## 3 Linkage of Geometric Properties of Data Points and \(L_{1}\) Spline  
In this section, based on the analytic results for the solution at the middle point in each 5-point window, we present two theorems that link the local linearity, convexity and oscillatory properties of the original data set with the local linearity, convexity and oscillatory properties of the locally calculated \(L_{1}\) spline. In particular, we show that the locally calculated \(L_{1}\) spline does not \"over-oscillate\". The capability of the 5-point local window method to preserve linearity is described in the following theorem. **Theorem 4**. : _(Proposition 3 of [22]) If any three consecutive points in a five-point window are collinear with slope \(\triangle z\), then \(b_{i}^{*}=\triangle z\) except in Cases 4 and 7._  
Proof. : See Cases 1, 2, 4, 5, 6, 11 and 12 in Section 2. Theorem 4 indicates that local linearity of the data is preserved in the 5-point-window \(L_{1}\) spline with the \"reasonable\" exception of when two lines intersect at the point \((x_{i},z_{i})\). \(C^{1}\) continuity of the spline prevents linearity from being preserved in both intervals bordering on a corner \((x_{i},z_{i})\). Convexity is not as simple as linearity. To study the convexity of the \(L_{1}\) spline, we need to consider not just a node \(x_{i}\) but the whole interval \([x_{i},x_{i+1}]\). In this interval, the \(L_{1}\) spline is determined by \(b_{i}^{*}\) and \(b_{i+1}^{*}\), which are calculated using the six data points \((x_{k},z_{k})\), \(k=i-2,i-1,i,i+1,i+2,i+3\) in the two overlapping 5-point windows for \(b_{i}^{*}\) and \(b_{i+1}^{*}\). In the remainder of this section, we focus on the \(L_{1}\) spline in the interval \([x_{i},x_{i+1}]\) and assume that these six data points (and, therefore, also their linear spline interpolant) are convex on \([x_{i-2},x_{i+3}]\). The analysis in the rest of this section will reveal that the spline in \([x_{i},x_{i+1}]\) is not always convex, but, when not, the oscillation is not large. **Lemma 5**. : _The following statements are equivalent:_  
\[ \begin{array}{ll}(\,\mathrm{i})&\mbox{The cubic spline function is convex on the interval $[x_{i},x_{i+1}]$;}\\ (\,\mathrm{ii})&(b_{i+1}^{*}-b_{i}^{*})\geq 3|(b_{i+1}^{*}-\triangle z_{i})+(b_{ i}^{*}-\triangle z_{i})|;\\ (\,\mathrm{iii})&0\leq-\frac{1}{2}(b_{i}^{*}-\triangle z_{i})\leq(b_{i+1}^{*} -\triangle z_{i})\leq-2(b_{i}^{*}-\triangle z_{i}).\end{array} \]  
**Remark. ** Condition (iii) in Lemma 5 is equivalent to Proposition 3.1 in [24]. Proof. : Recall from the definition in Section 1 that \(h_{i}=x_{i+1}-x_{i}\), \(i=0,1,\ldots,I-1\). The second derivative of the cubic spline function on \([x_{i},x_{i+1}]\) is  
\[-\frac{2}{h_{i}}(2b_{i}^{*}+b_{i+1}^{*}-3\triangle z_{i})+\frac{6 }{h_{i}^{2}}(b_{i}^{*}+b_{i+1}^{*}-2\triangle z_{i})(x-x_{i})\] \[=\frac{1}{h_{i}}(b_{i+1}^{*}-b_{i}^{*})+\frac{6}{h_{i}^{2}}(b_{i }^{*}+b_{i+1}^{*}-2\triangle z_{i})(x-x_{i}-\frac{h_{i}}{2})\,.\]  
Let \(x-x_{i}=\lambda h_{i}\), \(0\leq\lambda\leq 1\), then  
\[\frac{1}{h_{i}}(b_{i+1}^{*}-b_{i}^{*})+\frac{6}{h_{i}^{2}}(b_{i}^{ *}+b_{i+1}^{*}-2\triangle z_{i})(x-x_{i}-\frac{h_{i}}{2})\] \[=\frac{1}{h_{i}}(b_{i+1}^{*}-b_{i}^{*})+\frac{6}{h_{i}}(b_{i}^{*} +b_{i+1}^{*}-2\triangle z_{i})(\lambda-\frac{1}{2})\,.\]  
Hence the cubic spline function is convex on the interval \([x_{i},x_{i+1}]\) if and only if  
\[\frac{1}{h_{i}}(b_{i+1}^{*}-b_{i}^{*})+\frac{6}{h_{i}}(b_{i}^{*}+ b_{i+1}^{*}-2\triangle z_{i})(\lambda-\frac{1}{2})\geq 0,\quad\forall\lambda\in[0,1],\] \[\Leftrightarrow (b_{i+1}^{*}-b_{i}^{*})\geq 3|b_{i}^{*}+b_{i+1}^{*}-2\triangle z_{i}|\] \[\Leftrightarrow 0\leq-\frac{1}{2}(b_{i}^{*}-\triangle z_{i})\leq(b_{i+1}^{*}- \triangle z_{i})\leq-2(b_{i}^{*}-\triangle z_{i})\,.\]  
Every contiguous set of six points comes from two 5-point windows. Let Case \(\alpha\leftrightarrow\beta\) denote Case/Subcase \(\alpha\) for the left window (27 cases and 9 subcases) and Case/Subcase \(\beta\) for the right window (also 27 cases and 9 subcases).