Frames and subspaces

This thesis will consist of three parts. In the first part we find the closest probabilistic Parseval frame to a given probabilistic frame in the 2 Wasserstein Distance. It is known that in the traditional [symbol]2 distance the closest Parseval frame to a frame [phi] = {[symbol]i} N[i=1] [symbol] R[d] is [phi[�] = {[symbol] � i }N[i]=1 = {S [-1/2][[symbol]i]} N[i=1] where S is the frame operator of [phi]. We use this fact to prove a similar statement about probabilistic frames in the 2 Wasserstein metric. In the second part, we will associate a complex vector with a rank 2 real projection. Using this association we will answer many open questions in frame theory. In particular we will prove Edidin's theorem in phase retrieval in the complex case, answer a question on mutually unbiased bases, a question on equiangular lines, and a question on fusion frames. In the last part we will give a way to calculate the exact constant for the [symbol]1 � [symbol]2 inequality and use this method to prove a couple of interesting theorems