Monotonicity preserving rational spline histopolation (Q1765466)

Let \(a=x_{0}<\dots<x_{n}=b\) be real numbers. A histogram is a function having constant values \(z_{j}\) on each interval \([ x_{j-1},x_{j}[ .\) Let \(\mathcal{S}\) be the class of all continuously differentiable functions \(S: [ a,b] \to {\mathbb R}\) which are piecewise rational linear, so of the form \(S(x)=(a_{j}+b_{j}( x-x_{j-1}) )/(1+d_{j}( x-x_{j-1}) )\) for \(x\in [ x_{j-1},x_{j}[ \) where \( 1+d_{j}( x-x_{j-1}) >0.\) It is easy to see that each function \( S\in \mathcal{S}\) is either increasing or decreasing on \([ a,b] \). The main result of the paper states that for any {monotone histogram} there exists a unique function \( S\in \mathcal{S} \) such that \( \int_{x_{j-1}}^{x_{j}} S(x)\,dx=z_{j}( x_{j}-x_{j-1}) \) for \(j=1,\dots,n \) satisfying the boundary condition \(S( x_{0}) =\alpha \) and \(S( x_{n}) =\beta \), or alternatively, \(S^{\prime }( x_{0}) =\alpha \) and \(S^{\prime }( x_{n}) =\beta \). Some examples and numerical aspects of implementation are discussed. Analogous results for interpolation have been proved by the second author [Proc. Est. Acad. Sci., Phys. Math. 48, No. 1, 22--30 (1999; Zbl 0967.41003)]