Orthogonality criteria for multi-scaling functions

Necessary and sufficient conditions for the orthonormality of a multi-scaling function phi with integer dilation factor a and multiplicity r are established. Here phi := (phi(1),...,phi(r))(T) and satisfies [GRAPHICS] for some positive integer M, with P-0,..., P-M being r x r real matrices and P-0, P-M not equal 0. These conditions also apply to the setting of the usual uni-scaling functions and uniwavelets (i.e., multiplicity r = 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. (C) 1998 Academic Press.