Spectral Properties of Unbounded Jacobi Matrices with Almost Monotonic Weights

Journal of Approximation Theory

Trojan

2017

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“…In proving Theorem A and Theorem B we further develop the method used in [29,Theorem 4.1]. This approach allows us to better understand the reason why there might be a spectral gap.…”

Section: The Asymptotic Behaviour Of Generalized Eigenvectors Impliesmentioning

confidence: 99%

Periodic perturbations of unbounded Jacobi matrices I: Asymptotics of generalized eigenvectors

Journal of Approximation Theory

Trojan

2017

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“…The first main result of this article is Theorem 4, which generalizes the results obtained in [26] to the operator case. Its formulation involves an additional parameter sequence α = (α n : n ≥ 0).…”

Section: Introductionmentioning

confidence: 80%

“…The theory of block Jacobi matrices is much less developed than the scalar ones, i.e., corresponding to H = C. The aim of this paper is to provide extensions of results obtained in [26,28] for H = R to the case of arbitrary H. It is of interest as we provide new results even for H = C d with d ≥ 1, i.e., the most common (apart from R) studied case.…”

Section: Introductionmentioning

confidence: 91%

“…The method of the proofs of the presented theorems is based on an extension of the techniques used in [26,28]. In these articles, one examines the positivity or the convergence of sequences of quadratic forms on R 2 acting on the vector of two consecutive values of a generalized eigenvector u associated with λ ∈ ⊂ R; i.e.,…”

Section: Theorem 3 Let the Assumptions Of Theoremmentioning

confidence: 99%

See 1 more Smart Citation

Spectral Properties of Block Jacobi Matrices

2018

Constr Approx

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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].…”

Section: Introductionmentioning

confidence: 99%

Periodic perturbations of unbounded Jacobi matrices II: Formulas for density

Journal of Approximation Theory

2017

Self Cite

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.