Comonotone approximation of twice differentiable periodic functions

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

$$ \begin{array}{*{20}{c}} {\left\| {f - {T_n}} \right\| \leq \frac{{c\left( {k,s} \right)}}{{{n^2}}}{{{\upomega }}_k}\left( {f'',{1 \mathord{\left/{\vphantom {1 n}} \right.} n}} \right)} \\ {\left( {\left\| {f - {T_n}} \right\| \leq \frac{{c\left( {r + k,s} \right)}}{{{n^r}}}{{{\upomega }}_k}\left( {{f^{(r)}},{1 \mathord{\left/{\vphantom {1 n}} \right.} n}} \right),\quad f \in {C^{(r)}},\quad r \geq 2} \right),} \\ \end{array} $$

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.