## 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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