JournalsrmiVol. 9, No. 2pp. 333–371

Ondelettes generalisées et fonctions d'échelle à support compact

  • Pierre Gilles Lemarié-Rieusset

    Université d'Évry Val d'Essonne, Evry, France
Ondelettes generalisées et fonctions d'échelle à support compact cover
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Abstract

We show that to any multi-resolution analysis of L2(R)L^2 (\mathbb R) with multiplicity dd, dilation factor AA (where AA is an integer ≥ 2) and with compactly supported scaling functions we may associate compactly supported wavelets. Conversely, if ψϵ,j,k=Aj/2ψϵ(Ajxk)),1ϵE,j,kZ\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 L2(R)L^2 (\mathbb R) with continuous compactly supported mother functions ψϵ\psi_\epsilon, then it is provided by a multi-resolution analysis with dilation factor AA, multiplicity d=E/(A1)d = E/(A–1) and with compactly supported scaling functions (which have the same regularity as the wavelets ψϵ\psi_\epsilon). Those results can be extended to the cases of exponentially localized functions and of biorthogonal wavelets.

Cite this article

Pierre Gilles Lemarié-Rieusset, Ondelettes generalisées et fonctions d'échelle à support compact. Rev. Mat. Iberoam. 9 (1993), no. 2, pp. 333–371

DOI 10.4171/RMI/140