The Geometry of Structured Parseval Frames and Frame Potentials

In this dissertation, we study the geometric character of structured Parseval frames, which are families of vectors that provide perfect Hilbert space reconstruction. Equiangular Parseval frames (EPFs) satisfy that the magnitudes of the pairwise inner products between frame vectors are constant. These types of frames are useful in many applications. However, EPFs do not always exist and constructing them is often difficult. To address this problem, we consider two generalizations of EPFs, equidistributed frames and Grassmannian equal-norm Parseval frames, which include EPFs when they exist. We provide several examples of each type of Parseval frame. To characterize and locate these classes of frames, we develop an optimization program involving families of real analytic frame potentials, which are real-valued functions of frames. With the help of the {\L}ojasiewicz gradient inequality, we prove that the gradient descent of these functions on the manifold of Gram matrices of Parseval frames always converges to critical points. We then show that, under certain conditions, the frames corresponding to the Gram matrices of the critical points for different frame potentials possess desirable geometric properties. These properties include the equal-norm, equiangular, non-orthodecomposable, equidistributed and Grassmannian equal-norm cases. We also discuss the history of EPFs and frame potentials and provide a new characterization of EPFs in terms of the Fourier transform. Using this characterization, we reprove a known result regarding cyclic EPFs and difference sets.