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Continuous frames, function spaces, and the discretization problem. (English) Zbl 1093.42020
A collection of vectors $\{\psi_x: x \in X\}$ in a separable Hilbert space $H$ is called a continuous frame if there exist constants $0 < A \le B < \infty$ such that for all $f \in H$, $$ A\Vert f\Vert ^2 \le \int_X \vert \langle f, \psi_x \rangle\vert ^2d\mu(x) \le B\Vert f\Vert ^2, $$ where $X$ is a locally compact Hausdorff space and $\mu$ 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 $\alpha$-modulation spaces.

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