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Wavelets with exponential localization. (Ondelettes à localisation exponentielle.) (French) Zbl 0758.42020
The author constructs a new ondelette basis of $$L^ 2(\mathbb{R}^ n)$$. For $$n=1$$, that is a basis of the form $$\{\psi_{jk}(x)=2^{j/2}\psi(2^ jx-k)$$; $$j,k\in\mathbb{Z}\}$$, where $$\psi(x)$$ is a function with certain properties of regularity, behavior at $$\infty$$, and oscillation. The main part of the paper is the construction, using splines, of a suitable function $$\psi$$ of class $$C^{m-2}$$ for which $$\psi$$ and its first $$m-2$$ derivatives are $$O(e^{-\varepsilon| t|})$$ for a certain $$\varepsilon>0$$ and $$\int_ \mathbb{R} t^ k\psi(t)dt=0$$ for $$0\leq k\leq m- 1$$. For $$n>1$$, the basis requires a second function which the author constructs using ideas of his, S. Mallat and Y. Meyer [“Multi-scale analysis”, CEREMADE, Univ. Paris-Dauphine, Paris; per bibl.].

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