Finite gap Jacobi matrices, III. Beyond the Szegő class
(2012) In Constructive Approximation 35(2). p.259-272- Abstract
- Let e⊂R be a finite union of ℓ+1 disjoint closed intervals, and denote by ω j the harmonic measure of the j left-most bands. The frequency module for e is the set of all integral combinations of ω 1,…,ω ℓ . Let {a˜n,b˜n}∞n=−∞ be a point in the isospectral torus for e and p˜n its orthogonal polynomials. Let {an,bn}∞n=1 be a half-line Jacobi matrix with an=a˜n+δan , bn=b˜n+δbn . Suppose
∑n=1∞∣δan∣2+∣δbn∣2<∞
and ∑Nn=1e2πiωnδan , ∑Nn=1e2πiωnδbn have finite limits as N→∞ for all ω in the frequency module. If, in addition, these partial sums grow at most subexponentially with respect to ω, then for z∈ℂ∖ℝ, pn(z)/p˜n(z) has a limit as n→∞. Moreover, we show that there are non-Szegő class J’s for which this holds.
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/3448235
- author
- Christiansen, Jacob Stordal LU ; Simon, Barry and Zinchenko, Maxim
- publishing date
- 2012
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Szegő asymptotics, Orthogonal polynomials, Almost periodic sequences, Slowly decaying perturbations
- in
- Constructive Approximation
- volume
- 35
- issue
- 2
- pages
- 259 - 272
- publisher
- Springer
- external identifiers
-
- scopus:84857364817
- ISSN
- 0176-4276
- DOI
- 10.1007/s00365-012-9152-4
- language
- English
- LU publication?
- no
- id
- 258409a1-1b82-45b3-bb19-f3305ac8133c (old id 3448235)
- alternative location
- http://link.springer.com/article/10.1007%2Fs00365-012-9152-4
- date added to LUP
- 2016-04-01 11:07:10
- date last changed
- 2022-03-20 02:56:13
@article{258409a1-1b82-45b3-bb19-f3305ac8133c, abstract = {{Let e⊂R be a finite union of ℓ+1 disjoint closed intervals, and denote by ω j the harmonic measure of the j left-most bands. The frequency module for e is the set of all integral combinations of ω 1,…,ω ℓ . Let {a˜n,b˜n}∞n=−∞ be a point in the isospectral torus for e and p˜n its orthogonal polynomials. Let {an,bn}∞n=1 be a half-line Jacobi matrix with an=a˜n+δan , bn=b˜n+δbn . Suppose<br/><br> ∑n=1∞∣δan∣2+∣δbn∣2<∞<br/><br> and ∑Nn=1e2πiωnδan , ∑Nn=1e2πiωnδbn have finite limits as N→∞ for all ω in the frequency module. If, in addition, these partial sums grow at most subexponentially with respect to ω, then for z∈ℂ∖ℝ, pn(z)/p˜n(z) has a limit as n→∞. Moreover, we show that there are non-Szegő class J’s for which this holds.}}, author = {{Christiansen, Jacob Stordal and Simon, Barry and Zinchenko, Maxim}}, issn = {{0176-4276}}, keywords = {{Szegő asymptotics; Orthogonal polynomials; Almost periodic sequences; Slowly decaying perturbations}}, language = {{eng}}, number = {{2}}, pages = {{259--272}}, publisher = {{Springer}}, series = {{Constructive Approximation}}, title = {{Finite gap Jacobi matrices, III. Beyond the Szegő class}}, url = {{http://dx.doi.org/10.1007/s00365-012-9152-4}}, doi = {{10.1007/s00365-012-9152-4}}, volume = {{35}}, year = {{2012}}, }