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Bernstein-Szegő measures, Banach algebras, and scattering theory


Authors: Jeffrey S. Geronimo and Plamen Iliev
Journal: Trans. Amer. Math. Soc. 369 (2017), 5581-5600
MSC (2010): Primary 47B36, 42C05
DOI: https://doi.org/10.1090/tran/6841
Published electronically: March 6, 2017
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Abstract: We give a simple and explicit description of the Bernstein-Szegő type measures associated with Jacobi matrices which differ from the Jacobi matrix of the Chebyshev measure in finitely many entries. We also introduce a class of measures $ \mathcal {M}$ which parametrizes the Jacobi matrices with exponential decay and for each element in $ \mathcal {M}$ we define a scattering function. Using Banach algebras associated with increasing Beurling weights, we prove that the exponential decay of the coefficients in a Jacobi matrix is completely determined by the decay of the negative Fourier coefficients of the scattering function. Combining this result with the Bernstein-Szegő type measures we provide different characterizations of the rate of decay of the entries of the Jacobi matrices for measures in $ \mathcal {M}$.


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Additional Information

Jeffrey S. Geronimo
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332–0160 – and – Tao Aoqing Visiting Professor, Jilin University, Changchun, People’s Republic of China
Email: geronimo@math.gatech.edu

Plamen Iliev
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332–0160
Email: iliev@math.gatech.edu

DOI: https://doi.org/10.1090/tran/6841
Keywords: Bernstein-Szeg\H{o} measures, Banach algebras, scattering theory, orthogonal polynomials.
Received by editor(s): September 23, 2014
Received by editor(s) in revised form: September 5, 2015
Published electronically: March 6, 2017
Additional Notes: The first author was partially supported by Simons Foundation Grant #210169.
The second author was partially supported by Simons Foundation Grant #280940.
Article copyright: © Copyright 2017 American Mathematical Society