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Continuous frames, function spaces, and the discretization problem. (English) Zbl 1093.42020

A collection of vectors {ψ x :xX} in a separable Hilbert space H is called a continuous frame if there exist constants 0<AB< such that for all fH,

Af 2 X |f,ψ x | 2 dμ(x)Bf 2 ,

where X is a locally compact Hausdorff space and μ a positive Radon measure on X. The paper under review presents a method of constructing Banach spaces associated with a given continuous frame. Examples of spaces constructed via this method include classical coorbit spaces, inhomogeneous Besov and Triebel-Lizorkin spaces, and α-modulation spaces.

MSC:
42C15General harmonic expansions, frames
42C40Wavelets and other special systems
46B25Classical Banach spaces in the general theory of normed spaces
46B45Banach sequence spaces