Comonotone approximation of twice differentiable periodic functions - Dzyubenko, H. A. (Ukrains’kyi Matematychnyi Zhurnal, 2009)
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∥ ≤ c(k,s)n^2 ω_k(f″,1/n) (∥f−T_n∥ ≤ 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.