We present a complete theory of the asymptotics of the zeros of OPUC with
Verblunsky coefficients $\alpha_n = \sum_{\ell=1}^L C_\ell b_\ell^n +
O((b\Delta)^n)$ where $\Delta <1$ and $\abs{b_\ell} = b<1$.Comment: Keywords: orthogonal polynomials, Jacobi matrices, CMV matrice

Barrios Rolanía

Barry Simon

Geronimus

Mhaskar

Nevai

arXiv.org e-Print Archive

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Caltech Authors

English

Fine Structure of the Zeros of Orthogonal Polynomials, II. OPUC With
  Competing Exponential Decay

We review recent work on zeros of orthogonal polynomials

Simon, Barry

A Riemann-Hilbert approach to some theorems on Toeplitz operators and orthogonal polynomials,

Aizenman's theorem for orthogonal polynomials on the unit circle, to appear in Const.

Almost periodic Schrodinger operators, II. The integrated density of states,

Asymptotic behaviour of solutions of linear recurrences and sequences of Mobius-transformations,

Asymptotics of orthogonal polynomials inside the unit circle and Szego-Pade approximants,

Eigenvalue estimates for non-normal matrices and the zeros of random orthogonal polynomials on the unit circle,

Fine structure of the zeros of orthogonal polynomials, 1. A tale of two pictures,

Fine structure of the zeros of orthogonal polynomials, II. OPUC with competing exponential decay, to appear in

Fine structure of the zeros of orthogonal polynomials, III. Periodic recursion coefficients, to appear in

Fine structure of the zeros of orthogonal polynomials, IV. A priori bounds and clock behavior, in preparation.

General Orthogonal Polynomials,

General Orthogonal Polynomials, Cambridge Uni versity Press,

Hochstadt,.On-ihe theory of Hill's matrices and related inverse spectral problem's,

Internal Lifschitz tails,

Limits of zeros of orthogonal polynomials on the circle,

Local fluctuation of the spectrum of a multidimensional Ander son tight binding model,

Local fluctuation of the spectrum of a multidimensional Anderson tight binding model,

Localization at large disorder and at ex treme energies: An elementary derivation,

Localization at large disorder and at extreme energies: An elementary derivation,

Localization at weak disorder: Some elementary bounds,

Meromorphic Szego functions and asymptotic series for Verblun sky coefficients,

Meromorphic Szego functions and asymptotic series for Verblunsky coefficients,

Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle,

Observation of magnetic level repulsion in Fe6:Li molecular antiferromagnetic rings,

On Bernstein-Szego orthogonal polynomials on several in tervals, II. Orthogonal polynomials with periodic recurrence coeffcients,

On Bernstein-Szego orthogonal polynomials on several intervals, II. Orthogonal polynomials with periodic recurrence coeffcients,

On interpolation. III. Interpolatory theory of poly nomials,

On interpolation. III. Interpolatory theory of polynomials,

On the density of states of Schrodinger oper ators with a random potential,

On the density of states of Schrodinger operators with a random potential,

On the distribution of zeros of polynomials 652 orthogonal on the unit circle,

On the distribution of zeros of polynomials orthogonal on the unit circle,

On the meatrure of gaps and spectra for discrete 1D Schrodinger operators,

On the zeros of generalized Jacobi polynomials. The heritage of P. L. Chebyshev: a Festschrift in honor of the 70th birthday of T.

On the zeros of Jacobi polynomials,

On the zeros of orthogonal polynomials: The elliptic case,

Operators with singular continuous spectrum. IV. Hausdorff dimensions, rank one perturbations, and localization,

Orthogonal polynomials and their zeros,

Orthogonal polynomials from a complex perspective,

Orthogonal polynomials from a complex perspective, in Orthogonal Polynomials: Theory and Practice

Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory,

Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, AMS Colloquium Series,

Orthogonal Polynomials on the Unit Circle, Part 2: Spectral The ory, AMS Colloquium Series,

Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory, AMS Colloquium Series,

Orthogonal Polynomials,

Quadrature formula and zeros of para-orthogonal polynomials on the unit circle,

Random Matrices, 2nd ed.,

Repulsion of energy levels&quot; in complex atomic spectra,

Spectra of random selfadjoint operators,

Szego difference equations, transfer matrices and orthogonal polynomials on the unit circle,

Szego orthogonal polynomials with respect to an analytic weight in canonical representation and strong asymptotics,

The local structure of the spectrum of the one-dimensional Schrodinger operator,

The spectrum of Jacobi matrices,

The statistical distribution of the zeros of random paraorthog onal polynomials on the unit circle, to appear in

The statistical distribution of the zeros of random paraorthogonal polynomials on the unit circle, to appear in

Theory of Nonlinear Lattices, 2nd edition,

Uber nach Poly nomen fortschreitende Reihen, Sitzungsberichte der Bayerischen Akademie der Wissenschaften,

Uniform asymptotics of derivatives of orthQgonay-polynomials based on generalized Jacobi weights,

Zeros of polynomials orthogonal on several intervals,

Zeros of Random Orthogonal Polynomials on the Unit Circle, Ph.D. dissertation

Fine Structure of the Zeros of Orthogonal Polynomials:  A Review

We prove that the Szeg\H{o} function, $D(z)$, of a measure on the unit circle
is entire meromorphic if and only if the Verblunsky coefficients have an
asymptotic expansion in exponentials. We relate the positions of the poles of
$D(z)^{-1}$ to the exponential rates in the asymptotic expansion. Basically,
either set is contained in the sets generated from the other by considering
products of the form, $z_1 ... z_\ell \bar z_{\ell-1}... \bar z_{2\ell-1}$ with
$z_j$ in the set. The proofs use nothing more than iterated Szeg\H{o} recursion
at $z$ and $1/\bar z$

Meromorphic Szego functions and asymptotic series for Verblunsky
  coefficients

We discuss asymptotics of the zeros of orthogonal polynomials on the real line and on the unit circle when the recursion coefficients are periodic. The zeros on or near the absolutely continuous spectrum have a clock structure with spacings inverse to the density of zeros. Zeros away from the a.c. spectrum have limit points mod p and only finitely many of them

Fine structure of the zeros of orthogonal polynomials III: Periodic recursion coefficients

We present an expository introduction to orthogonal polynomials on the unit
circle

Bulletin (new series) of the American Mathematical Society

A .B .A l e k s a n d r o v ,Multiplicity of boundary values of inner functions,

A Borg-type theorem associated with orthogonal polynomials on the unit circle, preprint,

A certain inverse problem for Sturm-Liouville operators,

A convergence equivalence related to polynomials orthogonal on the unit circle,

A norm inequality for a “ﬁnite-section” Wiener-Hopf equation,

A Riemann-Hilbert approach to some theorems on Toeplitz operators and orthogonal polynomials, in preparation.

A simple proof of “Favard’s theorem” on the unit circle,

A special class of polynomials orthogonal on the unit circle including the associated polynomials,

A theorem of Gabor Szeg˝ o,

A theorem of Gabor Szego˝,

A view of three decades of linear ﬁltering theory,

Aizenman’s theorem for orthogonal polynomials on the unit circle, to appear in Const.

Aizenman’s theorem for orthogonal polynomials on the unit circle,t oa p p e a ri n Const.

Algebraic aspects of increasing subsequences,

Algebraic curves and nonlinear diﬀerence equations, Uspekhi Mat.

An inverse problem associated with polynomials orthogonal on the unit circle,

Appendix to “Theta-functions and nonlinear equations” by B.A.

Asymptotic behaviour of orthogonal polynomials on the unit circle with asymptotically periodic reﬂection coeﬃcients,

Asymptotic behaviour of orthogonal polynomials on the unit circle with asymptotically periodic reﬂection coeﬃcients, II. Weak asymptotics,J .A p -prox.

Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda chains,

Asymptotic properties of polynomials orthogonal on the circle with weights not satisfying the Szeg˝ o condition,

Asymptotic properties of polynomials orthogonal on the circle with weights not satisfying the Szego˝ condition,

Asymptotics for the ratio of leading coeﬃcients of orthonormal polynomials on the unit circle,

Averages of characteristic polynomials in random matrix theory,

Beitra¨ge zur Theorie der Toeplitzschen Formen, I, II,

Bounds on the density of states in disordered systems,

Bounds on the density of states in disordered systems,Z .P h y s .B44

Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions,

Characterization of measures associated with orthogonal polynomials on the unit circle,

Classiﬁcation theorems for general orthogonal polynomials on the unit circle,

Deformation of minimal polynomials and approximation of several intervals by an inverse polynomial mapping,

die im Innern des Einheitskreises beschr¨ ankt sind, I, II,J . Reine Angew.

Extrapolation, Interpolation, and Smoothing of Stationary Time Series. With Engineering Applications,

Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle, Linear Algebra Appl.

Half-line Schr¨ odinger operators with no bound states,

Half-line Schro¨dinger operators with no bound states,

Inverse images of polynomial mappings and polynomials orthogonal on them,

Lax pairs for the Ablowitz-Ladik system via orthogonal polynomials on the unit circle, to appear in

M´ emoire sur l’approximation des fonctions de tr` es-grands nombres, et sur une classe ´ etendue de d´ eveloppements en s´ erie,

Matrix models for circular ensembles,

Maximum entropy and the moment problem,

Me´moire sur l’approximation des fonctions de tre`s-grands nombres, et sur une classe e´tendue de de´veloppements en se´rie,

Meromorphic Szeg˝ o functions and asymptotic series for Verblunsky coeﬃcients,

Meromorphic Szego˝ functions and asymptotic series for Verblunsky coefficients,

Multiplicity of boundary values of inner functions,

Nonlinear equations of Korteweg-de Vries type, ﬁnite-band linear operators and Abelian varieties; Uspekhi Mat.

Norbert Wiener and the development of mathematical engineering, in “The Legacy of Norbert Wiener: A Centennial Symposium,”

Norbert Wiener and the development of mathematical engineering,i n“ T h e Legacy of Norbert Wiener: A Centennial Symposium,”

O p t i m a lt a i le s t i m a t e sf o r directed last passage site percolation with geometric random variables,

o, ¨ Uber den asymptotischen Ausdruck von Polynomen, die durch eine Orthogonalit¨ atseigenschaft deﬁniert sind,

o, Beitr¨ age zur Theorie der Toeplitzschen Formen, I, II,M a t h

o, Orthogonal Polynomials,

On a generalization of some investigations of

On a generalization of some investigations of G.

On a problem of extrapolation of A.N.

On certain Hermitian forms associated with the Fourier series of a positive function,

On orthogonal polynomials,

On polynomials orthogonal on the circle, on trigonometric moment problem, and on allied Carath´ eodory and Schur functions,

On polynomials orthogonal on the circle, on trigonometric moment problem, and on allied Carathe´odory and Schur functions,

On positive harmonic functions (second paper),

On positive harmonic functions: A contribution to the algebra of Fourier series,

On Rakhmanov’s theorem for Jacobi matrices,

On some equations in finite differences and the corresponding system of orthogonal polynomials,

On some equations in ﬁnite diﬀerences and the corresponding system of orthogonal polynomials,Z a p .M a t .O t d e lF i z . - M a t .F a k .iK h a r ’ k o v .M a t .O b ˇ sˇ

On Szeg˝ o’s limit theorem,

On Szego˝’s limit theorem,

On the asymptotics of the ratio of orthogonal polynomials,

On the asymptotics of the ratio of orthogonal polynomials, II,M a t h .

On the determination of a diﬀerential equation from its spectral function,

On the distribution of the length of the longest increasing subsequence of random permutations,

On the distribution of the length of the second row of a Young diagram under Plancherel measure,

On the trigonometric moment problem,

Optimal tail estimates for directed last passage site percolation with geometric random variables,

Orthogonal and extremal polynomials on several intervals,

Orthogonal polynomials on arcs of the unit circle,

Orthogonal polynomials on arcs of the unit circle, II. Orthogonal polynomials with periodic reﬂection coeﬃcients,

Orthogonal polynomials on the circumference and arcs of the circumference,

Orthogonal Polynomials on the Unit Circle, Part 1:

Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory,A M SC o l -loquium Series,

Orthogonal polynomials on the unit circle, Szeg˝ o diﬀerence equations and spectral theory of unitary matrices, second Doctoral thesis,

Orthogonal polynomials on the unit circle, Szego˝ difference equations and spectral theory of unitary matrices, second Doctoral thesis,

Orthogonal polynomials with ratio asymptotics,

Orthogonal Polynomials, Pergamon Press,

Orthogonal Polynomials: Estimates, Asymptotic Formulas, and Series of Polynomials Orthogonal on the Unit Circle and on an Interval, Consultants Bureau,

Perturbation of orthogonal polynomials on the unit circle—a survey,

Polynomials associated with measures in the complex plane,

Polynomials associated with measures in the complex plane,J .M a t h

Polynomials Orthogonal on a Circle and Their Applications,

Polynomials Orthogonal on a Circle and Their Applications,A m e r . M a t h .S o c .T r a n s l a t i o n1954

Polynomials orthogonal on the unit circle with random recurrence coefﬁcients,

Quasitriangular matrices,

Random Matrices,

OPUC on One Foot