Advanced

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
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
2013-09-06 18:53:37
date last changed
2017-01-01 04:10:46
@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},
  keyword      = {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},
  volume       = {35},
  year         = {2012},
}