Orthogonality Criteria for Multi-scaling Functions

AbstractNecessary and sufficient conditions for the orthonormality of a multi-scaling function φ with integer dilation factoraand multiplicityrare established. Here φ := (φ1, … , φr)Tand satisfies φ(x) =[formula]Pkφ(ax − k), for some positive integerM, withP0, … ,PMbeingr × rreal matrices andP0,PM≠ 0. These conditions also apply to the setting of the usual uni-scaling functions and uni-wavelets (i.e., multiplicityr= 1). The orthonormality of all the multi-scaling functions and multi-wavelets constructed in the wavelet literature can be verified. Extensive demonstrative examples are given