Generalized Frame Potential and Problems Related to SIC-POVMs
Benedetto, John J
Okoudjou, Kasso A
MetadataShow full item record
Frame theory generalizes the idea of bases in Hilbert space, and the frame potential is an important tool when studying frame theory. In this thesis, we first explore the minimization problem of a generalized definition of frame potential, namely the p-frame potential, and show there exists a universal optimizer under certain conditions by applying a method involving ultraspherical polynomials and spherical designs. Next, we further discuss the topic on Grassmannian frames, which are special cases of minimizers of p-frame potentials. We present the construction of equiangular lines in lower dimensions since numerical result showed their connections with Grassmannian frames. We also derive properties of the (6,4)-Grassmannian frame. Then, we obtain lower bounds for the generalized frame potentials in the complex setting. The frame potentials may provide a different approach to determine the existence of Gabor frames that are equiangular. This relates the potential minimization problem to the unsolved Zauner conjecture. In addition, we study the properties of Gramian matrices of Gabor frames in an attempt to search for Gabor frames with a small number of different inner products. We also calculate the number of different inner products in Gabor frames generated by Alltop sequences and Björck sequences. In addition, we also present examples related to a generalized support uncertainty inequality and shift-invariant spaces on LCA groups.