Compactness criteria in function spaces
Volume 94 / 2002
Colloquium Mathematicum 94 (2002), 37-50
MSC: 46B50, 42B35.
DOI: 10.4064/cm94-1-3
Abstract
The classical criterion for compactness in Banach spaces of functions can be reformulated into a simple tightness condition in the time-frequency domain. This description preserves more explicitly the symmetry between time and frequency than the classical conditions. The result is first stated and proved for ${\bf L}^2 ({\mathbb R}^d)$, and then generalized to coorbit spaces. As special cases, we obtain new characterizations of compactness in Besov–Triebel–Lizorkin, modulation and Bargmann–Fock spaces.
