Surgery on frames

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Title: Surgery on frames
Author: Nguyen, Nga Quynh
Abstract: 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 ? .
URI: http : / /hdl .handle .net /1969 .1 /ETD -TAMU -2994
Date: 2009-05-15


Surgery on frames. Available electronically from http : / /hdl .handle .net /1969 .1 /ETD -TAMU -2994 .

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