Abstract:

Quantum harmonic analysis on phase space uses representations of the Heisenberg group to define analogs of the Fourier transform and of convolutions for bounded operators, and where the Schatten classes of compact operators play the role of the Lebesgue spaces. We will briefly discuss quantum harmonic analysis analogs of the Hausdorff-Young inequality, the convolution theorem, Young's inequality and Wiener's approximation theorem.

Recently, quantum harmonic analysis has been linked to time-frequency analysis, Toeplitz operators, identifying local structures in time-series datasets, and the semiclassical limit from the Hartree equation to the Vlasov–Poisson system. We are going to present some of these connections.

In this talk, we will discuss in detail the quantum harmonic analysis version of Fourier restriction and its close relation to Fourier restriction of functions on phase space.

This is joint work with Helge J. Samuelsen and Eirik Skrettingland (Norwegian University of Science and Technology, Trondheim).