An algebraic approach to discrete dilations. ation to discrete wavelet transforms (Q1976476)

The authors investigate the connection between continuous and discrete wavelet transforms on the basis of algebraic arguments. The discrete approach is formulated in terms of the action of a semidirect product \(A\times \Gamma\) on \(l^2(\Gamma)\), with \(\Gamma\) a lattice and \(A\) an abelian group acting on \(\Gamma\). Some actions have their corresponding deformed dilations characterized by compatibility relations of a cohomological structure. In particular, some known multiresolution analyses are obtained as 1-coboundaries from trivial 1-cocycles. Next the authors show that the discrete wavelet transform corresponds to a generalized intertwiner between the action of the continuous affine group acting on \(L^2(R)\) and an action of the discrete affine semigroup acting on \(l^2(Z)\).