Reenable Function Vectors

Reenable function vectors are usually given in the form of an innnite product of their reenement (matrix) masks in the frequency domain and approximated by a cascade algorithm in both time and frequency domains. We provide necessary and suu-cient conditions for the convergence of the cascade algorithm. We also give necessary and suucient conditions for the stability and orthonormality of reenable function vectors in terms of their reenement matrix masks. Regularity of function vectors gives smoothness orders in the time domain, and decay rates at innnity in the frequency domain. Regularity criteria are established in terms of the vanishing moment order of the matrix mask.