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26
OPUC on one foot
 Bull. Amer. Math. Soc
, 2005
"... Abstract. We present an expository introduction to orthogonal polynomials on the unit circle (OPUC). 1. ..."
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Cited by 34 (10 self)
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Abstract. We present an expository introduction to orthogonal polynomials on the unit circle (OPUC). 1.
Recurrence and asymptotics for orthogonal rational functions on an interval
 IMA Journal of Numerical Analysis
"... Let {α1, α2,...} be a sequence of real numbers outside the interval [−1, 1] and µ a positive bounded Borel measure on this interval. We introduce rational functions ϕn(x) with poles {α1,..., αn} orthogonal on [−1, 1] and establish some ratio asymptotics for these orthogonal rational functions, i.e. ..."
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Cited by 30 (22 self)
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Let {α1, α2,...} be a sequence of real numbers outside the interval [−1, 1] and µ a positive bounded Borel measure on this interval. We introduce rational functions ϕn(x) with poles {α1,..., αn} orthogonal on [−1, 1] and establish some ratio asymptotics for these orthogonal rational functions, i.e. we discuss the convergence of ϕn+1(x)/ϕn(x) as n tends to infinity under certain assumptions on the measure and the location of the poles. From this we derive asymptotic formulas for the recurrence coefficients in the three term recurrence relation satisfied by the orthonormal functions. 1
Perturbations of orthogonal polynomials with periodic recursion coefficients
, 2007
"... We extend the results of Denisov–Rakhmanov, Szegő–Shohat– Nevai, and Killip–Simon from asymptotically constant orthogonal polynomials on the real line (OPRL) and unit circle (OPUC) to asymptotically periodic OPRL and OPUC. The key tool is a characterization of the isospectral torus that is well ada ..."
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Cited by 27 (15 self)
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We extend the results of Denisov–Rakhmanov, Szegő–Shohat– Nevai, and Killip–Simon from asymptotically constant orthogonal polynomials on the real line (OPRL) and unit circle (OPUC) to asymptotically periodic OPRL and OPUC. The key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations.
Lax pairs for the AblowitzLadik system via orthogonal polynomials on the unit
"... Abstract. In [14] Nenciu and Simon found that the analogue of the Toda system in the context of orthogonal polynomials on the unit circle is the defocusing AblowitzLadik system. In this paper we use the CMV and extended CMV matrices defined in [5] and [13, 14], respectively, to construct Lax pair r ..."
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Cited by 23 (4 self)
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Abstract. In [14] Nenciu and Simon found that the analogue of the Toda system in the context of orthogonal polynomials on the unit circle is the defocusing AblowitzLadik system. In this paper we use the CMV and extended CMV matrices defined in [5] and [13, 14], respectively, to construct Lax pair representations for this system. 1.
Jost functions and Jost solutions for Jacobi matrices, II. Decay and Analyticity
"... Abstract. We provide necessary and sufficient conditions for a Jacobi matrix to produce orthogonal polynomials with Szegő asymptotics off the real axis. A key idea is to prove the equivalence of Szegő asymptotics and of Jost asymptotics for the Jost solution. We also prove L2 convergence of Szegő as ..."
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Cited by 21 (16 self)
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Abstract. We provide necessary and sufficient conditions for a Jacobi matrix to produce orthogonal polynomials with Szegő asymptotics off the real axis. A key idea is to prove the equivalence of Szegő asymptotics and of Jost asymptotics for the Jost solution. We also prove L2 convergence of Szegő asymptotics on the spectrum. 1.
Halfline Schrödinger operators with no bound states
, 2003
"... We consider Schödinger operators on the halfline, both discrete and continuous, and show that the absence of bound states implies the absence of embedded singular spectrum. More precisely, in the discrete case we prove that if ∆+V has no spectrum outside of the interval [−2,2], then it has purely ..."
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Cited by 19 (7 self)
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We consider Schödinger operators on the halfline, both discrete and continuous, and show that the absence of bound states implies the absence of embedded singular spectrum. More precisely, in the discrete case we prove that if ∆+V has no spectrum outside of the interval [−2,2], then it has purely absolutely continuous spectrum. In the continuum case we show that if both −∆+V and −∆−V have no spectrum outside [0, ∞), then both operators are purely absolutely continuous. These results extend to operators with finitely many bound states.
A canonical factorization for meromorphic Herglotz functions on the unit disk and sum rules for Jacobi matrices
 J. Funct. Anal
"... ..."
Rank one perturbations and the zeros of paraorthogonal polynomials on the unit circle, preprint
"... Abstract. We prove several results about zeros of paraorthogonal polynomials using the theory of rank one perturbations of unitary operators. In particular, we obtain new details on the interlacing of zeros for successive POPUC. 1. ..."
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Cited by 14 (6 self)
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Abstract. We prove several results about zeros of paraorthogonal polynomials using the theory of rank one perturbations of unitary operators. In particular, we obtain new details on the interlacing of zeros for successive POPUC. 1.
Meromorhic Szegő functions and asymptotic series for Verblunsky coefficients
, 2005
"... Abstract. We prove that the Szegő 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. Basic ..."
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Cited by 13 (7 self)
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Abstract. We prove that the Szegő 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, z1... zℓ¯zℓ−1... ¯z2ℓ−1 with zj in the set. The proofs use nothing more than iterated Szegő recursion at z and 1/¯z. 1.