Bernstein-Szegő measures, Banach algebras, and scattering theory

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 M \mathcal {M} which parametrizes the Jacobi matrices with exponential decay and for each element in M \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 M \mathcal {M} .