Resumen de Ondelettes generalisées et fonctions d'échelle à support compact

Pierre Gilles Lemarié-Rieusset

• We show that to any multi-resolution analysis of L2(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 (?e,j,k = Aj/2 ?e (Ajx - k)), 1 = e = E and j, k Î Z, is a Hilbertian basis of L2(R) with continuous compactly supported mother functions ?e, 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 ?e). Those results can be extended to the cases of exponentially localized functions and of biorthogonal wavelets.