2019
Том 71
№ 11

All Issues

Yushchenko L. P.

Articles: 1
Brief Communications (Ukrainian)

On One Counterexample in Convex Approximation

Yushchenko L. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1715-1721

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.