Transference and Szego's theorem for measure preserving representations
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[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT REQUEST OF AUTHOR.] We prove analogues of the classical Szeg̈o's theorem concerning approximation by polynomials on the unit circle, and Jensen's inequality involving the summability of the logarithm, for functions in generalized Hardy spaces Hp([gamma],[mu]). The latter are defined in terms of strongly continuous, measure preserving representations in Lp([gamma],[mu]), where ([gamma],[mu]) is a locally compact measure space. In a similar framework, Asmar and Montgomery-Smith earlier obtained an analogue of the F. and M. Riesz theorem, which generalizes deep results due to de Leeuw and Glicksberg, and Forelli. Our approach employs a variety of methods, including elements of transference theory, representation theory, operator theory, and spectral theory of functions. We develop the notions of convolutions, spectra, analytic decompositions, generalized trigonometric and analytic trigonometric polynomials on ([gamma],[mu]), and study their approximation properties. This enables us to obtain broad generalizations of some results of the classical theory, which corresponds to the special case where measure preserving representations are generated by translations on the unit circle.
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