Monotonicity preserving interpolatory subdivision schemes (Q1300808)

Let \(\{(t^{(0)}_i, x^{(0)}_i)\in \mathbb{R}^2\}^N_{i= 0}\) be a given finite bounded data set, where the points are uniformly distributed with meshsize \(h> 0\). The data are also assumed to be monotone in the sense that \(x^{(0)}_i\) is monotone. Let \(t^{(k)}_i= 2^{-k}ih\), \(i= 0,\dots, 2^kN\). The purposes of this paper are: 1. characterize a class of subdivision schemes that are interpolatory and monotonicity preserving if the data are monotone, 2. restrict this class of subdivision schemes to schemes that generate continuously differentiable limit functions and are fourth-order accurate. The article has ten sections and one appendix. The class of schemes to be investigated is defined in Section 2 by the following relations \[ x^{(k+ 1)}_{2i}= x^{(k)}_i;\;x^{(k+ 1)}_{2i+ 1}= \textstyle{{1\over 2}}(x^{(k)}_i+ x^{(k)}_{i+ 1})+ \textstyle{{1\over 2}} s^{(k)}_i G(r^{(k)}_i, R^{(k)}_{i+ 1}),\tag{1} \] where \(s^{(k)}_i= x^{(k)}_{i+ 1}- x^{(k)}_i\), \(r^{(k)}_i= s^{(k)}_{i- 1}/ s^{(k)}_i\), \(R^{k)}_i= 1/r^{(k)}_i\), and \(G\) is a function which is antisymmetric, i.e., \(G(r, R)= -G(R, r)\). Monotonicity preservation and convergence of the scheme (1) to continuous and continuously differentiable functions are investigated in the Sections 3, 4 and 5. As for the problem of generating continuously differentiable limit functions, Section 6 is devoted to the construction of rational schemes. Theorem 11 finds the only rational function which has fourth-order approximation. Some additional properties of (1), which are discussed in Section 7, are used to obtain a fourth-order approximation. Convergence to continuously differentiable limit functions having approximation order four, is studied in Sections 8 and 9. Section 10 contains some generalizations.