Piecewise Convex–Concave Approximation in the Minimax Norm

Abstract

Suppose that $\mathbf {f} \in \mathbb {R}^{n}$ is a vector of n error-contaminated measurements of n smooth function values measured at distinct, strictly ascending abscissæ. The following projective technique is proposed for obtaining a vector of smooth approximations to these values. Find y minimizing ∥y −f∥_∞ subject to the constraints that the consecutive second-order divided differences of the components of y change sign at most q times. This optimization problem (which is also of general geometrical interest) does not suffer from the disadvantage of the existence of purely local minima and allows a solution to be constructed in only $O(nq \log n)$ operations. A new algorithm for doing this is developed and its effectiveness is proved. Some results of applying it to undulating and peaky data are presented, showing that it is fast and can give very good results, particularly for large densely packed data, even when the errors are quite large.