AN ARITHMETIC CHARACTERIZATION OF THE CONJUGATE QUADRATURE FILTERS ASSOCIATED TO ORTHONORMAL WAVELET BASES

Let h = {h(n)} be a sequence of complex numbers with finite length such that H(omega) = SIGMA h(n) exp(-2piinomega) satisfies the identity \H(omega)\2 + \H(omega + 1/2)\2 = 1 and H(0) = 1, i.e., h is the impulse response of a conjugate quadrature filter. In this paper, we give a characterization, by the real roots of H(omega), of the sequences h that generate an orthonormal wavelet basis in the sense of the theory developed by Meyer and Daubechies. This result leads to a counterexample to Pollen's conjecture.