## 2 Analytical Properties of 5-Point-Window \(L_{1}\) Splines
\begin{table}
\begin{tabular}{|c|c c c|c|c|c|c|} \hline Case & \multicolumn{4}{c|}{Sign of} & Same as & \multicolumn{2}{c|}{Linearity} & Convexity & \multicolumn{1}{c|}{Oscillation} \\  & \(\triangle z_{1}-\triangle z_{0}\) & \(\triangle z_{2}-\triangle z_{1}\) & \(\triangle z_{3}-\triangle z_{2}\) & Case & \multicolumn{2}{c|}{\(/\) Concavity} & \multicolumn{1}{c|}{Oscillation} \\ \hline
1 & 0 & 0 & 0 & & Yes & Yes & No \\
2 & 0 & 0 & + & & Yes & Yes & No \\
3 & 0 & 0 & \(-\) & 2 & Yes & Yes & No \\
4 & 0 & + & 0 & & Yes & Yes & No \\
5 & 0 & + & + & & Yes & Yes & No \\
6 & 0 & + & \(-\) & & Yes & No & No \\
7 & 0 & \(-\) & 0 & 4 & Yes & Yes & No \\
8 & 0 & \(-\) & + & 6 & Yes & No & No \\
9 & 0 & \(-\) & \(-\) & 5 & Yes & Yes & No \\
10 & + & 0 & 0 & 2 & Yes & Yes & No \\
11 & + & 0 & + & & Yes & Yes & No \\
12 & + & 0 & \(-\) & & Yes & No & No \\
13 & + & + & 0 & 5 & Yes & Yes & No \\
14 & + & + & + & & No & Yes & No \\
15 & + & + & \(-\) & & No & No & No \\
16 & + & \(-\) & 0 & 6 & Yes & No & No \\
17 & + & \(-\) & + & & No & No & Yes \\
18 & + & \(-\) & \(-\) & 15 & No & No & No \\
19 & \(-\) & 0 & 0 & 2 & Yes & Yes & No \\
20 & \(-\) & 0 & + & 12 & Yes & No & No \\
21 & \(-\) & 0 & \(-\) & 11 & Yes & Yes & No \\
22 & \(-\) & + & 0 & 6 & Yes & No & No \\
23 & \(-\) & + & + & 15 & No & No & No \\
24 & \(-\) & + & \(-\) & 17 & No & No & Yes \\
25 & \(-\) & \(-\) & 0 & 5 & Yes & Yes & No \\
26 & \(-\) & \(-\) & + & 15 & No & No & No \\
27 & \(-\) & \(-\) & \(-\) & 14 & No & Yes & No \\ \hline \end{tabular}
\end{table} 
Table 1: 27 cases in the 5-point window method. **Case 5 and 6. ** In Case 5, \(\triangle z_{0}=\triangle z_{1}<\triangle z_{2}<\triangle z_{3}\), \(c_{1}=0\) and \(c_{2}>0\). In Case 6, \(\triangle z_{0}=\triangle z_{1}<\triangle z_{2}\), \(\triangle z_{2}>\triangle z_{3}\), \(c_{1}=0\) and \(c_{2}<0\). In both cases, from Lemma 3,  
\[\frac{\mathrm{d}G_{1}(b_{2})}{\mathrm{d}b_{2}}+\frac{\mathrm{d}G_{2 }(b_{2})}{\mathrm{d}b_{2}}>\frac{5}{3}-\frac{5}{3}=0 \text{if }b_{2}-\triangle z_{1}>0,\] \[\frac{\mathrm{d}G_{1}(b_{2})}{\mathrm{d}b_{2}}+\frac{\mathrm{d}G _{2}(b_{2})}{\mathrm{d}b_{2}}\leq-\frac{5}{3}<0 \text{if }b_{2}-\triangle z_{1}<0.\]  
The unique optimal solution in both cases is \(b_{2}^{*}=\triangle z_{1}\). **Case 11. ** In this case, \(\triangle z_{0}<\triangle z_{1}=\triangle z_{2}<\triangle z_{3}\), \(c_{1}<0\) and \(c_{2}>0\). From Lemma 3,  
\[\frac{\mathrm{d}G_{1}(b_{2})}{\mathrm{d}b_{2}}+\frac{\mathrm{d}G _{2}(b_{2})}{\mathrm{d}b_{2}}>0 \text{if }b_{2}-\triangle z_{1}>0,\] \[\frac{\mathrm{d}G_{1}(b_{2})}{\mathrm{d}b_{2}}+\frac{\mathrm{d}G _{2}(b_{2})}{\mathrm{d}b_{2}}<0 \text{if }b_{2}-\triangle z_{1}<0.\]  
The unique optimal solution is \(b_{2}^{*}=\triangle z_{1}\). **Case 12. ** In this case, \(\triangle z_{0}<\triangle z_{1}=\triangle z_{2}\), \(\triangle z_{2}>\triangle z_{3}\), \(c_{1}<0\) and \(c_{2}<0\). From Lemma 3,  
\[\frac{\mathrm{d}G_{1}(b_{2})}{\mathrm{d}b_{2}}+\frac{\mathrm{d}G _{2}(b_{2})}{\mathrm{d}b_{2}}>0 \text{if }b_{2}-\triangle z_{1}>\min\{\frac{2-\sqrt{10}}{\sqrt{10}}c_{1},\frac{2-\sqrt{10}}{\sqrt{10}}c_{2}\},\] \[\frac{\mathrm{d}G_{1}(b_{2})}{\mathrm{d}b_{2}}+\frac{\mathrm{d}G _{2}(b_{2})}{\mathrm{d}b_{2}}=0 \text{if }0<b_{2}-\triangle z_{1}\leq\min\{\frac{2-\sqrt{10}}{\sqrt{10 }}c_{1},\frac{2-\sqrt{10}}{\sqrt{10}}c_{2}\},\] \[\frac{\mathrm{d}G_{1}(b_{2})}{\mathrm{d}b_{2}}+\frac{\mathrm{d}G _{2}(b_{2})}{\mathrm{d}b_{2}}<0 \text{if }b_{2}-\triangle z_{1}<0.\]  
Any solution that lies in \([\triangle z_{1},\triangle z_{1}+\min\{\frac{2-\sqrt{10}}{\sqrt{10}}c_{1}, \frac{2-\sqrt{10}}{\sqrt{10}}c_{2}\}]\) is optimal. Since \(\triangle z_{1}=\delta_{2}=\triangle z_{2}\), the unique solution is \(b_{2}^{*}=\delta_{2}=\triangle z_{1}\). **Case 14. ** In this case, \(\triangle z_{0}<\triangle z_{1}<\triangle z_{2}<\triangle z_{3}\), \(c_{1}<0\) and \(c_{2}>0\).