On One Counterexample in Convex Approximation - Yushchenko, L. P. (Ukrains’kyi Matematychnyi Zhurnal, 2000)
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: \(array*20c n → ∞ array array*20c \{x [ - 1,1] array |f(x) - p_n (x)| ω _4 (ρ _n (x),f) = ∞ \) , where ω4(t, f) is the fourth modulus of continuity of the function fand \(ρ _n ( x ): = 1n^2 + 1n1 - x^2 \) . We generalize this result to q-convex functions.