Coconvex multi approximation in the uniform norm

Abstract

In this research, we study the results about coconvex multi approximation. Recently this subject received a lot of attention. The first result in the topic is a direct theorem for the error bound of the convex approximation for functions of two continuous derivatives that were introduced by Kopotun. In 2006 Kopotun, Leviatan and Shevchuk gave definitive answers to open problems of Jackson’s estimates by the moduli of smoothness of Ditzian- Totiktype, also in 2019 they stated many open problems in the coconvex case and mentioned recent developments on this subject. In convex multi approximation, we have a function f ε C[-1, 1] d change its convexity at a finite number of times in [-1, 1] d and we take care of error bound of approximation of the function f using multi algebraic polynomials, in other words, multi algebraic polynomials change its convex particularly at the same points where f is. We can get some of Jackson’s estimates in which the intended constants depend on the place of the convexity change points. In order to clarify the complexities, we discuss in this research invers inequality for the coconvex approximation of the function f using multi algebraic polynomials.