Revista Matemática Iberoamericana


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Volume 9, Issue 2, 1993, pp. 333–371
DOI: 10.4171/RMI/140

Published online: 1993-08-31

Ondelettes generalisées et fonctions d'échelle à support compact

Pierre Gilles Lemarié-Rieusset[1]

(1) Université d'Évry Val d'Essonne, Evry, France

We show that to any multi-resolution analysis of $L^2 (\mathbb R)$ with multiplicity $d$, dilation factor $A$ (where $A$ is an integer ≥ 2) and with compactly supported scaling functions we may associate compactly supported wavelets. Conversely, if $\psi_{\epsilon, j, k} = A^{j/2}\psi_\epsilon (A^jx–k)), 1 ≤ \epsilon ≤ E, j, k \in \mathbb Z$ is a Hilbertian basis of $L^2 (\mathbb R)$ with continuous compactly supported mother functions $\psi_\epsilon$, then it is provided by a multi-resolution analysis with dilation factor $A$, multiplicity $d = E/(A–1)$ and with compactly supported scaling functions (which have the same regularity as the wavelets $\psi_\epsilon$). Those results can be extended to the cases of exponentially localized functions and of biorthogonal wavelets.

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Lemarié-Rieusset Pierre Gilles: Ondelettes generalisées et fonctions d'échelle à support compact. Rev. Mat. Iberoam. 9 (1993), 333-371. doi: 10.4171/RMI/140