From: Brenda Sue Cook (brenda@math.missouri.edu)
Date: Wed Aug 15 2007 - 13:31:44 CDT
Ph.D. Defense
University of Missouri-Columbia
Elena Koutcherik
Department of Mathematics
Transference and Szego's Theorem For Measure Preserving Representations
Abstract
We obtain analogues of the classical Szeg\"o'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 $H^p(\Om, \mu)$.
The latter are
defined in terms of strongly continuous, measure preserving
transformations in $L^p(\Om, \mu)$,
where $(\Om, \mu)$ is a locally compact measure space.
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
$(\Om, \mu)$,
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.
Dissertation Advisor: Nakhle Asmar
Friday, August 17
11:00AM
104 A&S
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