On One Counterexample in Convex Approximation

Abstract

We prove the existence of a function fcontinuous and convex on [−1, 1] and such that, for any sequence {p _n} _{n= 1} ^∞of algebraic polynomials p _nof degree ≤ nconvex on [−1, 1], the following relation is true: $\begin{array}{*{20}c} {\lim \sup } \\ {n \to \infty } \\ \end{array} \begin{array}{*{20}c} {\max } \\{x \in [ - 1,1]} \\ \end{array} \frac{{|f(x) - p_n (x)|}}{{\omega _4 (\rho _n (x),f)}} = \infty$, where ω₄(t, f) is the fourth modulus of continuity of the function fand $\rho _n \left( x \right): = \frac{1}{{n^2 }} + \frac{1}{n}\sqrt {1 - x^2 }$. We generalize this result to q-convex functions.