## 1 Introduction  
Shape-preserving techniques for interpolating and approximating multiscale data, that is, data with sudden large changes in magnitude and/or spacing, are important for modeling of natural and urban terrain, geophysical features, biological objects, robotic paths and many other irregular surfaces,processes and functions. Over the past decade, a new class of univariate and bivariate splines, namely, \(L_{1}\) splines, that have superior shape-preserving properties for interpolating and approximating multiscale data has arisen ([1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21]). The \(L_{1}\)-norm minimization principles on which \(L_{1}\) splines are based result in non-differentiable convex generalized geometric programs that, so far, have been more complex and more computationally expensive to solve than the programs by which other variants of splines, e.g., conventional and tension splines, T-splines, _etc._, are solved, but the shape preservation provided by \(L_{1}\) splines is significantly better than the shape preservation provided by these alternative approaches. \(L_{1}\) splines have typically been calculated by minimization of global spline functionals, that is, spline functionals that extend over the whole range of the data to be interpolated. However, there have been three reports in the literature of \(L_{1}\) splines on local windows. The first such report is in [17], where bivariate \(L_{1}\) splines were calculated by a non-iterative \"domain decomposition\" procedure on overlapping \(80\times 80\) windows and \(40\times 40\) subsets of these windows were pieced together to create global surfaces. With parallel computation, the domain-decomposition procedure results in sharply reduced computing time. In 2007, a result for univariate \(L_{1}\) splines on much smaller windows arose. In [2], [PERSON], [PERSON] and [PERSON] showed that, given five points on a Heaviside function with two to the left of the discontinuity and three to the right, the \(L_{1}\) spline for these five points is linear over the set of three points ([2], Proposition 9). Even though preservation of linearity is not all of what we desire in shape preservation, it is a large part thereof. This linearity-preservation result suggests that calculation of \(L_{1}\) splines on small, 5-point windows, is geometrically meaningful. An immediate generalization of Proposition 9 of [2] is that, if, in a set of five points, three consecutive points on one end are collinear, then the \(L_{1}\) spline through those three points is, except in the case of a V-shaped corner, linear. Such a result does not hold when the five points are embedded in a larger data set and a global \(L_{1}\) spline functional is minimized. The best result that can be achieved in the case of a global \(L_{1}\) spline functional is the following. **Theorem 1**. : (Theorem 2 of [7]) _If four consecutive data points \((x_{i},z_{i})\), \((x_{i+1}\), \(z_{i+1})\), \((x_{i+2},z_{i+2})\) and \((x_{i+3},z_{i+3})\) lie on a straight line, then a cubic \(L_{1}\) spline \(z(x)\) preserves linearity over the middle interval \([x_{i+1},x_{i+2}]\). If \(\beta_{i-1}^{*}\
eq\pm\frac{5}{3}\), then \(z(x)\) preserves linearity over the first interval \([x_{i},x_{i+1}]\). If \(\beta_{i+2}^{*}\
eq\pm\frac{5}{3}\), then \(z(x)\) preserves linearity over the last interval \([x_{i+2},x_{i+3}]\). (Here \(\beta_{i-1}^{*}\) and \(\beta_{i+2}^{*}\) are components of the optimal dual solution in [7]. )_  
In this case, one needs four (rather than just three) consecutive collinear points and the \(L_{1}\) spline is guaranteed to be linear only in the second interval (the interval between the second and the third of the four points). The \(L_{1}\) spline is linear in the first and third of the three cells only if additional conditions, ones that do not have clear geometric meaning, are fulfilled. Proposition 9 of [2] is thus a significant improvement over Theorem 2 of [7].