REFINABLE FUNCTION VECTORS

Refinable function vectors are usually given in the form of an infinit e product of their refinement (matrix) masks in the frequency domain a nd approximated by a cascade algorithm in both time and frequency doma ins. We provide necessary and sufficient conditions for the convergenc e of the cascade algorithm. We also give necessary and sufficient cond itions for the stability and orthonormality of refinable function vect ors 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.