Some compactly supported Riesz wavelets associated to any Ed(2)(ℤ) dilation

Abstract

Let [Formula: see text] be a [Formula: see text] dilation matrix with integer entries and with determinant [Formula: see text]. The existence of Riesz wavelets with compact support associated to [Formula: see text] is proved. Our proof is constructive and the generators of these Riesz wavelets may be taken to be symmetric, with high regularity and many vanishing moments. In our construction, we first study the structure of the quotient group [Formula: see text]. Afterwards and perhaps the main advance in this paper is the fact that we obtain a family of trigonometric polynomials on [Formula: see text] with zeros only in [Formula: see text]. At this point, we are able to get scaling functions with compact support of a multiresolution analysis. In addition, Riesz wavelets with compact support hold by standard multiresolution techniques. Finally, we give some examples of Riesz wavelets where we emphasize on a numerical estimation of their regularity.