Asymptotic analysis of Daubechies polynomials

To study wavelets and filter banks of high order, we begin with the zeros of ${\mathbf {B}}_{p}(y)$. 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 $p$ additional zeros at $z = -1$, and this construction makes it orthogonal and maximally flat. The dilation equation leads to orthogonal wavelets with $p$ vanishing moments. Symmetric biorthogonal wavelets (generally better in image compression) come similarly from a subset of the zeros of ${\mathbf {B}}_{p}(y)$. 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 $|z-z^{-1}| = 2$. All zeros have $|y| \le 1/2$, and the rightmost zeros approach $y = 1/2$ (corresponding to $z= \pm i$ ) with speed $p^{- 1/2}$. The curve $|4y(1-y)| = {(4 \pi p)}^{{1}/{2p}} \, |1-2y|^{ 1/p}$ gives a very accurate approximation for finite $p$. The wide dynamic range in the coefficients of ${\mathbf {B}}_{p}(y)$ makes the zeros difficult to compute for large $p$. Rescaling $y$ by $4$ allows us to reach $p = 80$ by standard codes.