Wavelets with exponential localization. (Ondelettes à localisation exponentielle.) (French) Zbl 0758.42020

Lemarie, Pierre-Gilles

J. Math. Pures Appl., IX. Sér. 67, No. 3, 227-236 (1988).

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

Keywords:

wavelet basis of \(L^ 2(\mathbb{R}^ n)\); multi-scale analysis