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 ${\psi }_{ϵ,j,k}=A^{j/2}{\psi }_{ϵ}(A^{j}x–k)),1\le ϵ\le E,j,k\in \mathbb{Z}$ is a Hilbertian basis of $L^2(\mathbb{R})$ with continuous compactly supported mother functions ${\psi }_{ϵ}$, 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 ${\psi }_{ϵ}$). Those results can be extended to the cases of exponentially localized functions and of biorthogonal wavelets.