Invariant projectors, dilatation matrices, wavelets and multi-resolution analysis
By means of a translation and a dilatation matrix \(A\) two biorthogonal bases in \(L^ 2(\mathbb{R}^ n)\) are determined by \(\psi_{\varepsilon,j,k}(x)= |\text{det }A|^{{j\over 2}} \psi_ \varepsilon(A^ j x- k)\) and \(\psi^*_{\varepsilon,j,k}(x)= |\text{det }A|^{{j\over 2}} \psi^*_ \varepsilon(A^ j x- k)\), where \(\{\psi_ \varepsilon\mid 1\leq \varepsilon\leq E\}\) is a set of functions (wavelets), \(j\in \mathbb{Z}\), \(k\in \mathbb{Z}^ n\); dilatation means that \(A\) must have eigenvalues with moduli \(>1\). Moreover, completeness, biorthogonality \(\langle\psi_{\varepsilon,j,k},\psi^*_{\varepsilon',j',k'}\rangle= \delta_{\varepsilon\varepsilon'}\delta_{jj'}\delta_{kk'}\), and almost-orthogonality are required; the latter demands \(\|\sum \lambda_{\varepsilon,j,k}\psi_{\varepsilon,j,k}\|^ 2_ 2\leq c \sum| \lambda_{\varepsilon,j,k}|^ 2\) for some constant \(c\) and every almost null sequence \((\lambda_{\varepsilon,j,k})\), summing going by \(\varepsilon\in \{1,\dots,E\}\), \(j\in \mathbb{Z}\), \(k\in \mathbb{Z}^ n\), and the same for \(\psi^*_{\varepsilon,j,k}\). By \(P_ 0 f=\sum \langle f,\psi^*_{\varepsilon,j,k}\rangle \psi_{\varepsilon,j,k}\) a projector \(P_ 0\) of \(L^ 2(\mathbb{R}^ n)\) on \(V_ 0= \bigoplus_{j<0} \overline W_ j\) is given, parallel to \((V^*_ 0)^ \perp\) and invariant with respect to translations, where \(W_ j\) is spanned on \(\psi_{\varepsilon,j,k}\), \(1\leq \varepsilon\leq E\), \(k\in \mathbb{Z}^ n\). In the paper simple criteria are given for the existence of a Riesz type basis in \(V_ 0\), for establishing the almost-orthogonality property (using so-called ``vaguelettes'') and for a basis of wavelets to generate the multi-resolution analysis [by \textit{S. Mallat}, Trans. Am. Math. Soc. 315, No. 1, 69-87 (1989; Zbl 0686.42018)]. Sections: I. Projectors, invariant by entire translations, II. Lemma about vaguelettes, III. Basis of wavelets and multi-resolution analysis, IV. The case of dimension 1, V. Counterexamples, Annex A: Weights and algebras of Beurling, Annex B: Geometry of dilatati.
scientific article; zbMATH DE number 647729