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A unified characterization of reproducing systems generated by a finite family. II. (English) Zbl 1039.42032
For $$y\in \mathbb{R}^n$$, define the translation operator on $$L^2(\mathbb{R}^n)$$ by $$(T_yf)(x)=f(x-y).$$ Given a family $$\{g_p\}_{p\in P}$$ of functions in $$L^2(\mathbb{R}^n)$$ and a corresponding family $$\{C_p\}_{p\in P}$$ of real and invertible $$n\times n$$ matrices, normalized tight frames for $$L^2(\mathbb{R}^n)$$ of the form $$\{T_{C_pk}g_p\}_{p\in P, k\in \mathbb{Z}^n}$$ are characterized.
The result unifies the known results for Gabor systems and affine systems. Several interesting cases are considered in detail, e.g., wavelet systems generated by matrices of special types. Under a technical assumption, dual systems of the form $\{T_{C_pk}g_p\}_{p\in P, k\in \mathbb{Z}^n}, \{T_{C_pk}\gamma_p\}_{p\in P, k\in \mathbb{Z}^n}$ are characterized; in the Gabor case, this leads to a version of the Wexler-Raz theorem for systems generated by $$L$$ functions.
See also Part I [D. Labate, J. Geom. Anal. 12, No. 3, 469–491 (2002; Zbl 1029.42026)].

##### MSC:
 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 42C15 General harmonic expansions, frames
##### Keywords:
affine systems; Gabor systems; tight frames; dual systems; wavelets
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