Nathan Wagner will give a talk at 4/11/2023.

The classical Riesz-Kolmogorov theorem gives a characterization of the precompact subsets of $L^p$ in terms of a uniform spatial decay condition and a uniform continuity condition (which can be interpreted in the case of $L^2$ as a uniform decay in frequency on the Fourier side). In the first talk, we will briefly describe the background of the theorem and state new variants of this result. These variations include statements in the abstract setting of Hilbert spaces with continuous Parseval frames, as well as a variety of function spaces, including the Paley-Wiener space and Besov-Sobolev spaces of analytic functions on the unit ball. We will also briefly mention some applications of these results to Toeplitz operators and “umbrella theorems.” In the second talk, we will present outlines of some of the proofs. These talks are based on joint work with Mishko Mitkovski, Cody Stockdale, and Brett Wick.