Coconvex Approximation of Functions with More than One Inflection Point

  • H. A. Dzyubenko
  • V. D. Zalizko

Abstract

Assume that fC[−1, 1] belongs to C[−1, 1] and changes its convexity at s > 1 different points y i, \(\overline {1,s} \) , from (−1, 1). For nN, n ≥ 2, we construct an algebraic polynomial P n of order ≤ n that changes its convexity at the same points y i as f and is such that $$|f(x) - P_n (x)|\;\; \leqslant \;\;C(Y)\omega _3 \left( {f;\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right),\;\;\;\;\;x\;\; \in \;\;[ - 1,\;1],$$ where ω3(f; t) is the third modulus of continuity of the function f and C(Y) is a constant that depends only on \(\mathop {\min }\limits_{i = 0,...,s} \left| {y_i - y_{i + 1} } \right|,\;\;y_0 = 1,\;\;y_{s + 1} = - 1\) , y 0 = 1, y s + 1 = −1.
Published
25.03.2004
How to Cite
DzyubenkoH. A., and ZalizkoV. D. “Coconvex Approximation of Functions With More Than One Inflection Point”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, no. 3, Mar. 2004, pp. 352-65, https://umj.imath.kiev.ua/index.php/umj/article/view/3759.
Section
Research articles