Frames with desired angle properties

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT REQUEST OF AUTHOR.] The purpose of this dissertation is to study frames with desired angle properties. More precisely, we study the subspace packing problem, harmonic biangular tight frames, and regular two-distance sets. After a brief review of finite frame theory in Chapter 1, we study the subspace packing problem with respect to the chordal distance in Chapter 2. We show that a solution to this problem is necessarily a fusion frame for the underlying space. We then continue to exploit the idea of using maximal sets of mutually unbiased bases and block designs to construct several infinite families of solutions to the problem. In Chapter 3, motivated by the characterization of harmonic equiangular tight frames in terms of difference sets, we study harmonic biangular tight frames and their connection with other combinatorial structures including divisible difference sets, partial difference sets, and Gaussian difference sets. Chapter 4 is dedicated to studying spherical two-distance sets. These are sets of unit vectors for which the inner products admit two different values. In this chapter, we investigate a special case of such sets when we require one inner product value to appear the same number of times in each row of the Gram matrix. We call this type of set a regular two-distance set. We present various properties of such sets as well as focus on the case where the sets form tight frames for the underlying space. Several connections among regular two-distance sets, equiangular lines, and quasi-symmetric designs are also discussed.Includes bibliographical references