2017
Том 69
№ 9

All Issues

Coconvex Pointwise Approximation

Dzyubenko H. A., Gilewicz J., Shevchuk I. A.

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Abstract

Assume that a function fC[−1, 1] changes its convexity at a finite collection Y := {y 1, ... y s} of s points y i ∈ (−1, 1). For each n > N(Y), we construct an algebraic polynomial P n of degree ≤ n that is coconvex with f, i.e., it changes its convexity at the same points y i as f and $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c{\omega }_{2} \left( {f,\frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in \left[ { - 1,1} \right],$$ where c is an absolute constant, ω2(f, t) is the second modulus of smoothness of f, and if s = 1, then N(Y) = 1. We also give some counterexamples showing that this estimate cannot be extended to the case of higher smoothness.

English version (Springer): Ukrainian Mathematical Journal 54 (2002), no. 9, pp 1445-1461.

Citation Example: Dzyubenko H. A., Gilewicz J., Shevchuk I. A. Coconvex Pointwise Approximation // Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1200-1212.

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