Two families of compactly supported Parseval framelets in L2(Rd)

For any dilation matrix with integral entries A ∈ Rd×d, d ≥ 1, we construct two families of Parseval wavelet frames in L2(Rd). Both families have compact support and any desired number of vanishing moments. The first family has | detA| generators. The second family has any desired degree of regularity. For the members of this family, the number of generators depends on the dilation matrix A and the dimension d, but never exceeds | detA| + d. Our construction involves trigonometric polynomials developed by Heller to obtain refinable functions, the Oblique Extension Principle, and a slight generalization of a theorem of Lai and Stöckler.