|
Description:
|
In this dissertation , we investigate methods of modifying a tight frame sequence
on a finite subset of the frame so that the result is a tight frame with better properties .
We call this a surgery on the frame . There are basically three types of surgeries :
transplants , expansions , and contractions . In this dissertation , it will be necessary to
consider surgeries on not -necessarily -tight frames because the subsets of frames that
are excised and replaced are usually not themselves tight frames on their spans , even
if the initial frame and the final frame are tight . This makes the theory necessarily
complicated , and richer than one might expect .
Chapter I is devoted to an introduction to frame theory . In Chapter II , we
investigate conditions under which expansion , contraction , and transplant problems
have a solution . In particular , we consider the equiangular replacement problem .
We show that we can always replace a set of three unit vectors with a set of three
complex unit equiangular vectors which has the same Bessel operator as the Bessel
operator of the original set . We show that this can not always be done if we require
the replacement vectors to be real , even if the original vectors are real . We also prove
that the minimum angle between pairs of vectors in the replacement set becomes
largest when the replacement set is equiangular . Iterating this procedure can yield a
frame with smaller maximal frame correlation than the original . Frames with optimal
maximal frame correlation are called Grassmannian frames and no general method
is known at the present time for constructing them . Addressing this , in Chapter III
we introduce a spreading algorithm for finite unit tight frames by replacing vectors three -at -a -time to produce a unit tight frame with better maximal frame correlation
than the original frame . This algorithm also provides a “good” orientation for the
replacement sets . The orientation part ensures stability in the sense that if a selected
set of three unit vectors happens to already be equiangular , then the algorithm gives
back the same three vectors in the original order . In chapter IV and chapter V , we
investigate two special classes of frames called push -out frames and group frames .
Chapter VI is devoted to some mathematical problems related to the ”cocktail party
problem ” . |