Bernstein-Szegő measures, Banach algebras, and scattering theory

Geronimo, Jeffrey; Iliev, Plamen

doi:10.1090/tran/6841

Bernstein-Szegő measures, Banach algebras, and scattering theory
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by Jeffrey S. Geronimo and Plamen Iliev PDF

Trans. Amer. Math. Soc. 369 (2017), 5581-5600 Request permission

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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