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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
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