Zeros of a mask function and the orthogonality of the related scaling function

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) over cap>(xi) = Pi(j-1)(infinity) m(2(-j)xi), phi(x) = integral(-infinity)( +infinity) <(phi)over cap>(xi)(e2 pi ix xi) d xi. The orthogonality of phi( x), i.e., integral(-infinity)(+infinity) phi(x-m) <(phi)over bar> (x - n) d x = delta(m,n), is related to the zeros of m(xi). In 1995, A. Cohen and R. D. Ryan, "Wavelets and Multiscale Signal Processing," Chapman & Hall, prove d that if m(xi) has no zeros in [-1/6, 1/6], then phi(x) is an orthogonal f unction. In (X. Zhou and W. Su, Appl. Comput. Harmon. Anal. 8, 197-202 (200 0)) we proved that if m(xi) has no zeros in [-1/10, 1/10] and \ m(1/6)\ - \ m(-1/6)\ > 0, then phi(x) is also an orthogonal function. A natural questi on, then, is whether this procedure can be extended to arbitrarily small in tervals, i.e., whether for any Delta is an element of (0, 1/2) there exists a finite set Z(Delta) such that the orthogonality of cp(x) is ensured by t he requirement that \ m(xi)\ > 0 for \xi \ less than or equal to Delta and for some xi is an element of Z(Delta). In this paper, we show that this is true if and only if Delta exceeds a strictly positive Delta(0) which we der ive explicitly. (C) 2000 Academic Press.