A linear approach to shape preserving spline approximation (Q5934301)

This paper deals with the approximation of a given large scattered univariate or bivariate data set that possesses certain shape properties, such as convexity, monotonicity, or range restrictions. The data are approximated by B-splines or tensor-product B-splines preserving the shape characteristics of the data. The problem of shape preserving spline approximation is simplified to the following optimization problem: \[ \min_d \{\|Ad-f\|: Cd\geq b\}. \] The vector \(f\) contains the \(M\) given data values, and the vector \(d\) contains the \(N\) unknown spline coefficients. The \(M\times N\)-matrix \(A\) is determined by the spline approximation, and the constraint matrix \(C\) is of dimension \(L\times N\). The authors discuss the following questions: Which norm should be chosen? How should be shape constraints be linearized? Can one give a sequence of approximants that come arbitrarily close to an interpolant?