Crystallographic Haar Wavelets

(Γ,a)-crystallographic multiwavelets are a finite set of functions Y = { yⁱ}_i=1^LPsi= { psi ^{i}}_{i=1}^{L}, which generate an orthonormal basis, a Riesz basis or a Parseval frame for L ²(ℝ ^d ), under the action of a crystallographic group Γ, and powers of an appropriate expanding affine map a, taking the place of the translations and dilations in classical wavelets respectively. Associated crystallographic multiresolution analysis of multiplicity n ((Γ,a)-MRA) are defined in a natural way. A complete characterization of scaling function vectors which generates Haar type (Γ,a)-MRA’s in terms of (Γ,a)-multireptiles is given. Examples of (Γ,a)-MRA crystallographic wavelets of Haar type in dimension 2 and 3 are provided.