Quantum Frames and Uncertainty Principles arising from Symplectomorphisms

In this thesis, a new generalized signal transform along with a new uncertainty principle is elaborated. Starting from a coordinate system, associated to a specific symplectomorphism on phase space, the coordinates are used to define curvilinear flows along which the phase space picture of a prototype function is translated. As the uncertainty principle restricts the amount to which the phase space picture of functions can be concentrated, with each such function is associated a phase space cell. Using the phase space translates of these cells, the notion of a quantum frame is defined, by means of which a reservoir of interesting functions may be decomposed. To define optimal phase space cells, two complementing uncertainty principles, associated with coordinate systems in phase space, are introduced, one of which measures the deviation from the chosen frame, while the other optimizes with respect to the canonically conjugate coordinates and leads to more concentrated waveforms.