Finite gap Jacobi matrices, III. Beyond the Szegő class
(2012) In Constructive Approximation 35(2). p.259272 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 leftmost 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 halfline 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 nonSzegő class J’s for which this holds.
Please use this url to cite or link to this publication:
http://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
 01764276
 DOI
 10.1007/s0036501291524
 language
 English
 LU publication?
 no
 id
 258409a11b8245b3bb19f3305ac8133c (old id 3448235)
 alternative location
 http://link.springer.com/article/10.1007%2Fs0036501291524
 date added to LUP
 20130906 18:53:37
 date last changed
 20180107 04:50:41
@article{258409a11b8245b3bb19f3305ac8133c, 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 leftmost 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 halfline 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 nonSzegő class J’s for which this holds.}, author = {Christiansen, Jacob Stordal and Simon, Barry and Zinchenko, Maxim}, issn = {01764276}, keyword = {Szegő asymptotics,Orthogonal polynomials,Almost periodic sequences,Slowly decaying perturbations}, language = {eng}, number = {2}, pages = {259272}, publisher = {Springer}, series = {Constructive Approximation}, title = {Finite gap Jacobi matrices, III. Beyond the Szegő class}, url = {http://dx.doi.org/10.1007/s0036501291524}, volume = {35}, year = {2012}, }