Comparing the degree of constrained and unconstrained trigonometric approximation

Let $r,s\in N$. For a continuous $2\pi$-periodic function, changing its monotonicity $2s$ times in a period, and whose degree of approximation by trigonometric polynomials of degree $<n$, is $\le n^{-r}$, $n\ge 1$, we investigate its degree of approximation by such polynomials that, in addition, follow the changes of monotonicity. Obviously, the unconstrained degree is smaller than the constrained one, but for $r>2s-2$, there is a constant $c(s,r)$ such that the constrained degree is $\le c(s,r)n^{-r}$, $n\ge 1$. On the other hand we show that, in general, this is invalid for $r\le 2s-2$.