I am trying to use set of $c_k$ with $k \in \mathbb{N}$ such that $f(x) = \sum_{k=1}^{M} c_k \sin kx \approx x$.
$c_k$ is determined by setting $$f^{(2k+1)}(0) =0,\,\, k=1,..,n$$ $$f'(0) =1$$ $$f^{(2k-1)}(\pi) =0,\,\, k=1,..,M-n-1$$
These conditions provide $M$ linear equations to solve for $c_k$.
Question is, is it guaranteed that $f(x) \leq x$ for $0 \leq x \leq \pi$ for any $n,M$?
(the last $f^{(2k-1)}(\pi)$ condition does not exist when $M = n+1$ - and this is allowed as well. Also, $M\geq 2$, $n\geq 1$.)