Suppose we have the following linear program (LP1), $$\min_{f \in \mathcal{C}} \int_{\mathbb{R}} f(t) \cos(2 \pi x_0 t) dt \\ \text{subject to } \int_{\mathbb{R}}f(t)dt = 1 \\\forall x\in [0,1]: \int_{\mathbb{R}} \cos(2\pi x t) f(t)dt \geq 0 \text{ ,}$$

Where $\mathcal{C}$ is some "nice" family of functions (which I am not sure what it is. In the comments it was mentioned that $f$ has to be an analytic function with bounded spectrum). Now define for all $n \in \mathbb{N}$ the following "approximate" linear program (LP2):

$$\min \sum_{i=0}^n a_i \cos(2 \pi x_0i) \\ \text{subject to } \sum_{i=0}^n a_i = 1 \\\forall k \in \{1/n, 2/n, \ldots, 1\} : \sum_{i=0}^n \cos(2\pi i k)a_i \geq 0.$$

Let $f^*$ be the optimal function for (LP1), and let $f_n$ be the optimal function for (LP2) (i.e. $f_n(x) = \sum_{i=0}^n a^*_ig(x)$, for $a_i^*$ optimizing the linear program).

My question is: under what conditions (if any) we can prove that $\lim_{n\rightarrow \infty} f_n$ converges and $f_{\infty} = f^*$? What can we say about the rate of convergence (obviously by the way the question is phrased this is impossible to answer, but are there examples where the answer is known)? Are there known results for the case where the linear program has many constraints?