Coconvex Pointwise Approximation

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

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.
Published
25.09.2002
How to Cite
DzyubenkoH. A., GilewiczJ., and ShevchukI. A. “Coconvex Pointwise Approximation”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 54, no. 9, Sept. 2002, pp. 1200-12, https://umj.imath.kiev.ua/index.php/umj/article/view/4159.
Section
Research articles