Positivity of refinable functions defined by nonnegative finite masks

Let (a(j)|j=0,1,…,N) with a(0),a(N)≠0 be a given nonnegative mask. Assume that the subdivision scheme with this mask is convergent. Let the associated refinable function be ϕ. So the support of ϕ is contained in [0,N]. Melkman conjectured in 1997 that unless the scheme is interpolatory and N>2 the refinable function ϕ is positive on (0,N). In the present paper we confirm this conjecture. A lower bound of ϕ on [2−m,N−2−m] is also given.