Compactness Criteria in Function Spaces

The classical criterion for compactness in Banach spaces of functions can be reformulated into a simple tightness condition in the time-frequency do- main. This description preserves more explicitly the symmetry between time and frequency than the classical conditions. The result is first stated and proved for L 2 (R d ), and then generalized to coorbit spaces. As special cases, we obtain new characterizations of compactness in Besov-Triebel-Lizorkin spaces, modulation spaces and Bargmann-Fock spaces.

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