Summary: We study the two-scale dilation equation \(f(x)=\sum^n_{k=0} c_kf(2x-k)\), where the coefficients \(c_k\) are real and \(\sum_k c_{2k}= \sum_k c_{2k+1}=1\). By expressing the dilation equation in matrix product form, we prove a necessary and sufficient condition for the cascade algorithm (introduced by Daubechies and Lagarias) to converge uniformly to a continuous solution. We also establish several basic relations between the convergence of infinite products of matrices and the existence and regularity of solutions to the two-scale dilation equations.