Abstract
Let \(\mu\) be a nontrivial probability measure on the unit circle \(\partial\mbox{\bf D},\ w\) the density of its absolutely continuous part, \(\alpha_n\) its Verblunsky coefficients, and \(\Phi_n\) its monic orthogonal polynomials. In this paper we compute the coefficients of \(\Phi_n\) in terms of the \(\alpha_n\). If the function \(\log w\) is in \(L^1(d\theta)\), we do the same for its Fourier coefficients. As an application we prove that if \(\alpha_n\in\ell^4\) and if \(Q(z) \equiv\sum_{m=0}^N q_m z^m\) is a polynomial, then with \(\bar Q(z) \equiv\sum_{m=0}^N \bar q_m z^m\) and S the left-shift operator on sequences we have
We also study relative ratio asymptotics of the reversed polynomials \(\Phi_{n+1}^*(\mu)/\Phi_n^*(\mu)-\Phi_{n+1}^*(\nu)/\Phi_n^*(\nu)\) and provide a necessary and sufficient condition in terms of the Verblunsky coefficients of the measures \(\mu\) and \(\nu\) for this difference to converge to zero uniformly on compact subsets of \(\mbox{\bf D}\).
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Golinskii, L., Zlatos, A. Coefficients of Orthogonal Polynomials on the Unit Circle and Higher-Order Szego Theorems. Constr Approx 26, 361–382 (2007). https://doi.org/10.1007/s00365-006-0650-7
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DOI: https://doi.org/10.1007/s00365-006-0650-7