"Quantum Time-Frequency Analysis and Pseudodifferential Operators"Luef, FranzWe introduce Quantum Time--Frequency Analysis, which expands the approach of Quantum Harmonic Analysis to include modulations of operators in addition to translations. This is done by a projective representation of double-phase space, and we consider the associated matrix coefficients and integrated representation. This leads to the polarised Cohen's class of two operators, which is an isomorphism from Hilbert-Schmidt operators to a reproducing kernel Hilbert space, and has orthogonality relations similar to many objects in classical time--frequency analysis. By considering a class of windows for the polarised Cohen's class that is smaller than the class of Hilbert-Schmidt operators, then we find spaces of modulation spaces of operators, and we consider the properties of these spaces, including discretization results. In many cases, using rank--one examples of operators, we recover familiar objects and results from classical time--frequency analysis and the theory of pseudo differential operators. This is joint work with Henry McNulty. |
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