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Finite gap Jacobi matrices, III. Beyond the Szegő class

Christiansen, Jacob Stordal LU ; Simon, Barry and Zinchenko, Maxim (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:
author
; and
publishing date
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&lt;∞<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}},
}