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Necessary 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.

When the scale factor a = 2 is used to define scaling functions and wavelets, it is well-known that any compactly supported orthonormal (o.n.) wavelet, which is symmetric or antisymmetric, is some integer translate and possible sign change of the Haar function. The objective of this paper is to exhibit the construction of compactly supported o.n. symmetric scaling functions with scale factor a = 3 and the two corresponding compactly supported o.n. wavelets, one being symmetric and the other being antisymmetric. Examples of low-order ones are given. On the other hand, we also show that even with a = 3, compactly supported orthonormal scaling functions and wavelets with minimum supports cannot be symmetric or antisymmetric. unless they are the trivial Haar-type functions. A lower bound of the size of the support for a given regularity exponent to achieve symmetry and antisymmetry is derived.