We show that to any multi-resolution analysis of with multiplicity , dilation factor (where is an integer ≥ 2) and with compactly supported scaling functions we may associate compactly supported wavelets. Conversely, if is a Hilbertian basis of with continuous compactly supported mother functions , then it is provided by a multi-resolution analysis with dilation factor , multiplicity and with compactly supported scaling functions (which have the same regularity as the wavelets ). 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