An algebraic approach to discrete dilations. Application to discrete wavelet transforms

We investigate the connections between continuous and discrete wavelet tran sforms on the basis of algebraic arguments. The discrete approach is formul ated abstractly in terms of the action of a semidirect product A x Gamma on l(2)(Gamma), with Gamma a lattice and A an albelian semigroup acting on Ga mma. We show that several such actions may be considered, and investigate t hose which may be written as deformations of the canonical one. The corresp onding deformed dilations (the pseudodilations) turn out to be characterize d by compatibility relations of a cohomological nature The connection with multiresolution wavelet analysis is based on families of pseudodilations of a different type.