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

Świderski, Grzegorz; Trojan, Bartosz

doi:10.1016/j.jat.2017.01.003

Cited by 18 publications

(33 citation statements)

References 23 publications

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“…But according to [17] one has 0 ∈ supp(µ) = R. A more general example may be found in [26,Example 4]. Observe that Corollary 3 gives not only sufficient conditions when the conjecture is correct but also provides the value of the limit.…”

Section: Corollary 3 Under the Conditions Of Corollary 1 Or Corollarmentioning

confidence: 99%

“…Finally, [26,Corollary 2] implies that S n (x) converges uniformly to g(x), which is a continuous function without zeros. It shows (18).…”

Section: Then For Eachmentioning

confidence: 99%

“…Furthermore, we do not use complex analysis in our proofs. But more importantly, one can take N > 1, which allows to cover many perturbations considered in the literature (see [26,Section 4.3] for details).…”

Section: Introductionmentioning

confidence: 99%

“…The approximating measures can be obtained from classical theorems concerning the bounded case. Then, by detailed analysis of the proofs from [26] implying convergence of Turán determinants, we…”

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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Section: Corollary 3 Under the Conditions Of Corollary 1 Or Corollarmentioning

confidence: 99%

“…Finally, [26,Corollary 2] implies that S n (x) converges uniformly to g(x), which is a continuous function without zeros. It shows (18).…”

Section: Then For Eachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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“…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: 97%

Spectral Properties of Block Jacobi Matrices

Świderski

2018

Constr Approx

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Uni-asymptotic Linear systems and Jacobi Operators

Moszyński

2019

Integr. Equ. Oper. Theory

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A family {Us} s∈S of bounded linear operators in a normed space X is uni-asymptotic, when all its trajectories {Usx} s∈S with x = 0 have the same norm-asymptotic behavior (see 1.5); {Us} s∈S is tight, when the operator norm and the minimal modulus of Us have the same asymptotic behavior (see 1.6). We prove that uni-asymptoticity is equivalent to tightness if dim X < +∞, and that the finite dimension is essential. Some other conditions equivalent to uni-asymptoticity are provided, including asymptotic formulae for the operator norm and for the trajectories, expressed in terms of determinants det Us (see Theorem 1.7). We find a connection of these abstract results with some results and notions from spectral theory of Jacobi operators, e.g., with the H-class property for transfer matrix sequence.Mathematics Subject Classification. 47D06, 47B36, 81Q10.1 In this paper we identify indexed family {fs} s∈S with the function on S given by: S s −→ fs. 23 Page 2 of 15 M. Moszyński IEOTAbove S represents the set of "admissible moments of time", and X-the set of "admissible states" for the process. Hence, in this paper, for fixed x ∈ X, we use the name trajectory (or orbit), when we consider the family {U s x} s∈S of elements of X, obtained by a family {U s } s∈S of operators. Let us stress, however, that the actual role of the mathematical objects X, S, {U s } s∈S and {U s x} s∈S can be essentially different than their typical role as above-that is, as in mathematical descriptions of some real processes, e.g, in physics etc. As we shall see in Sect. 2 (see 2.28), those objects can be also useful for mathematical description of some "purely mathematical" processes for purely mathematical goals.In this paper we consider only the linear case, i.e., X is a Banach space here and all the operators U s are linear (and we shall mainly assume finite dimension of X). However, some notions introduced here have also sense in non-linear case, and they could be worth future research.Section 1 is devoted to some abstract studies of so-called uni-asymptotic families of operators, i.e., such families {U s } s∈S that all its non-zero trajectories {U s x} s∈S have "the same norm-asymptotic behavior". This precisely means, that for any non-zero initial conditions x, y ∈ X we have3). Although the word "asymptotic" may not fit well to the above definition for the general S, it makes sense for some ordered S-s, e.g., for S = N n0 (see 0.1). The name "uni-asymptotic" is used also for linear spaces consisting of some functions f = {f (s)} s∈S from S into X, with a natural analogic meaning. This section contains the main results of the paper-Theorems 1.5 and 1.7 on some equivalent conditions for uni-asymptoticity in finite dimension. One of the most important observations is that the uni-asymptoticity is equivalent to tightness, where {U s } s∈S is tight, when the operator norm and the minimal modulus of U s have the same asymptotic behavior (see 1.6). Examples showing the necessity of the dim X < ∞ assumption are also provided. They show as well, th...

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

Abstract: Abstract. We study asymptotics of generalized eigenvectors associated with Jacobi matrices. Under weak conditions on the coefficients we identify when the matrices are self-adjoint and we show that they satisfy strong non-subordinacy condition.

Cited by 18 publications

References 23 publications

Periodic perturbations of unbounded Jacobi matrices II: Formulas for density

Periodic perturbations of unbounded Jacobi matrices II: Formulas for density

Spectral Properties of Block Jacobi Matrices

Uni-asymptotic Linear systems and Jacobi Operators

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