Dynamics in the Szegő class and polynomial asymptotics
(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
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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.
- author
- Christiansen, Jacob S. LU
- organization
- publishing date
- 2019-03-19
- 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
- 2022-04-25 22:35:18
@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}}, }