On the number of Daubechies scaling functions and a conjecture of Chyzak et al. (English) Zbl 0974.42028

Wang, Yang

Exp. Math. 10, No. 1, 87-89 (2001).

Let \(N\in\mathbb{Z}\), \(N\geq 1\). A so-called Daubechies scaling function \(\varphi\in L^2(\mathbb{R})\) with support in \([1-N, N]\) satisfies the dilation equation \[ \varphi(x)= \sum^N_{k=1- N}h_k \varphi(2x- k), \] where \[ \sum^N_{k=1- N} h_k= 2,\;\sum^N_{k=1- N} h_k h_{k- 2j}= 2\delta_{0,j},\;\sum^N_{k- 1-N} (-1)^k h_{1-k} k^j= 0 \] for all \(j= 0,\dots, N-1\). Using Riesz factorization of Laurent polynomials, the author shows that there exist at most \(2^{N-1}\) and at least \(2^{\lfloor N/2\rfloor}\) distinct Daubechies scaling functions with support in \([1-N, N]\).

Reviewer: Manfred Tasche (Rostock)

MSC:

42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems
12D05	Polynomials in real and complex fields: factorization

Keywords:

Riesz factorization; Laurent polynomials; Daubechies scaling functions

References:

[1]	Chyzak F., Experiment. Math. 10 (1) pp 67– (2001)
[2]	DOI: 10.1137/1.9781611970104 · Zbl 0776.42018 · doi:10.1137/1.9781611970104
[3]	DOI: 10.1137/0522089 · Zbl 0763.42018 · doi:10.1137/0522089
[4]	DOI: 10.1137/0523059 · Zbl 0788.42013 · doi:10.1137/0523059