Applied problems in frame theory
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This thesis is a study of two applied problems in frame theory: phase retrieval and quantum detection. These problems are inspired by engineering applications in signal processing and information theory. ... The goal of the injectivity problem is to classify frames which are injective with respect to self-adjoint Hilbert-Schmidt operators. By associating vectors x 2 Hn with vectors ~x in a larger space, we are able to use standard linear algebra and functional analysis techniques to provide characterizations for the injectivity problem in complex and real Hilbert spaces, as well as construct solutions. Given an injective frame, the goal of the state estimation problem is to construct a self-adjoint trace one operator T such that the vector with coordinates hTxk; xki is equal to a predetermined measurement vector. We give equivalent conditions for solvability of the state estimation problem and provide best approximate solutions when no exact solution is possible. We also show results about density of both problems.