Comonotone approximation of twice differentiable periodic functions

Authors

  • H. A. Dzyubenko

Abstract

In the case where a -periodic function f is twice continuously differentiable on the real axis and changes its monotonicity at different fixed points y_i ∈ [− π, π), i = 1,…, 2s, s ∈ ℕ (i.e., on , there exists a set Y := {y_i } i∈ℤ of points y_i = y_{i+2s} + 2π such that the function f does not decrease on [y_i , y_{i−1}] if i is odd and does not increase if i is even), for any natural k and n, n ≥ N(Y, k) = const, we construct a trigonometric polynomial T_n of order ≤n that changes its monotonicity at the same points y_i ∈ Y as f and is such that ∥f−T_n∥ ≤ \frac{c(k,s)}{n^2} ω_k(f″,1/n) (∥f−T_n∥ ≤ \frac{c(r+k,s)}{n^r} ω_k(f^{(r)},1/ n),f ∈ C^{(r)},\; r ≥ 2), where N(Y, k) depends only on Y and k, c(k, s) is a constant depending only on k and s, ω k (f, ⋅) is the modulus of smoothness of order k for the function f, and ‖⋅‖ is the max-norm.

Published

25.04.2009

Issue

Section

Research articles

How to Cite

Dzyubenko, H. A. “Comonotone Approximation of Twice Differentiable Periodic Functions”. Ukrains’kyi Matematychnyi Zhurnal, vol. 61, no. 4, Apr. 2009, pp. 435-51, https://umj.imath.kiev.ua/index.php/umj/article/view/3032.