Series Representation of Daubechies' Wavelets (Journal of Computational Mathematics, 1997)
This paper gives a kind of series representation of the scaling functions $\phi_N$ and the associated wavelets $\psi _N$ constructed by Daubechies. Based on Poission summation formula, the functions $\phi_N(x+N-1)$, $\phi_N(x+N)$, $\cdots$, $\phi_N(x+2N-2) (0 \leq x\leq 1)$ are linearly represented by $\phi_N(x)$, $\phi_N(x+1)$, $\cdots$, $\phi_N(x+2N-2)$ and some polynomials of order less than $N$, and $\Phi _0(x):=(\phi_N(x), \phi_N(x+1), \cdots,\phi_N(x+N-2))^t$ is translated into a solution of a nonhomogeneous vector-valued functional equation $$f(x)=A_d f(2x-d)+P _d(x),x\in [\frac{d}{2},\frac{d+1}{2}], d=0,1,$$ where $A_0,A_1$ are $(N-1)\times (N-1)$-dimensional matrices, the components of $P_0(x), P_1(x)$ are polynomials of order less than $N$. By iteration, $\Phi_0(x)$ is eventually represented as an $(N-1)$-dimensional vector series $\sum\limits_{k=0} ^\infty u _k(x)$ with vector norm $\|U _k(x)\|\leq C\beta ^k$, where $\beta =\beta_N