Bernstein-Szegő measures, Banach algebras, and scattering theory
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- by Jeffrey S. Geronimo and Plamen Iliev PDF
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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 $\mathcal {M}$.References
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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
- MR Author ID: 72750
- Email: geronimo@math.gatech.edu
- Plamen Iliev
- Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332–0160
- MR Author ID: 629581
- Email: iliev@math.gatech.edu
- 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. - © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 5581-5600
- MSC (2010): Primary 47B36, 42C05
- DOI: https://doi.org/10.1090/tran/6841
- MathSciNet review: 3646771