Introduction to Quantum Harmonic Analysis and Gabor Frames

Explore the key concepts of quantum harmonic analysis (QHA) in this 52-minute lecture delivered by Henry McNulty (substituting for Franz Luef) at the Workshop on "Quantum Harmonic Analysis" held at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) in May 2025. Discover fundamental QHA concepts including operator convolutions and the Fourier-Wigner transform, with particular emphasis on their interpretation through time-frequency analysis techniques such as short-time Fourier transform, localization operators, and spreading functions. Learn how the interaction between QHA and time-frequency analysis has introduced important concepts like mixed-state localization operators, Cohen's class of an operator, and operator short-time Fourier transform. Examine Gabor frames and their essential properties, understanding how they naturally emerge when seeking reconstruction formulas for functions using sampled values of the short-time Fourier transform. Connect Gabor frames to QHA, exploring how the Janssen representation of the Gabor frame operator represents a special case of Poisson summation formula for an operator and its Fourier-Wigner transform. Conclude with an exploration of Gabor g-frames as a natural extension of Gabor frames from the QHA perspective.