Wavelets bases adapted to a self-similar quasicrystal

Given any self-similar quasicrystal Lambda in R-n with inflation theta>1, we construct bases of L-2(R-n) having the following structure: theta(nj/2)psi(lambda)(theta(j)x-lambda), lambda E Lambda\theta Lambda, j is an element of Z, where the mother wavelets psi(lambda), lambda is an element of Lambda\theta Lambda, are smooth and with exponential decay or compact support. We also show that wavelets psi(lambda) constitute a relatively compact set in some Sobolev space and that they depend continuously on lambda when Lambda is equipped with an appropriate topology. (C) 1998 American Institute of Physics. [S0022-2488(98)01808-8].