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 $H(\omega ) = \Sigma h_n \exp ( - 2\pi in\omega )$ satisfies the identity $|H(\omega )|^2 + |H(\omega + \frac{1}{2})|^2 = 1$ 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.