## 3 Linkage of Geometric Properties of Data Points and \(L_{1}\) Spline
After applying Lemma 5 to all convex situations and eliminating equivalent cases, we can identify that the \(L_{1}\) spline is convex on \([x_{i},x_{i+1}]\) in Cases 1\(\leftrightarrow\)1, 1\(\leftrightarrow\)2, 2\(\leftrightarrow\)5, 10\(\leftrightarrow\)2, 11\(\leftrightarrow\)5, 5\(\leftrightarrow\)14-3, is not convex in Cases 2\(\leftrightarrow\)4, 4\(\leftrightarrow\)11, 5\(\leftrightarrow\)14-1, 5\(\leftrightarrow\)14-2, 5\(\leftrightarrow\)14-4, 14-1\(\leftrightarrow\)14-4, 14-2\(\leftrightarrow\)14-4, and is not determined in Cases 5\(\leftrightarrow\)13, 14-1\(\leftrightarrow\)14-1, 14-1\(\leftrightarrow\)14-2, 14-1\(\leftrightarrow\)14-3, 14-2\(\leftrightarrow\)14-3. However, the \(L_{1}\) spline does not have extraneous oscillation on \([x_{i},x_{i+1}]\) as is shown in the remainder of this section. **Lemma 6**. : \(b_{i}^{*}\in[b_{i}^{l},b_{i}^{u}]\)_, where \(b_{i}^{l}=\min\{\triangle z_{i-1},\triangle z_{i}\}\) and \(b_{i}^{u}=\max\{\triangle z_{i-1},\triangle z_{i}\}\)._  
Proof. : The proof comes directly from the analysis of the 27 cases. **Remark**. Lemma 6 does not hold for global \(L_{1}\) splines. Consider, for example, the 11 data points \((-5,4)\), \((-4,3)\), \((-3,2)\), \((-2,1)\), \((-1,0)\), \((0,0)\), \((1,0)\), \((2,-1)\), \((3,-2)\), \((4,-3)\) and \((5,-4)\). By the 5-point-window method, \(b_{5}^{*}\) (the derivative at \(x=0\)) is 0. In contrast, the \(b_{5}^{*}\) of the global \(L_{1}\) spline is 0.37304. **Lemma 7**. : _If \(b_{i}^{*}\leq\triangle z_{i}\leq b_{i+1}^{*}\), then the cubic \(L_{1}\) spline is bounded above by the linear spline \(\zeta(x)\) on the interval \([x_{i},x_{i+1}]\)._  
Proof. : Given \(b_{i}^{*}\leq\triangle z_{i}\leq b_{i+1}^{*}\), the cubic \(L_{1}\) spline on \([x_{i},x_{i+1}]\) can be written as  
\[z(x)=z_{i}+b_{i}^{*}(x-x_{i})-\frac{1}{h_{i}}(2b_{i}^{*}+b_{i+1}^{*}-3\triangle z _{i})(x-x_{i})^{2}+\frac{1}{h_{i}^{2}}(b_{i}^{*}+b_{i+1}^{*}-2\triangle z_{i}) (x-x_{i})^{3}\,.\]  
Let \(x-x_{i}=\lambda h_{i}\), \(0\leq\lambda\leq 1\), then  
\[\zeta(x)-z(x) =(\triangle z_{i}-b_{i}^{*})(x-x_{i})+\tfrac{1}{h_{i}}(2b_{i}^{* }+b_{i+1}^{*}-3\triangle z_{i})(x-x_{i})^{2}\] \[-\tfrac{1}{h_{i}^{2}}(b_{i}^{*}+b_{i+1}^{*}-2\triangle z_{i})(x- x_{i})^{3}\] \[=\frac{1}{h_{i}}(b_{i+1}^{*}-b_{i}^{*})+\frac{6}{h_{i}^{2}}(b_{i }^{*}+b_{i+1}^{*}-2\triangle z_{i})(x-x_{i}-\frac{h_{i}}{2})\,.\]0 \[=\frac{1}{h_{i}}(b_{i+1}^{*}-b_{i}^{*})+\frac{6}{h_{i}^{2}}(b_{i }^{*}+b_{i+1}^{*}-2\triangle z_{i})(x-x_{i}-\frac{h_{i}}{2})\,.\]1 \[=\frac{1}{h_{i}}(b_{i+1}^{*}-b_{i}^{*})+\frac{6}{h_{i}^{2}}(b_{i }^{*}+b_{i+1}^{*}-2\triangle z_{i})(x-x_{i}-\frac{h_{i}}{2})\,.\]2  
**Theorem 8**. : _If the linear spline is convex on the interval \([x_{i-1},x_{i+2}]\), that is, \(\triangle z_{i-1}\leq\triangle z_{i}\leq\triangle z_{i+1}\), then the cubic \(L_{1}\) spline is bounded above by the linear spline on the interval \([x_{i},x_{i+1}]\)._  
Proof. : The proof comes from Lemmas 6 and 7. The results in this section indicate that the \(L_{1}\) splines produced by the 5-point-window method with the proposed choice procedure for resolving nonuniqueness preserve linearity and convexity in many cases and do not oscillate excessively. From Lemma 6, we see that the \(b_{i}^{*}\) calculated by this method is always bounded by \(\triangle z_{i-1}\) and \(\triangle z_{i}\). This property is a prime factor in restricting oscillation of \(L_{1}\) splines and may in the future lead to additional theoretic results about the properties of \(L_{1}\) splines for non-over-oscillating interpolation of oscillatory data.