## 2 Analytical Properties of 5-Point-Window \(L_{1}\) Splines
\right.\]Therefore, \(p(q)=0\) and  
\[G(q;0)=\frac{5}{3}q\,,\]  
which implies that  
\[\frac{\mathrm{d}G(q;0)}{\mathrm{d}q}=\frac{5}{3}\,.\]  
When \(c=0\) and \(q<0\),  
\[\frac{\partial\phi(p,q;c)}{\partial p}=\left\{ \begin{array}{ll} \frac{2(\sqrt{10}-1)}{3}+\frac{10p^{2}+20 pq+6q^{2}}{6(p+q)^{2}}>0,& \quad\text{if }p>-2q,\\ \frac{2(\sqrt{10}-1)}{3}+1>0,&\quad\text{if }-\frac{1}{2}q\leq p\leq-2q,\\ \frac{2(\sqrt{10}-1)}{3}-\frac{10p^{2}+20 pq+6q^{2}}{6(p+q)^{2}}>0,& \quad\text{if }0<p<-\frac{1}{2}q,\\ -\frac{2(\sqrt{10}-1)}{3}-\frac{10p^{2}+20 pq+6q^{2}}{6(p+q)^{2}}<0,& \quad\text{if }p<0.\end{array} \right.\]  
Therefore, \(p(q)=0\) and  
\[G(q;0)=-\frac{5}{3}q\,,\]  
which implies that  
\[\frac{\mathrm{d}G(q;0)}{\mathrm{d}q}=-\frac{5}{3}\,.\]  
The proofs for \(c>0\) and \(c<0\) are similar to the proof for \(c=0\) and are omitted. Now, we let  
\[ \begin{array}{ll}p_{1}=b_{1}-\triangle z_{1}\,,&\quad p_{2}=b_{3}-\triangle z _{2}\,,\\ q_{1}=b_{2}-\triangle z_{1}\,,&\quad q_{2}=b_{2}-\triangle z_{2}\,,\\ c_{1}=\triangle z_{0}-\triangle z_{1}\,,&\quad c_{2}=\triangle z_{3}-\triangle z _{2}\end{array}  \tag{13}\]  
With this notation, we have  
\[G_{1}(b_{2})+G_{2}(b_{2})=G(q_{1};c_{1})+G(q_{2};c_{2})=G(b_{2}-\triangle z_{ 1};c_{1})+G(b_{2}-\triangle z_{2};c_{2}) \tag{14}\]  
**Remark**. Later in this paper, we will use \(\triangle z_{1}-\triangle z_{0}\), \(\triangle z_{2}-\triangle z_{1}\) and \(\triangle z_{3}-\triangle z_{2}\) to classify cases of linearity, convexity and oscillation. However, for clarity of the analysis in much of the remainder of this section, we use \(c_{1}\) to denote \(\triangle z_{0}-\triangle z_{1}\) instead of \(\triangle z_{1}-\triangle z_{0}\) because \(G_{1}\) and \(G_{2}\) are defined in a symmetric manner in (9) and (10) and \(b_{0}\) and \(b_{4}\) are determined by \(b_{1}\) and \(b_{3}\), which are in turn determined by \(b_{2}\) (progression outward from the middle point). From Lemma 3, \(G_{1}(b_{2})+G_{2}(b_{2})\) is convex and continuous for \(b_{2}\in R\) and is differentiable except at \(b_{2}=\triangle z_{1}\) and \(b_{2}=\triangle z_{2}\). If \(b_{2}<\min\{\triangle z_{1}+\frac{2-\sqrt{10}}{\sqrt{10}}|c_{1}|,\triangle z_{2 }+\frac{2-\sqrt{10}}{\sqrt{10}}|c_{2}|\}\), then  
\[\frac{\mathrm{d}G_{1}(b_{2})}{\mathrm{d}b_{2}}+\frac{\mathrm{d}G_{2}(b_{2})}{ \mathrm{d}b_{2}}<0\,.\]  
If \(b_{2}>\max\{\triangle z_{1}-\frac{2-\sqrt{10}}{\sqrt{10}}|c_{1}|,\triangle z_{ 2}-\frac{2-\sqrt{10}}{\sqrt{10}}|c_{2}|\}\), then  
\[\frac{\mathrm{d}G_{1}(b_{2})}{\mathrm{d}b_{2}}+\frac{\mathrm{d}G_{2}(b_{2})}{ \mathrm{d}b_{2}}>0\,.\]Therefore, the scalars  
\[\min\{\triangle z_{1}+\frac{2-\sqrt{10}}{\sqrt{10}}|c_{1}|,\triangle z _{2}+\frac{2-\sqrt{10}}{\sqrt{10}}|c_{2}|\}\] \[=\min\{\triangle z_{1}+\frac{2-\sqrt{10}}{\sqrt{10}}|\triangle z_{ 0}-\triangle z_{1}|,\triangle z_{2}+\frac{2-\sqrt{10}}{\sqrt{10}}|\triangle z_{ 3}-\triangle z_{2}|\} \tag{15}\]  
and  
\[\max\{\triangle z_{1}-\frac{2-\sqrt{10}}{\sqrt{10}}|c_{1}|, \triangle z_{2}-\frac{2-\sqrt{10}}{\sqrt{10}}|c_{2}|\}\] \[=\max\{\triangle z_{1}-\frac{2-\sqrt{10}}{\sqrt{10}}|\triangle z_{ 0}-\triangle z_{1}|,\triangle z_{2}-\frac{2-\sqrt{10}}{\sqrt{10}}|\triangle z_ {3}-\triangle z_{2}|\} \tag{16}\]  
form a lower and upper bound, respectively, for \(b_{2}^{*}\), the optimal \(b_{2}\). Since \(G_{1}(b_{2})+G_{2}(b_{2})\) is convex, we could (if the lower bound is less than the upper bound) use any line search method to find \(b_{2}^{*}\). However, simply using line search methods at this point does not reveal geometric properties of the spline and does not lead to efficient calculation of \(b_{2}^{*}\). The geometric properties of the set of 5 data points can be classified by looking at \(\triangle z_{1}-\triangle z_{0}\), \(\triangle z_{2}-\triangle z_{1}\) and \(\triangle z_{3}-\triangle z_{2}\). For example, \(\triangle z_{1}-\triangle z_{0}=0\) means that the the first three points lie on a straight line; \(\triangle z_{1}-\triangle z_{0}>0\) means that the first three points are convex.