In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed Orthogonal polynomials on the unit circle are not discussed

Orthogonal Polynomials

In the choice of an estimator for the spectrum of a stationary time series from a finite sample of the process, the problems of bias control and consistency, or "smoothing," are dominant. In this paper we present a new method based on a "local" eigenexpansion to estimate the spectrum in terms of the solution of an integral equation. Computationally this method is equivalent to using the weishted average of a series of direct-spectrum estimates based on orthogonal data windows (discrete prolate spheroidal sequences) to treat both the bias and smoothing problems. Some of the attractive features of this estimate are: there are no arbitrary windows; it is a small sample theory; it is consistent; it provides an analysis-of-variance test for line components; and it has high resolution. We also show relations of this estimate to maximum-likelihood estimates, show that the estimation capacity of the estimate is high, and show applications to coherence and polyspectrum estimates.

Spectrum estimation and harmonic analysis

We consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn(x)dx = e−nV(x)dx on the line as n ∞. The potentials V are assumed to be real analytic, with sufficient growth at infinity. The principle results concern Plancherel-Rotach-type asymptotics for the orthogonal polynomials down to the axis. Using these asymptotics, we then prove universality for a variety of statistical quantities arising in the theory of random matrix models, some of which have been considered recently in [31] and also in [4]. An additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials (see also [4]).



The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem following [19, 20]. The Riemann-Hilbert problem is analyzed in turn using the steepest-descent method introduced in [12] and further developed in [11, 13]. A critical role in our method is played by the equilibrium measure dμV for V as analyzed in [8]. © 1999 John Wiley & Sons, Inc.

Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory

We consider asymptotics of orthogonal polynomials with respect to weights w(x)dx= e Q(x) dx on the real line, where Q(x)=∑ 2m k=0 qkx k , q2m> 0, denotes a polynomial of even order with positive leading coefficient. The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem following [22, 23]. We employ the steepest-descent-type method introduced in [18] and further developed in [17, 19] in order to obtain uniform Plancherel-Rotach-type asymptotics in the entire complex plane, as well as asymptotic formulae for the zeros, the leading coefficients, and the recurrence coefficients of the orthogonal polynomials. c 1999 John Wiley & Sons, Inc.

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Strong asymptotics of orthogonal polynomials with respect to exponential weights

Here I = S j (a2j 1,a2j) andI(y) is the characteristic function of the set I. In the Gaussian Unitary Ensemble (GUE) the probability that no eigenvalues lie in I is equal to �(a). Also �(a) is a tau-function and we present a new simplified derivation of the system of nonlinear completely integrable equations (the aj's are the independent variables) that were first derived by Jimbo, Miwa, Mori, and Sato in 1980. In the case of a single interval these equations are reducible to a Painleve V equation. For large s we give an asymptotic formula for E2(n;s), which is the probability in the GUE that exactly n eigenvalues lie in an interval of length s. These notes provide an introduction to that aspect of the theory of random matrices dealing with the distribution of eigenvalues. To first orient the reader, we present in Sec. II some numerical experiments that illustrate some of the basic aspects of the subject. In Sec. III we introduce the invariant measures for the three "circular ensembles" involving unitary matrices. We also define the level spacing distributions and express these distributions in terms of a particular Fredholm determinant. In Sec. IV we explain how these measures are modified for the orthogonal polynomial ensembles. In Sec. V we discuss the universality of these level spacing distribution functions in a particular scaling limit. The discussion up to this point (with the possible exception of Sec. V) follows the well-known path pioneered by Hua, Wigner, Dyson, Mehta and others who first developed this theory (see, e.g., the reprint volume of Porter (34) and Hua (17)). This, and much more, is discussed in Mehta's book (25)—the classic reference in the subject. An important development in random matrices was the discovery by Jimbo, Miwa, Mori, and Sato (21) (hereafter referred to as JMMS) that the basic Fredholm determinant mentioned above is a �-function in the sense of the Kyoto School. Though it has been some twelve years since (21) was published, these results are not widely appreciated by the practitioners of random matrices. This is due no doubt to the complexity of their paper. The methods of JMMS are methods of discovery; but now that we know the result, simpler proofs can be constructed. In Sec. VI we give such a proof of the JMMS equations. Our proof is a simplification and generalization of Mehta's (27) simplified proof of the single interval case. Also our methods build on the earlier work of Its, Izergin, Korepin, and Slavnov (18) and Dyson (12). We include in this section a discussion of the connection between the JMMS equations and the integrable Hamiltonian systems that appear in the geometry of quadrics and spectral theory as developed by Moser (31). This section concludes with a discussion of the case of a single interval (viz., probability that exactly n eigenvalues lie in a given interval). In this case the JMMS equations can be reduced to a single ordinary differential equation—the Painleve V equation. Finally, in Sec. VII we discuss the asymptotics in the case of a large single interval of the various level spacing distribution functions (4,38,28). In this analysis both the Painleve representation and new results in Toeplitz/Wiener- Hopf theory are needed to produce these asymptotics. We also give an approach based on the asymptotics of the eigenvalues of the basic linear integral operator (14,25,35). These results are then compared with the continuum model calculations of Dyson (12).

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Introduction to Random Matrices

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Mean convergence of Lagrange interpolation. III

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Distribution of zeros of orthogonal polynomials

Characterization Theorems for Orthogonal Polynomials.- Orthogonal Polynomials in Coding Theory and Algebraic Combinatorics.- Orthogonal Polynomials, Pade Approximations and Julia Sets.- The Three Term Recurrence Relation and Spectral Properties of Orthogonal Polynomials.- On the Role of Orthogonal Polynomials on the Unit Circle in Digital Signal.- A Survey on the Theory of Orthogonal Systems and Some Open Problems.- Orthogonal Polynomials and Functional Analysis.- Using Symbolic Computer Algebraic Systems to Derive Formulas Involving Orthogonal Polynomials and Other Special Functions.- Computational Aspects of Orthogonal Polynomials.- The Recursion Method and the Schroedinger Equation.- Birth and Death Processes and Orthogonal Polynomials.- Orthogonal Polynomials in Connection with Quantum Groups.- The Approximate Approach to Orthogonal Polynomials for Weights on (-?,?).- Orthogonal Polynomials Associated with Root Systems.- Some Extensions of the Beta Integral and the Hypergeometric Function.- Orthogonal Matrix Polynomials.- Orthogonal Polynomial from a Complex Perspective.- Nth Root Asymptotic Behavior of Orthonormal Polynomials.- An Introduction to Group Representations and Orthogonal Polynomials.- Asymptotics for Orthogonal Polynomials and Three - Term Recurrences.- List of of Participants.- Scientific Program.

Orthogonal polynomials : theory and practice

A characterization of the classical orthogonal poly-nominals, R.P.Agarwal and G.V.Milovanovic monotone approximation by pseudopolynomicals, G.A.Anastassiou complex-valued spline functions defined on an open complex set, bounded, and multiply connected, MAtteia on the norm of the best approximation operator, M.W.Bartlett and J.J.Swetits remainders for boolean interpolation, G.Baszenski and F.J.Delvos some spectral approximations of mono-dimensional fourth-order problems, C.Bernardi and Y.Manday able summability of entire functions of exponential type, W.T.Butterworth mean and uniform convergence of quadrature rules for evaluating the finite Hilbert Transform, G.Criscuolo and G.Mastroianni some properties of a rational operator of Bernstein-type, B.D.Vecchia anti-periodic interpolation of uniform meshes, F.J.Delvos a nodal spline generalization of the Lagrange Interpolant, J.M.De Villiers and C.H.Rohwer entire functions associated with eq where q is of slower than polynomial growth, K.A.Driver quadrature formulae with Birkhoff-type data on equidistant nodes for 2-periodic functions, D.P.Dryanov et al rate of convergence of moving least squares interpolation methods - the univariate case, R.Farwig on proximinal subspaces of continuous function spaces, M.Feder discretization of convolution and reconstruction of band-limited functions from irregular sampling, H.G.Feichtinger Chebyshev Polynomials are not always optimal, B.Fischer and R.Freund Banach Spaces with a basic inequality property and the best compact approximation property, R.J.Fleming and J.E.Jamison M-ideals and a basic inequality in Banach Spaces, R.J.Fleming and J.E.Jamison strong asymptotics and the limit distribution of the zeros of Jacobi Polynomials P, W.Gawronski and B.Shawyer rational approximation of a class of infinite dimensional systems, K.Glover et al the Posse-Markov-Stieltjies Inequality for Turan type quadratic sums, L Gori on the convergence of Quasi-Gaussian Functionals, L.Gori and E.Santi equivalence of a weighted modulus of smoothness and a modified weighted K-functional, M.Heilman and M.W.Muller best L-approximation by convex functions of several variables, S.Legg and D.Townsend approximation properties of beta operators, M.K.Khan some properties of a Bernstein-type operator of Blimann, Butzer and Hahn, R.A.Kahn best L-approximations, K.Kitahara generalized series expansions of a function, A.Knopfmacher the dimension of the space of periodic splines on the regular hexagonal lattice, F.Krebs on Fejer Means with respect to orthogonal polynomials - a hypergroup theoretic approach, R.Lasser and J.Obermaier degree of approximation by polynomials with restricted co-efficients, D.Leviatan. (Part contents) ...

Progress in approximation theory

It is shown that the Christoffel functions arising from the Szegd extremum problem associated with a finite positive Borel measure on the interval [- wr, wr) satisfy

Paul Nevai

Papers

Mean convergence of Lagrange interpolation. III

Distribution of zeros of orthogonal polynomials

Orthogonal polynomials : theory and practice

Progress in approximation theory

Szegö’s extremum problem on the unit circle