Dörfler, Monika;Feichtinger, Hans G.; Gröchenig, Karlheinz

Compactness criteria in function spaces

Colloq. Math. Vol.94 No.1 (2002) p.37-50, Zbl:1017.46014, MR1930200


The classical compactness criteria going back to A.Weil for $L^p$-spaces, with $1 <= p < infty$ characterize bounded subsets $S$ of $L^p$ as exactly those which are equicontinuous (in the $L^p$-norm sense) and tight. In other words, for every positive $eps > 0$ it is possible to approximate, up to that $eps$, all the members of $M$ by functions having common compact support. When applied to the Fourier invariant space $L^2$ it is not difficult to find out that equicontinuity on the time side corresponds to tightness in the frequency domain. Thus for $L^2$ the relative compactness can be characterized by tightness in the time as well as in the frequency variable.

Starting from this observation, and following the general idea that the time -frequency point of view, resp. the use of the short-time Fourier transform is the right tool for many questions arising in analysis, we come up with a time-frequency tightness characterization for relative compactness in $L^2$.

This result then serves as a tool for a much more general approach, preserving the main ideas, but allowing to replace the STFT by various other transformations (corresponding to integrable group representations). The
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