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Compactness criteria in function spaces. (English) Zbl 1017.46014

The well-known compactness criterion in \(L_2 ({\mathbb R}^n)\) can be reformulated in terms of the continuous wavelet transform \[ W_g f (x,s) = s^{- \frac{n}{2}} \int_{{\mathbb R}^n} f(t) g\biggl( \frac{t-x}{s}\biggr) dt, \quad x \in {\mathbb R}^n, s >0. \] The authors extend this observation to abstract coorbit spaces (Theorem 4). They apply the outcome to some concrete distinguished examples: modulation spaces in \({\mathbb R}^n\), spaces of type \(B^s_{pq} ({\mathbb R}^n)\), \(F^s_{pq} ({\mathbb R}^n)\), and Bargman-Fock spaces on \({\mathbb C}^n\).

MSC:

46B50 Compactness in Banach (or normed) spaces
42B35 Function spaces arising in harmonic analysis
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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