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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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##### References:
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