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Orthonormal bases of compactly supported wavelets. (English) Zbl 0644.42026
We construct compactly supported functions \(_ N\psi\) \((N\in\mathbb N)\) such that the family of “wavelets” \(\{_ N\psi _{mn};\, m,n\in\mathbb Z\}\), where \(_ N\psi _{mn}(x)=2^{-m/2}\psi (2^{-m}x-n),\) constitutes an orthonormal basis of \(L^ 2(\mathbb R)\). These functions \({}_ N\psi\) generalize the Haar basis; the case \(N=1\) reduces to the Haar basis. Typically support \((_ N\psi)=[-N+1,N]\); exists \(\mu >0\) so that \(_ N\psi \in C^{\mu N}\) for all \(N\in\mathbb N\). The function \(_ N\psi\) can thus be chosen to be of arbitrarily high regularity, at the price of increasing the support width of \(_ N\psi\). The paper starts by reviewing the concept of multiresolution analysis, as well as several algorithms used in vision analysis. The construction of the present orthonormal bases of compactly supported wavelets then follows from a synthesis of these different approaches.
Reviewer: Ingrid Daubechies

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
65T60 Numerical methods for wavelets
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