Detection of C²‐singularities using uniform spline approximations

Abstract

MOS Classification: 41A15; 41A27; 41A36; 41A40For the detection of C 2 -singularities, we present lower estimates for the error in Schoenberg variation-diminishing spline approximation with equidistant knots in terms of the classical second-order modulus of smoothness. To this end, we investigate the behaviour of the iterates of the Schoenberg operator. In addition, we show an upper bound of the second-order derivative of these iterative approximations. Finally, we provide an example of how to detect singularities based on the decay rate of the approximation error.