Ondelettes generalisées et fonctions d'échelle à support compact

Abstract

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.