Symmetric MRA tight wavelet frames with three generators and high vanishing moments

Let phi be a compactly supported symmetric real-valued refinable function in L-2(R) with a finitely supported symmetric real-valued mask on Z. Under the assumption that the shifts of phi are stable, in this paper we prove that one can always construct three wavelet functions psi(1), psi(2), and psi(3) such that (i) All the wavelet functions psi(1), psi(2), and psi(3) are compactly supported, real-valued and finite linear combinations of the functions phi (2. -k), k is an element of Z; (ii) Each of the wavelet functions psi(1), psi(2), and psi(3) is either symmetric or antisymmetric; (iii) {psi(1), psi(2), psi(3)} generates a tight wavelet frame in L-2(R), that is, parallel tofparallel to(2) = SigmaSigmaSigma(l=1 jis an element ofZ kis an element ofZ)(3)\<f, psi(j,k)(l)>\(2) f is an element of L-2(R), where psi(j,k)(l) : = 2(j/2)psi(l)(2(j) . -k), l = 1, 2, 3 and j, k is an element of Z; (iv) Each of the wavelet functions psi(1), psi(2), and psi(3) has the highest possible order of vanishing moments, that is, its order of vanishing moments matches the order of the approximation order provided by the refinable function phi.