Compactly supported orthogonal symmetric scaling functions

Daubechies (1988, Comm. Pure Appl. Math. 41, 909-996) showed that, except for the Hear function, there exist no compactly supported orthogonal symmetric scaling functions for the dilation q = 2. Nevertheless, such scaling functions do exist for dilations q > 2 (as evidenced by Chui and Lien's construction (1995, Appl. Comput. Harmon. Anal. 2, 68-84) for q = 3); these functions are the main object of this paper. We construct new symmetric scaling functions and introduce the "Batman" family of continuous symmetric scaling functions with very small supports. We establish the exact smoothness of the "Batman" scaling functions using the joint spectral radius technique. (C) 1999 Academic Press.