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03/11/03 Colloquium
DR. DEMETRIO
LABATE
WASHINGTON UNIVERSITY - ST. LOUIS
A Unified Theory of Reproducing Function Systems
Abstract: By a reproducing method for a Hilbert
space ℌ we mean the use of two countable families 
in ℌ, so that the first analyzes a
function 
ℌ by forming the inner products 
and the second reconstructs
h from this information: 
. A variety of such systems have been used successfully in
both pure and applied mathematics. They have the following feature in common:
they are generated by a single or a finite collection of functions by applying
to the generators an appropriate set of dilations, modulations, and
translations. The Gabor system, for example, involve a countable
collection of modulations and translations; the affine systems (that
produce a variety of wavelets) involve translations and dilations. Considerable amount of research has been
conducted in order to characterize those generators of such systems. In this
talk we present an approach that unifies all of these characterizations by
means of a relatively simple system of equalities. We also describe how our methods
apply to various affine-like, wave packets and Gabor
systems.
