On One Counterexample in Convex Approximation

  • L. P. Yushchenko

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.
Published
25.12.2000
How to Cite
Yushchenko, L. P. “On One Counterexample in Convex Approximation”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 52, no. 12, Dec. 2000, pp. 1715-21, https://umj.imath.kiev.ua/index.php/umj/article/view/4576.
Section
Short communications