On One Counterexample in Convex Approximation

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: , where ₄(t, f) is the fourth modulus of continuity of the function fand . We generalize this result to q-convex functions.