Comonotone approximation of twice differentiable periodic functions
Abstract
In the case where a 2π-periodic function f is twice continuously differentiable on the real axis ℝ and changes its monotonicity at different fixed points y_i ∈ [− π, π), i = 1,…, 2s, s ∈ ℕ (i.e., on ℝ, 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.Downloads
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.