The purpose of this manuscript is to explore and develop a number of the basic
aspects of a geometric, or operator-theoretic, approach to discrete frame theory on
Hilbert space which arises from the simple observation that a frame sequence is
simply an inner direct summand of a Riesz basis. In other words, frames have a
natural geometric interpretation as sequences of vectors which dilate (geometrically)
to bases. This approach leads to simplified proofs of some of the known results in
frame theory, and also leads to some new results and applications for frames.
A frame is a sequence {xn} of vectors in a Hilbert space H with the property
that there are constants A, B 0 such that
A\\x\\2 J£\x,xj\2B\\x\\2
for all x in the Hilbert space. A Riesz basis is a bounded unconditional basis; In
Hilbert space this is equivalent to being the image of an orthonormal basis under
a bounded invertible operator; another equivalence is that it is a Schauder basis
which is also a frame. An inner direct summand of a Riesz basis is a sequence {xn}
in a Hilbert space H for which there is a second sequence {yn} in a second Hilbert
space M such that the orthogonal direct sum {xn 0 yn} is a Riesz basis for the
direct sum Hilbert space H 0 M.
Frame sequences have been used for a number of years by engineers and applied
mathematicians for purposes of signal processing and data compression. There are
papers presently in the literature which concern interpretations of discrete frame
transforms from a functional analysis point of view. However, our approach seems
to be different in an essential way in that the types of questions we address are
somewhat different, and there is a fundamental difference in perspective arising
from a dilation vantage point. This article is concerned mainly with the pure
mathematics underlying frame theory. Most of the new results we present are in
fact built up from basic principles.
Received by the editor September 30, 1997.
The first author was a participant in the Workshop in Linear Analysis and Prob-
ability at Texas A&M University and the second author was supported in part by
NSF Grant DMS-9706810.
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