Consider some yet-to-be-determined, periodic, real-valued function, $f(x)$ (where $x \in [-L,L]$), defined in terms of the first $2M + 1$ Fourier coefficients, i.e. $$f(x) = \sum_{n = -M}^M c_n e^{-i\frac{n \pi x}{L}},$$ where $2L$ is the period of the function. Given that $0 \leq f(x) \leq k$ over the entire range, what are the restrictions on the possible values that the coefficients can take (e.g. in terms of k, or recursively?). Yes, this is not the full Fourier series, but rather a finite approximation to it where $M$ can become arbitrarily large. Put differently, how do we constrain the values of these Fourier coefficients such that the resultant function is bounded and positive?
So far, I know that we require the $c_0$ term to be at least as large as the sum of the other terms in the sum, however I'm not sure this keeps the function bounded below $k$. Furthermore, each coefficient need not be real valued (except for $c_0$), only that $c_n = c_{-n}^*$ such that the resultant function is real valued. Therefore, I imagine the answer could be that each coefficient will be constrained to be in some circle around the origin of the Argand plane of radius $r_n$ for each coefficient $c_n$. But I'm not sure, and especially not sure how to find this radius!
Bonus: same question but with $\lim(M \rightarrow \infty).$
$f$ is $2L$-periodic, and we see by the orthogonality relation, $$ c_n = \frac{1}{2L}\int_{-L}^L f(x)e^{-in\pi x/L} dx. $$ Now we can use the bound $0 \leq f(x) \leq k$ to see $$ |c_n| = \left|\frac{1}{2L}\int_{-L}^L f(x)e^{-in\pi x/L} dx\right| \leq \frac{1}{2L}\int_{-L}^L \left|f(x)e^{-in\pi x/L}\right|dx = \frac{1}{2L}\int_{-L}^L |f(x)| dx \leq \frac{1}{2L}\int_{-L}^L kdx = k. $$ Hence we have the condition $$ |c_n| \leq k \text{ for all } n \in \{-M, -M+1, \cdots, M-1, M\}. $$ I think this works in the infinite case too, but some care might be needed due to the infinite sum.