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16
Fine structure of the zeros of orthogonal polynomials, I. A tale of two pictures
 Comm. Pure Appl. Math
"... Abstract. We present a complete theory of the asymptotics of the zeros of OPUC with Verblunsky coefficients αn = �L ℓ=1 Cℓbn ℓ + O((b∆) n) where ∆ < 1 and bℓ  = b < 1. 1. ..."
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Cited by 28 (13 self)
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Abstract. We present a complete theory of the asymptotics of the zeros of OPUC with Verblunsky coefficients αn = �L ℓ=1 Cℓbn ℓ + O((b∆) n) where ∆ < 1 and bℓ  = b < 1. 1.
Two extensions of Lubinsky’s universality theorem
 J. Anal. Math. 105
, 2008
"... Abstract. We extend some remarkable recent results of Lubinsky and Levin– Lubinsky from [−1, 1] to allow discrete eigenvalues outside σess and to allow σess first to be a finite union of closed intervals and then a fairly general compact set in R (one which is regular for the Dirichlet problem). 1 ..."
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Cited by 20 (9 self)
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Abstract. We extend some remarkable recent results of Lubinsky and Levin– Lubinsky from [−1, 1] to allow discrete eigenvalues outside σess and to allow σess first to be a finite union of closed intervals and then a fairly general compact set in R (one which is regular for the Dirichlet problem). 1
Bulk universality and clock spacing of zeros for ergodic Jacobi matrices with a.c. spectrum, Analysis and PDE
"... Abstract. By combining some ideas of Lubinsky with some soft analysis, we prove that universality and clock behavior of zeros for OPRL in the a.c. spectral region is implied by convergence of 1 n Kn(x, x) for the diagonal CD kernel and boundedness of the analog associated to second kind polynomials. ..."
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Cited by 14 (6 self)
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Abstract. By combining some ideas of Lubinsky with some soft analysis, we prove that universality and clock behavior of zeros for OPRL in the a.c. spectral region is implied by convergence of 1 n Kn(x, x) for the diagonal CD kernel and boundedness of the analog associated to second kind polynomials. We then show that these hypotheses are always valid for ergodic Jacobi matrices with a.c. spectrum and prove that the limit of 1 n Kn(x, x) is ρ∞(x)/w(x) where ρ ∞ is the density of zeros and w is the a.c. weight of the spectral measure. 1.
Finite gap Jacobi matrices, I. The isospectral torus
, 2008
"... Let e ⊂ R be a finite union of disjoint closed intervals. In the study of OPRL with measures whose essential support is e, a fundamental role is played by the isospectral torus. In this paper, we use a covering map formalism to define and study this isospectral torus. Our goal is to make a coherent ..."
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Cited by 12 (12 self)
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Let e ⊂ R be a finite union of disjoint closed intervals. In the study of OPRL with measures whose essential support is e, a fundamental role is played by the isospectral torus. In this paper, we use a covering map formalism to define and study this isospectral torus. Our goal is to make a coherent presentation of properties and bounds for this special class as a tool for ourselves and others to study perturbations. One important result is the expression of Jost functions for the torus in terms of theta functions.
Weak convergence of CD kernels and applications
"... Abstract. We prove a general result on equality of the weak limits of the zero counting measure, dνn, of orthogonal polynomials (defined by a measure dµ) and 1 n Kn(x, x)dµ(x). By combining this with Máté–Nevai and Totik upper bounds on nλn(x), we prove some general results on ∫ 1 I nKn(x, x)dµs → 0 ..."
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Cited by 10 (7 self)
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Abstract. We prove a general result on equality of the weak limits of the zero counting measure, dνn, of orthogonal polynomials (defined by a measure dµ) and 1 n Kn(x, x)dµ(x). By combining this with Máté–Nevai and Totik upper bounds on nλn(x), we prove some general results on ∫ 1 I nKn(x, x)dµs → 0 for the singular part of dµ and ∫ w(x) ρE(x)− I n Kn(x, x)  dx → 0, where ρE is the density of the equilibrium measure and w(x) the density of dµ. 1.
The ChristoffelDarboux Kernel
"... A review of the uses of the CD kernel in the spectral theory of orthogonal polynomials, concentrating on recent results. ..."
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Cited by 9 (5 self)
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A review of the uses of the CD kernel in the spectral theory of orthogonal polynomials, concentrating on recent results.
The GelfondSchnirelman Method In Prime Number Theory
 Canad. J. Math
"... The original GelfondSchnirelman method, proposed in 1936, uses polynomials with integer coe#cients and small norms on [0, 1] to give a Chebyshevtype lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for t ..."
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Cited by 4 (4 self)
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The original GelfondSchnirelman method, proposed in 1936, uses polynomials with integer coe#cients and small norms on [0, 1] to give a Chebyshevtype lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for the integral of Chebyshev's #function, expressed in terms of the weighted capacity. This extends previous work of Nair and Chudnovsky, and connects the subject to the potential theory with external fields generated by polynomialtype weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support. 1. Lower bounds for arithmetic functions Let #(x) be the number of primes not exceeding x. The celebrated Prime Number Theorem (PNT), suggested by Legendre and Gauss, states that ##.
Approximation results for reflectionless Jacobi matrices
"... Abstract. We study spaces of reflectionless Jacobi matrices. The main theme is the following type of question: Given a reflectionless Jacobi matrix, is it possible to approximate it by other reflectionless and, typically, simpler Jacobi matrices of a special type? For example, can we approximate by ..."
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Cited by 4 (4 self)
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Abstract. We study spaces of reflectionless Jacobi matrices. The main theme is the following type of question: Given a reflectionless Jacobi matrix, is it possible to approximate it by other reflectionless and, typically, simpler Jacobi matrices of a special type? For example, can we approximate by periodic operators? 1.
Finite gap Jacobi matrices, III. Beyond the Szegő class, in preparation
"... Abstract. Let e ⊂ R be a finite union of ℓ + 1 disjoint closed intervals and denote by ωj the harmonic measure of the j leftmost bands. The frequency module for e is the set of all integral combinations of ω1,..., ωℓ. Let {ãn, ˜ bn} ∞ n=1 be a point in the isospectral torus for e and ˜pn its ortho ..."
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Cited by 3 (2 self)
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Abstract. Let e ⊂ R be a finite union of ℓ + 1 disjoint closed intervals and denote by ωj the harmonic measure of the j leftmost bands. The frequency module for e is the set of all integral combinations of ω1,..., ωℓ. Let {ãn, ˜ bn} ∞ n=1 be a point in the isospectral torus for e and ˜pn its orthogonal polynomials. Let {an, bn} ∞ n=1 be a halfline Jacobi matrix with an = ãn +δan, bn = ˜ bn +δbn. Suppose δan  2 + δbn  2 < ∞ n=1