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A characterization of functions that generate wavelet and related expansion. (English) Zbl 0896.42022
Let \(\Phi=\{\phi^1,\ldots,\phi^L\}\) be a family of scaling functions in \(L^2({\mathbb{R}}^n)\) and define \(\phi_{k,{\mathbf k}}^\ell({\mathbf x})=2^{jn/2}\phi^\ell(2^j{\mathbf x}-{\mathbf k})\). Let \(\Psi=\{\psi^1,\ldots,\psi^L\}\) be a corresponding set of wavelets with \(\psi_{j,{\mathbf k}}^\ell\) defined similarly to \(\phi_{k,{\mathbf k}}^\ell\). This paper characterizes those families \(\Phi\) and \(\Psi\) such that for each \(f\in L^2({\mathbb{R}}^n)\) one has \(f=\sum_{\ell=1}^L\sum_{j\in{\mathbb{Z}}}\sum_{{\mathbf k}\in{\mathbb{Z}}^n} (f,\phi_{k,{\mathbf k}}^\ell)\psi_{k,{\mathbf k}}^\ell\) where \((\cdot,\cdot)\) is the inner product in \(L^2({\mathbb{R}}^n)\) and the convergence is with respect to the derived norm. The characterization is expressed as identities for certain series in the Fourier transform functions \(\widehat{\phi}^\ell\) and \(\widehat{\psi}^\ell\) which should hold a.e.

MSC:
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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