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A unified characterization of reproducing systems generated by a finite family. (English) Zbl 1029.42026
Let \(C\) be a nonsingular \(n\times n\) matrix with real entries. Given a collection of functions \(\{g_p\}_{p\in P}\) in \(L^2(\mathbb{R}^n)\), tight frames of the form \(\{g_p(\cdot -Ck)\}_{k\in \mathbb{Z}^n, p\in P}\) are characterized. Furthermore, equivalent conditions for two systems of this type to be dual frames are given. The results are applied to wavelet systems and Gabor systems.

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42C15 General harmonic expansions, frames
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