On the Growth of the Polynomial Entropy Integrals for Measures in the Szegő Class
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چکیده
Let σ be a probability Borel measure on the unit circle T and {φn} be the orthonormal polynomials with respect to σ. We say that σ is a Szegő measure, if it has an arbitrary singular part σs, and R T log σ ′dm > −∞, where σ′ is the density of the absolutely continuous part of σ, m being the normalized Lebesgue measure on T. The entropy integrals for φn are defined as n = Z T |φn| log |φn|dσ It is not difficult to show that n = o( √ n). In this paper, we construct a measure from the Szegő class for which this estimate is sharp (over a subsequence of n’s).
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