The Smith-Barnwell condition and nonnegative scaling functions

w)I2 + [G(w + :)I' = 1. Its periodic extension then satisfies +*(w) 2 1 and hence a sampling function with q w) = +(w)/G*(w) belongs to L2(Z2). 5) Daubechies Wavelets: The scaling function for the simplest class of Daubechies wavelets, those with support on [0, 31, are defined by the dilation equations REFERENCES [ 11 I. Daubechies, " Orthonormal bases of compactly supported wavelets, " Comm. cg = U(,-1)/(u2 + I) , C , = (1-U)/(" ' + I) , c2 = (U + 1)/(u2 + l) , Since p (j) = C c k p (2 j-k) = ~ ~ ~ ~-~ ~ (k) it follows that ~ (l) , (0(2), ~ (3)) ' is an eigenvector of the matrix C = [ C ~ j-k ]. Since p(0) = 4 3) = 0, we need consider only the matrix c3 = u (u + 1)/(u2 + l) , V € R. Abstract-It is shown that any periodic function mo(f) with nonnega-tive Fourier coefficients that satisfies the Smith-Barnwell conditions nzo(0)=l, I m , (f) 1 * + I m o (f + *) I * = 1 is of the form mo(f)= exp(iilf)cos f k l with I , k odd integers. As a consequence we conclude that any nonnegative scaling function with orthonormal integer translates is of the form x ~ ~ , ~ + ,) with k € 2 tivity.