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$$∥f−T_n∥ ≤ \frac{c(k,s)}{n^2} ω_k(f″,1/n)$$$$(∥f−T_n∥ ≤ \frac{c(r+k,s)}{n^r} ω_k(f^{(r)},1/n),f ∈ C^{(r)},\; r ≥ 2),$$ 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.