## 1 Introduction
Proposition 9 of [2] shows that one can preserve linearity over a larger set of points by calculating the \(L_{1}\) spline using local 5-point windows rather than globally. This has a potential strategic implication, namely, that one may be able, by replacing a global minimization problem by a set of local minimization problems, to both further improve the shape preservation capabilities of \(L_{1}\) splines and at the same time reduce the computing time because the local problems are independent of each other and can be solved in parallel. This is the opportunity that this paper wishes to investigate. The authors [PERSON], [PERSON] and [PERSON] of [2] have followed up on their results of 2007 with an article [22] containing new analytical results about preservation of linearity by windowed, rotation-invariant parametric \(L_{1}\) splines of degrees 3 and higher. In contrast, this present paper considers linearity, convexity and oscillation for 5-point-window, rotation-dependent nonparametric cubic \(L_{1}\) splines. The precise purpose of this present paper is to provide analytical results that link linearity, convexity and oscillatory properties of the data on 5-point windows with linearity, convexity, oscillatory and uniqueness properties of the resulting \(L_{1}\) spline. In each 5-point window, a local \(L_{1}\) spline functional is used to determine the first derivative at the middle (or, near boundaries, other) point in this window. After the first derivatives at all of the data points have been determined, a \(C^{1}\) piecewise cubic interpolant, called the \(L_{1}\) spline (or \"locally calculated cubic \(L_{1}\) spline\"), is set up by Hermite interpolation in each interval. In Section 2, we investigate analytical properties of the spline functional that link local geometric properties of 5-point windows of the data with geometric properties of the local \(L_{1}\) splines on these windows. Based on the analytical results for 5-point windows, we investigate in Section 3 the properties of the \(C^{1}\) piecewise cubic interpolant that has derivatives determined by these 5-point-window \(L_{1}\) splines. In Section 4, we summarize the results presented in the previous sections and describe potential computational implications of these results. All of the quantities in this paper are real quantities. The nodes \(x_{i}\), \(i=0,1,\ldots,I\), are a strictly monotonic but otherwise arbitrary partition of the finite interval \([x_{0},x_{I}]\). Let \(h_{i}=x_{i+1}-x_{i}\), \(i=0,1,\ldots,I-1\). At each node \(x_{i}\), the function value \(z_{i}\) is given, \(i=0,1,\ldots,I\). The slope of the line segment connecting \((x_{i},z_{i})\) and \((x_{i+1},z_{i+1})\) is \(\triangle z_{i}:=\dfrac{z_{i+1}-z_{i}}{h_{i}}\), \(i=0,1,\ldots,I-1\). The \(L_{1}\) splines discussed in this paper are cubic polynomials in each interval \((x_{i},x_{i+1})\), \(i=0,1,\ldots,I-1\), and are \(C^{1}\) continuous at the nodes. The first derivative of the spline at node \(x_{i}\), \(i=0,1,\ldots,I\), is denoted by \(b_{i}\) (to be determined by minimization of the \(L_{1}\) spline functional). We use \(\delta_{i}\) to denote the slope of the chord between neighboring points:  
\[\delta_{i}=\dfrac{z_{i+1}-z_{i-1}}{x_{i+1}-x_{i-1}},\quad i=2,\ldots,I-2 \tag{1}\]  
We use \(\zeta\) to denote the linear spline:  
\[\zeta(x)=\dfrac{(x_{i+1}-x)z_{i}+(x-x_{i})z_{i+1}}{h_{i}},\quad x\in[x_{i},x_ {i+1}],\quad i=0,1,\ldots,I-1 \tag{2}\]