A criterion of orthogonality for a class of scaling functions

Suppose that m(xi) is a trigonometric polynomial with period 1 satisfying m (0) = 1 and \m(xi)\(2) + \m(xi + 1/2)\(2) = 1 for all xi in R. Let <(phi)ov er cap>(xi) = Pi(j=1)(infinity)(2(-j)xi), phi(x) = integral(-infinity)(+inf inity) <(phi)over cap>(xi)e(2 pi ix xi) d xi. In 1989, Cohen [3] proved tha t if m(xi) has no zeros in [-1/6, 1/6], then phi(x) is an orthogonal functi on, i.e., integral(-infinity)(+infinity) phi(x - m) <(phi)over bar>(x - n) dx = delta(m,n). In this paper, we prove a generalization: if m(xi) has no zeros in [-1/10, 1/10] > \m(1/6)\ + m(-1/6)\ > 0, then phi(x) is an orthogo nal function. (C) 2000 Academic Press.