I have a function f expressed in terms of Fourier sine series as $f(x) = \sum_{i=1}^M u_i \, sin(i \pi x)$ in a bounded interval, say $\Omega = [0,1]$. $f$ should satisfy the constraint: \begin{equation} -1 \leq f(x) \leq 1 \quad \forall x \in \Omega \end{equation} Is there any way to express the constraints on $a_i$?
What I have been doing so far is to grid the space with $N$ equally spaced points $x_1,\ldots,x_n$ and impose the constraint for each grid point $x_i$ so that I can express it as a linear set of inequalities on the coefficients:
\begin{equation} A \mathbf{u}\leq\mathbf{1} \end{equation}
where $A \in \mathbb{R}^{2N \times M}$ and $N\gg M$.
I have the feeling that most of these constraints are redundant. I was wondering if I can express this condition with less constraints or if it is possible to approximate the convex feasible region with fewer inequalities.