the corresponding minimal Jacobi operator.

Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane

We study the asymptotic behaviour of orthogonal polynomials in the complex plane that are associated to a certain normal matrix model. The model depends on a parameter and the asymptotic distribution of the eigenvalues undergoes a transition for a special value of the parameter, where it develops a corner-type singularity. In the double scaling limit near the transition we determine the asymptotic behaviour of the orthogonal polynomials in terms of a solution of the Painlev\'e IV equation. We determine the Fredholm determinant associated to such solution and we compute it numerically on the real line, showing also that the corresponding Painlev\'e transcendent is pole-free on a semiaxis.

The arcsine law and an asymptotic behavior of orthogonal polynomials

In the present papar we generalize “quantum-classical correspondence” for harmonic oscillators to the context of interacting Fock spaces. Under a simple condition for Jacobi sequences, it is shown that the Arcsine law is the unique probability distribution corresponding to the “Classical limits (large quantum number limits)”. As a corollary, we obtain that the squared n-th orthogonal polynomials for a probability distribution corresponding to such kinds of interacting Fock spaces, multiplied by the probability distribution and normalized, weakly converge to the Arcsine law as n tends to infinity.

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 $w(x)=(1-x)^{\alpha}(1+x)^{\beta}h(x)$, $\alpha,\beta>-1$ 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.

Level Curvature Distribution: from Bulk to the Soft Edge of Random Hermitian Matrices

Level curvature is a measure of sensitivity of energy levels of a disordered/chaotic system to perturbations. In the bulk of the spectrum random matrix theory predicts the probability distributions of level curvatures to be given by the Zakrzewski Delande expressions. Motivated by growing interest in statistics of extreme (maximal or minimal) eigenvalues of disordered systems of various nature, it is natural to ask about the associated level curvatures. We show how calculating the distribution for the curvatures of extreme eigenvalues in Gaussian unitary ensemble can be reduced to studying asymptotic behaviour of orthogonal polynomials appearing in the recent work of Nadal and Majumdar. The corresponding asymptotic analysis being yet outstanding, we instead will discuss solution of a related, but somewhat simpler problem of calculating the level curvature distribution averaged over all the levels in a spectral window close to the edge of the semicircle. The method is based on asymptotic analysis of kernels associated with Hermite polynomials and their Cauchy transforms, and is straightforwardly extendable to any rotationally-invariant ensemble of random matrices.

A new approach to relative asymptotic behavior for discrete Sobolev-type orthogonal polynomials on the unit circle

In Foulquié et al. (1999) [2] , Li and Marcellán (1996) [4] , Marcellán and Moral (2002) [5] , the relative asymptotic behavior of orthogonal polynomials with respect to a discrete Sobolev-type inner product on the unit circle was studied. In this paper, we propose an alternative approach to this problem based on the Uvarov spectral transformation.

On the asymptotics of orthogonal polynomials on the curve with a denumerable mass points

We investigate the asymptotic behavior of orthogonal polynomials with respect to a measure of the type \(\sigma =\alpha +\gamma \), where \(\alpha \) is a measure concentrated on a rectifiable Jordan curve and \(\gamma \) is an infinite discrete measure.

Some asymptotic properties for orthogonal polynomials with respect to varying measures

We study some properties of the zeros and the asymptotic behavior of orthogonal polynomials with respect to varying measures on the unit circle. In the proofs, some techniques of rational approximation are used.

Asymptotics of orthogonal polynomials inside the unit circle and Szegő–Padé approximants

We study the asymptotic behavior of orthogonal polynomials inside the unit circle for a subclass of measures that satisfy Szegő's condition. We give a connection between such behavior and a Montessus de Ballore-type theorem for Szegő–Padé rational approximants of the corresponding Szegő function.

Local asymptotics of the ratio of orthogonal polynomials in the neighbourhood of an end-point of the support of the orthogonality measure

A new method of studying the asymptotic behaviour of orthogonal polynomials in the neighbourhood of an end-point of the support of the orthogonality measure is put forward. The asymptotics in the neighbourhood of the end-point is found in terms of the asymptotics of the values of the polynomials at the point itself and the asymptotic behaviour of the coefficients of the recurrence relations. Applications of the result obtained are considered.

Asymptotic behavior of orthogonal polynomials corresponding to a measure with infinite discrete part off an arc

We study the asymptotic behavior of orthogonal polynomials. The measure is concentrated on a complex rectifiable arc and has an infinity of masses in the region exterior to the arc.

Asymptotic Behaviour of Orthogonal Polynomials on the Unit Circle with Asymptotically Periodic Reflection Coefficients: II. Weak Asymptotics

In this paper we study orthogonal polynomials with asymptotically periodic reflection coefficients. It's known that the support of the orthogonality measure of such polynomials consists of several arcs. We are mainly interested in the asymptotic behaviour on the support and derive weak convergence results for the orthogonal polynomials and also for the Christoffel function.

Strong asymptotics of orthogonal polynomials with varying measures and Hermite-Padé approximants

The strong asymptotic behaviour of orthogonal polynomials with respect to a general class of varying measures is given for the case of the unit circle and the real line. These results are used to obtain certain asymptotic relations for the polynomials involved in the construction of Hermite-Padé approximants of a Nikishin system of functions.

Asymptotic Behavior of Orthogonal Polynomials Corresponding to a Measure with Infinite Discrete Part off a Curve

We determine the asymptotic behavior of orthogonal polynomials associated to a measureα=β+γ, whereβis a measure concentrated on a rectifiable Jordan curve andγis an infinite discrete measure.

Asymptotic Behaviour of Orthogonal Polynomials on the Unit Circle with Asymptotically Periodic Reflection Coefficients

Let {an}n∈N0withan∈C,an+N=anand |an|<1 for alln∈N0, be a periodic sequence of reflection coefficients and let {Pn}n∈N0be the associated sequence of orthogonal polynomials generated byPn+1=zPn−ānP*n. Furthermore let {bn}n∈N0be an asymptotically periodic sequence of reflection coefficients which arises by a perturbation of the sequence {an}n∈N0and thus satisfies the conditions limν→∞bj+νN=ajforj=0, …, N−1, and |bn|<1 for alln∈N0. Let {Pn}n∈N0generated byPn+1=zPn−b&nP*nbe the disturbed orthogonal polynomials. Using the “periodic” polynomials {Pn}n∈N0as a comparison system we derive so-called comparative asymptotics for the disturbed polynomials on and off the support of the disturbed orthogonality measure, which consists essentially of several arcs of the unit circle. As a by-product of these results we obtain asymptotically a description of the location of the zeros of {Pn}n∈N0. Finally, a representation for the absolutely continuous part of the disturbed orthogonality measure is derived, and it is shown that there are at most finitely many point measures if thebn's converge geometrically fast to thean's.