Asymptotic analysis of Daubechies polynomials

Authors:
Jianhong Shen and Gilbert Strang

Journal:
Proc. Amer. Math. Soc. **124** (1996), 3819-3833

MSC (1991):
Primary 41A58

DOI:
https://doi.org/10.1090/S0002-9939-96-03557-5

MathSciNet review:
1346987

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Abstract | References | Similar Articles | Additional Information

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.

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Additional Information

**Jianhong Shen**

Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Email:
jhshen@math.mit.edu

**Gilbert Strang**

Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Email:
gs@math.mit.edu

DOI:
https://doi.org/10.1090/S0002-9939-96-03557-5

Received by editor(s):
June 25, 1995

Dedicated:
Dedicated to Gabor Szegö on the 100th anniversary of his birth

Communicated by:
James Glimm

Article copyright:
© Copyright 1996
American Mathematical Society