Refinable function vectors

Refinable function vectors are usually given in the form of an infinite product of their refinement (matrix) masks in the frequency domain and approximated by a cascade algorithm in both time and frequency domains. We provide necessary and sufficient conditions for the convergence of the cascade algorithm. We also give necessary and sufficient conditions for the stability and orthonormality of refinable function vectors in terms of their refinement matrix masks. Regularity of function vectors gives smoothness orders in the time domain and decay rates at infinity in the frequency domain. Regularity criteria are established in terms of the vanishing moment order of the matrix mask.