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A characterization of affine dual frames in $$L^2(\mathbb{R}^n)$$. (English) Zbl 0961.42018
The affine system (resp. quasiaffine system) generated by a finite family $$\Psi=(\Psi^1,\ldots,\Psi^L)$$ of $$L^2(R^n)$$ associated to a lattice $$\Gamma$$ of $$R^n$$ and a dilation matrix $$A$$ preserving $$\Gamma$$, is the collection: $X(\Psi) =\{ \Psi^l_{j,\gamma}(x)=|\det A|^{j/2}\Psi(A^jx-\gamma)~;~ j\in Z,\gamma\in\Gamma,1\leq l\leq L\},$ respectively $\begin{split} X^q(\Psi) = \{ \Psi^l_{j,\gamma};j\geq 0,\gamma\in\Gamma,1\leq l\leq L\}\cup\\ \{\widetilde{\Psi}^l_{j,\gamma}(x)=|\det A|^{j/2}\Psi(A^j(x-\gamma)); j<0,\gamma\in\Gamma,1\leq l\leq L\}.\end{split}$ The author gives a characterization of all (quasi)affine frames in $$L^2(R^n)$$ that have a (quasi)affine dual in terms of two simple equations in the Fourier transform domain. For $$X(\Psi)$$ and $$X(\Phi)$$, these are: $\sum_{l=1}^L\sum_{j\in Z}\widehat{\Psi}^l(A^{Tj}\xi)\overline{\widehat{\Phi}^l (A^{Tj}\xi)}=|\det P|$
$\sum_{l=1}^L\sum_{j=0}^{\infty}\widehat{\Phi}^l(A^{Tj}\xi)\overline{A^{T j} (\xi+s)} = 0$ for a.e. $$\xi\in R^n$$ and $$s\in P^{-T}Z^n\setminus A^TP^{-T}Z^n$$, where $$PZ^n=\Gamma$$. The quasi-part of the statement is immediate because, as previously proved (see the articles by C. K. Chui, X. Shi and J. Stoeckler [Adv. Comput. Math. 8, No. 1-2, 1-17 (1998; Zbl 0892.42019)] and A. Ron and Z. Shen [J. Funct. Anal. 148, No. 2, 408-447 (1997; Zbl 0891.42018)]), $$X^q(\Psi)$$ is a frame iff $$X(\Psi)$$ is a frame, and $$X^q(\Phi)$$ is a dual of $$X^q(\Psi)$$ iff $$X(\Phi)$$ is a dual of $$X(\Psi)$$.

##### MSC:
 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
##### Keywords:
wavelet frame; quasiaffine frame; affine frame; dual frame
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