Dual Riesz bases and the canonical operator

Let g(j,k)(X):=2(nj/2)g(2(j)x-k). A set G(0):={g(l), l=1,...,m} of functions in L-2 (R-n) is called an R-family if G := {g(j,k)(l); l=1,...m, j is an element of Z, k is an element of Z(n)} is a Riesz basis of L-2(R-n). If both G and its dual are generated by R-families, then G(0) is called a W-family. In this article we present conditions under which a Riesz basis is generated by a W-family. The main result is a method to obtain W-families generated by multiresolution analyses by perturbations of semiorthogonal W-families generated by multiresolution analyses. As an application we give examples of affine Riesz bases that are not semiorthogonal, but are generated by W-families.