I'm trying to construct the (left) boundary-corrected Daubechies scaling function, following the article
A. Cohen, I. Daubechies, P. Vial. "Wavelets on the Interval and Fast Wavelet Transforms". 1993.
The authors say, that "plots have again been obtained via the cascade algorithm", but my tries fail and I can not get the plots of Fig. 6 on page 23 by myself.
The way, I'm trying to do it, is first analytically to get the values of $\phi^{left}(t)$ in all integers. Having this vector of values, one can easily compute the values in all binary rationals using the presented construction \begin{align*} \phi^{left}(t) = \sqrt{2} \sum^{N - 1}_{l=0} H^{left}_{k,l} \phi^{left}(2t-l) + \sqrt{2} \sum^{N + 2k}_{m=N} h^{left}_{k,m} \phi(2t-m), \end{align*} where $\phi$ is known "usual" (not boundary-corrected) Daubechies Scaling Function translated by $-N+1$ such that $supp(\phi) = [-N+1, N]$. Moreover, we assume that the coefficients $H^{left}_{k,l}$ and $ h^{left}_{k,m}$ are known. One can show that $supp(\phi^{left})=[0, N + k]$.
But here is my problem with this approach: Let, for example, $N=2$ and $k=0$, so that $$\phi^{left}(t) = \sqrt{2} H^{left}_{0,0} \phi^{left}(2t) + \sqrt{2} H^{left}_{0,1} \phi^{left}(2t - 1) + \sqrt{2} h^{left}_{0,2} \phi(2t-2).$$
Then we start with the system $\phi^{left}|_{\mathbb{Z}}$:
$$ \begin{bmatrix} \phi^{left}(0)\\ \phi^{left}(1)\\ \phi^{left}(2) \end{bmatrix} = \begin{bmatrix} \sqrt{2} H^{left}_{0,0} & 0 & 0\\ 0 & \sqrt{2} H^{left}_{0,1} & \sqrt{2} H^{left}_{0,0}\\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} \phi^{left}(0)\\ \phi^{left}(1)\\ \phi^{left}(2) \end{bmatrix} + \begin{bmatrix} 0\\ \sqrt{2} h^{left}_{0,2} \phi(0)\\ 0 \end{bmatrix}, $$ where $H^{left}_{0,0}=0.6033325119$, $H^{left}_{0,1}=0.6908955318$ and $h^{left}_{0,2}=-0.3983129977$. The system has a solution $$\phi^{left}(0) = 0; \;\;\; \phi^{left}(1) = \frac{\sqrt{2} h^{left}_{0,2} \phi(0)}{1 - \sqrt{2} H^{left}_{0,1}} = -33.56346328045783; \;\;\; \phi^{left}(2) = 0,$$ what is obviously wrong based on the graph bellow (Fig.6 in (1) on page 23):
Any help or even idea how to construct this scaling function would be very helpful!
The problem was that I had a misprint in my formula. The correct form (corresponding to the formula from the article) is \begin{align*} \phi^{left}_k(t) = \sqrt{2} \sum^{N - 1}_{l=0} H^{left}_{k,l} \phi^{left}_l(2t) + \sqrt{2} \sum^{N + 2k}_{m=N} h^{left}_{k,m} \phi(2t-m). \end{align*} This equation is quite easy to solve directly.