Outer products and frame coefficients

Abstract

In this dissertation, we will examine two distinct areas of frame theory. The first will be the area of outer products. In particular, we will examine the spanning and independence properties of the collection [see abstract for equation] for a given set [see abstract for equation]. In the case that our collection of outer products is a Riesz sequence, we will examine the relation between the Riesz bounds of the outer products to those of the generating vectors. It is perhaps not surprising that an independent collection of vectors will produce an independent collection of outer products. What is surprising though, is that the outer products have the same or better Riesz bounds. However, linearly independent collections of vectors are by no means the only collections that produce independent outer products. We will see that almost all vector sequences produce independent outer product sequences, dimension and cardinality permitting. Next we will examine the distribution of frame coefficients. We will start this investigation by examining the number of indices for which the frame coefficient is non-zero. Then we will generalize and bound these coefficients away from zero. We will study products of frame coefficients and find that in cases of particular classes of frames, we can bound a particular sum away from zero. Finally, we will look at the distance from an arbitrary vector to the frame vectors for a given frame and find some surprising results for certain classes of frames.