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Dynamics in the Szegő class and polynomial asymptotics

Christiansen, Jacob S. LU (2019) In Journal d'Analyse Mathematique 137(2). p.723-749
Abstract


We introduce the Szegő class, Sz(E), for an arbitrary Parreau–Widom set E ⊂ ℝ and study the dynamics of its elements under the left shift. When the direct Cauchy theorem holds on ℂ\E, we show that to each J ∈ Sz(E) there is a unique element J′ in the isospectral torus, T
E
, so that the left-shifts of J are asymptotic to the orbit {J′
m
} on T
... (More)


We introduce the Szegő class, Sz(E), for an arbitrary Parreau–Widom set E ⊂ ℝ and study the dynamics of its elements under the left shift. When the direct Cauchy theorem holds on ℂ\E, we show that to each J ∈ Sz(E) there is a unique element J′ in the isospectral torus, T
E
, so that the left-shifts of J are asymptotic to the orbit {J′
m
} on T
E
. Moreover, we show that the ratio of the associated orthogonal polynomials has a limit, expressible in terms of Jost functions, as the degree n tends to ∞. This enables us to describe the large n behaviour of the orthogonal polynomials for every J in the Szegő class.

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Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Journal d'Analyse Mathematique
volume
137
issue
2
pages
723 - 749
publisher
Magnes Press
external identifiers
  • scopus:85063208158
ISSN
0021-7670
DOI
10.1007/s11854-019-0013-y
language
English
LU publication?
yes
id
6a471a2a-beca-4898-a77c-35a3bed28004
date added to LUP
2019-04-02 13:39:33
date last changed
2025-10-14 11:39:41
@article{6a471a2a-beca-4898-a77c-35a3bed28004,
  abstract     = {{<p><br>
                                                         We introduce the Szegő class, Sz(E), for an arbitrary Parreau–Widom set E ⊂ ℝ and study the dynamics of its elements under the left shift. When the direct Cauchy theorem holds on ℂ\E, we show that to each J ∈ Sz(E) there is a unique element J′ in the isospectral torus, T                             <br>
                            <sub>E</sub><br>
                                                         , so that the left-shifts of J are asymptotic to the orbit {J′                             <br>
                            <sub>m</sub><br>
                                                         } on T                             <br>
                            <sub>E</sub><br>
                                                         . Moreover, we show that the ratio of the associated orthogonal polynomials has a limit, expressible in terms of Jost functions, as the degree n tends to ∞. This enables us to describe the large n behaviour of the orthogonal polynomials for every J in the Szegő class.                         <br>
                        </p>}},
  author       = {{Christiansen, Jacob S.}},
  issn         = {{0021-7670}},
  language     = {{eng}},
  month        = {{03}},
  number       = {{2}},
  pages        = {{723--749}},
  publisher    = {{Magnes Press}},
  series       = {{Journal d'Analyse Mathematique}},
  title        = {{Dynamics in the Szegő class and polynomial asymptotics}},
  url          = {{http://dx.doi.org/10.1007/s11854-019-0013-y}},
  doi          = {{10.1007/s11854-019-0013-y}},
  volume       = {{137}},
  year         = {{2019}},
}