The Smith-Barnwell condition and nonnegative scaling functions

It is shown that any periodic function m/sub 0/( zeta ) with nonnegative Fourier coefficients that satisfies the Smith-Barnwell conditions m/sub 0/(0)=1, mod m/sub 0/( zeta )/sup 2/+ mod m/sub 0/( zeta + pi ) mod /sup 2/=1 is of the form m/sub 0/( zeta )=exp(il zeta /2) cos k zeta /2 with l, k odd integers. As a consequence it is concluded that any nonnegative scaling function with orthonormal integer translates is of the form x/sub (k,k+1)/ with k in Z.<>