Measures for orthogonal polynomials with unbounded recurrence coefficients

Aptekarev, A. I.; Geronimo, Jeffrey S.

doi:10.1016/j.jat.2016.02.009

Cited by 20 publications

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References 23 publications

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“…Some kinds of this result were obtained by different methods in works [4,5,9,12]. Here we give independent, new and simple proof of it.…”

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confidence: 63%

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On one condition of absolutely continuous spectrum for self-adjoint operators and its applications

Ianovich¹

2018

OpMath

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Abstract. In this work the method of analyzing of the absolutely continuous spectrum for self-adjoint operators is considered. For the analysis it is used an approximation of a self-adjoint operator A by a sequence of operators An with absolutely continuous spectrum on a given interval [a, b] which converges to A in a strong sense on a dense set. The notion of equi-absolute continuity is also used. It was found a sufficient condition of absolute continuity of the operator A spectrum on the finite interval [a, b] and the condition for that the corresponding spectral density belongs to the class Lp [a, b] (p ≥ 1). The application of this method to Jacobi matrices is considered. As one of the results we obtain the following assertion: Under some mild assumptions, suppose that there exist a constant C > 0 and a positive function g(x) ∈ Lp [a, b] (p ≥ 1) such that for all n sufficiently large and almost all x ∈ [a, b] the estimate≤ bn(P 2 n+1 (x) + P 2 n (x)) ≤ C holds, where Pn(x) are 1st type polynomials associated with Jacobi matrix (in the sense of Akhiezer) and bn is a second diagonal sequence of Jacobi matrix. Then the spectrum of Jacobi matrix operator is purely absolutely continuous on [a, b] and for the corresponding spectral density f (x) we have f (x) ∈ Lp [a, b].

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“…Some kinds of this result were obtained by different methods in works [4,5,9,12]. Here we give independent, new and simple proof of it.…”

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confidence: 63%

“…Such matrices A n were used for approximation in works [3,4,8,19]. Note that they were used also in work [6], where the absolutely continuous spectrum of Jacobi matrices was also considered, but in connection with commutation relations rather than with approximation.…”

Section: Lemma 22mentioning

confidence: 99%

On one condition of absolutely continuous spectrum for self-adjoint operators and its applications

Ianovich¹

2018

OpMath

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“…[3,5,6,12,13,14,25]. As far as the approximation of µ is concerned, the only result known to the author is [1]. In Section 3 we prove the following theorem.…”

Section: Introductionmentioning

confidence: 98%

“…Consider a sequence (p n : n ∈ N) of polynomials defined by (1) p −1 (x) = 0, p 0 (x) = 1, a n−1 p n−1 (x) + b n p n (x) + a n p n+1 (x) = xp n (x) (n ≥ 0) for sequences a = (a n : n ∈ N) and b = (b n : n ∈ N) satisfying a n > 0 and b n ∈ R. The sequence (1) is orthonormal in L 2 (µ) for a Borel measure µ on the real line. We are interested in the case when the sequence a is unbounded and the measure µ is unique.…”

Section: Introductionmentioning

confidence: 99%

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Periodic perturbations of unbounded Jacobi matrices II: Formulas for density

Świderski

2017

Journal of Approximation Theory

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Abstract. We give formulas for the density of the measure of orthogonality for orthonormal polynomials with unbounded recurrence coefficients. The formulas involve limits of appropriately scaled Turán determinants or Christoffel functions. Exact asymptotics of the polynomials and numerical examples are also provided. IntroductionConsider a sequence (p n : n ∈ N) of polynomials defined by(1)for sequences a = (a n : n ∈ N) and b = (b n : n ∈ N) satisfying a n > 0 and b n ∈ R. The sequence (1) is orthonormal in L 2 (µ) for a Borel measure µ on the real line. We are interested in the case when the sequence a is unbounded and the measure µ is unique.When it holds, we want to find conditions on the sequences a and b assuring absolute continuity of µ and a constructive formula for its density. In the case when the sequences a and b are bounded, there are several approaches to an approximation of the density of µ. One is obtained by means of N -shifted Turán determinants, i.e. expressions of the formfor positive N (see [19,8,24]). Another by Christoffel functions, i.e.(see [18,23]).2010 Mathematics Subject Classification. Primary: 42C05.

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Orthogonal Polynomials with Periodically Modulated Recurrence Coefficients in the Jordan Block Case II

Świderski

Trojan

2023

Constr Approx

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We study Jacobi matrices with N-periodically modulated recurrence coefficients when the sequence of N-step transfer matrices is convergent to a non-trivial Jordan block. In particular, we describe asymptotic behavior of their generalized eigenvectors, we prove convergence of N-shifted Turán determinants as well as of the Christoffel–Darboux kernel on the diagonal. Finally, by means of subordinacy theory, we identify their absolutely continuous spectrum as well as their essential spectrum. By quantifying the speed of convergence of transfer matrices we were able to cover a large class of Jacobi matrices. In particular, those related to generators of birth–death processes.

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Measures for orthogonal polynomials with unbounded recurrence coefficients

Abstract: Systems of orthogonal polynomials whose recurrence coefficients tend to infinity are considered. A summability condition is imposed on the coefficients and the consequences for the measure of orthogonality are discussed. Also discussed are asymptotics for the polynomials.

Cited by 20 publications

References 23 publications

On one condition of absolutely continuous spectrum for self-adjoint operators and its applications

On one condition of absolutely continuous spectrum for self-adjoint operators and its applications

Periodic perturbations of unbounded Jacobi matrices II: Formulas for density

Orthogonal Polynomials with Periodically Modulated Recurrence Coefficients in the Jordan Block Case II

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