Speaker: Dimitris Giannakis, Dartmouth College

Date: January 19 and 26, 2023

Abstract: Harmonic Hilbert spaces are reproducing kernel Hilbert spaces on locally compact abelian groups which are built as Fourier images of weighted L2 spaces on the dual group. Introduced by Delvos as function spaces on the real line, they have found applications in areas such as time-frequency analysis and interpolation of functions. Under appropriate choices of weights, harmonic Hilbert spaces become Banach algebras under pointwise multiplication which are Fourier images of convolution algebras on the dual group. The latter, have a long history of study in harmonic analysis by Wermer, Feichtinger, Brandenburg, and many others. In this talk, we discuss a class of harmonic Hilbert spaces induced by a class of subconvolutive weight functions on the dual group of a compact abelian group. We describe the properties of their Gelfand spectra and their connection with Markov semigroups induced by kernel integral operators. As an application, we discuss how a class of measure-preserving flows with pure point spectra can be consistently embedded into quantum systems on harmonic Hilbert spaces on tori.