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Higher-order Szegő theorems with two singular points

Simon, Barry and Zlatoš, Andrej (2005) Higher-order Szegő theorems with two singular points. Journal of Approximation Theory, 134 (1). pp. 114-129. ISSN 0021-9045. doi:10.1016/j.jat.2005.02.003. https://resolver.caltech.edu/CaltechAUTHORS:20170512-073745126

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Abstract

We consider probability measures, dμ = w(θ)^(dθ)_(2π) + dμ_s, on the unit circle, ∂D, with Verblunsky coefficients, {αj}_(j=0)^∞. We prove for θ_1 ≠ θ_2 in [0,2π) that ∫[1-cos(θ-θ_1)][1-cos(θ-θ_2)]log w(θ)^(dθ)_(2π > -∞if and only if ∑_(j=0)^∞ │{(δ-e^(-iθ2))(δ-e^(-iθ1))α}_j^2 +|α_j|^4 < ∞,where δ is the left shift operator (δβ)_j = β_(j+1). We also prove that ∫(1-cosθ)^2 log w (θ)^(dθ)_(2π) > - ∞ if and only if ∑_(j=0)^∞|α_(j+2) - 2α_(j+1) + α_j|^2 + |αj|^ 6 <∞.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1016/j.jat.2005.02.003DOIArticle
http://www.sciencedirect.com/science/article/pii/S0021904505000456PublisherArticle
https://arxiv.org/abs/math-ph/0409065arXivDiscussion Paper
ORCID:
AuthorORCID
Simon, Barry0000-0003-2561-8539
Additional Information:© 2005 Elsevier Inc. Received 16 September 2004, Accepted 9 February 2005, Available online 7 April 2005. Communicated by Leonid Golinskii We thank S. Denisovand S. Kupin for telling us of their joint work [3].
Subject Keywords:Orthogonal polynomials on the unit circle; Szegő theorem
Issue or Number:1
DOI:10.1016/j.jat.2005.02.003
Record Number:CaltechAUTHORS:20170512-073745126
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170512-073745126
Official Citation:Barry Simon, Andrej Zlatoš, Higher-order Szegő theorems with two singular points, Journal of Approximation Theory, Volume 134, Issue 1, May 2005, Pages 114-129, ISSN 0021-9045, https://doi.org/10.1016/j.jat.2005.02.003. (http://www.sciencedirect.com/science/article/pii/S0021904505000456)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:77390
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:12 May 2017 23:55
Last Modified:15 Nov 2021 17:30

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