Asymptotic analysis of Daubechies polynomials |
| |
Authors: | Jianhong Shen Gilbert Strang |
| |
Affiliation: | Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 ; Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 |
| |
Abstract: | To study wavelets and filter banks of high order, we begin with the zeros of . This is the binomial series for , truncated after terms. Its zeros give the zeros of the Daubechies filter inside the unit circle, by . The filter has additional zeros at , and this construction makes it orthogonal and maximally flat. The dilation equation leads to orthogonal wavelets with vanishing moments. Symmetric biorthogonal wavelets (generally better in image compression) come similarly from a subset of the zeros of . We study the asymptotic behavior of these zeros. Matlab shows a remarkable plot for . The zeros approach a limiting curve in the complex plane, which is the circle . All zeros have , and the rightmost zeros approach (corresponding to ) with speed . The curve gives a very accurate approximation for finite . The wide dynamic range in the coefficients of makes the zeros difficult to compute for large . Rescaling by allows us to reach by standard codes. |
| |
Keywords: | |
|
| 点击此处可从《Proceedings of the American Mathematical Society》浏览原始摘要信息 |
|
点击此处可从《Proceedings of the American Mathematical Society》下载全文 |
|