2019
Том 71
№ 11

All Issues

On One Counterexample in Convex Approximation

Yushchenko L. P.

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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.

English version (Springer): Ukrainian Mathematical Journal 52 (2000), no. 12, pp 1956-1963.

Citation Example: Yushchenko L. P. On One Counterexample in Convex Approximation // Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1715-1721.

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