ASYMPTOTIC ANALYSIS OF DAUBECHIES POLYNOMIALS

TO Study wavelets and filter banks of high order, we begin with the ze ros of B-p(y). This is the binomial series for (1 - y)(-p), truncated after p terms. Its zeros give the p - 1 zeros of the Daubechies filter inside the unit circle, by z + z(-1) = 2 - 4y. The filter has p addit ional zeros at z = -1, and this construction makes it orthogonal and m aximally flat. The dilation equation leads to orthogonal wavelets with p vanishing moments. Symmetric biorthogonal wavelets (generally bette r in image compression) come similarly from a subset of the zeros of B -p(y). We study the asymptotic behavior of these zeros; Matlab shows a remarkable plot for p = 70. The zeros approach a limiting curve \4y(1 - y)\ = 1 in the complex plane, which is the circle \z - z(-1)\ = 2. An zeros have \y\ less than or equal to 1/2, and the rightmost zeros a pproach y = 1/2 (corresponding to z = +/-i) with speed p(-1/2). The cu rve \4y(1 - y)\ = (4 pi p)(1/2p)\1 - 2y\(1/p) gives a very accurate ap proximation for finite p. The wide dynamic range in the coefficients o f B-p(y) makes the zeros difficult to compute for large p. Rescaling y by 4 allows us to reach p = 80 by standard codes.