Coconvex polynomial approximation of twice differentiable functions (Q1904370)

Questions about estimates of the degree of approximation of functions by algebraic polynomials which preserve certain features of shape of the function, have attracted much attention in the last two decades. This paper deals with the question of approximating a function which changes convexity finitely many times in \([-1,1 ]\), by means of polynomials which are coconvex with it. The author assumes that the function \(f\in C^2[0,1 ]\) and this enables him to use a flipping technique that was originally introduced for comonotone approximation by the reviewer and \textit{R. K. Beatson} [Can. Math. Bull. 26, 220-224 (1983; Zbl 0511.41019)]. The author is able to obtain the Jackson estimate involving the modulus of continuity of \(f''\) and also some results on simultaneous approximation of \(f\) and its derivatives. These are the first satisfactory results on coconvex approximation. The construction is quite involved and very interesting. It has the potential of being used to extend the results to other classes of functions.