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

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 \([1-N, N]\).

MSC:

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

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