A theorem for piecewise convex-concave data approximation. (Q1426773)

The author studies the problem of calculating the best approximation to a given univariate data by minimizing a strictly convex function of the errors subject to the condition that there are at most \(q\) (a given integer) sign changes in the second divided differences of the approximation. Following \textit{I. C. Demetriou} and \textit{M. J. D. Powell} [Approximation Theory and Optimization, Cambridge University Press, Cambridge, 109--132 (1997; Zbl 1031.65027)] a characterization theorem is provided that reduces the problem to an equivalent one, where the unknowns are the positions of the sign changes subject to the feasibility restrictions at the sign changes. Certain advantages of this approach are also highlighted.