Interpolating Gaussian mixture. How many points is sufficient for complete reconstruction Consider the following Gaussian mixture of $N$ components:
\begin{align}
f(x)=  \sum_{i=1}^N p (s_i)  e^{-\frac{(x-s_i)^2}{2}}\big/ \sqrt{2\pi}
\end{align}
where we assume that

*

*$\max_{i} |s_i| \le C$

*$p (s_i)$ is probability distribution (i.e.,  $\sum_i p(s_i) =1 $ and $1 \ge p(s_i) \ge 0 $

*$p(s_i)$ is symmetric. That is, if $s_i \neq 0$, then there exists $j$ such that $i \neq j$ and  $s_i=-s_j$ and $p(s_i)=p(s_j)$.

*we know the value of $N$
Otherwise, we have no knowledge of $s_i$'s or $p(s_i)$
My question is the following: Suppose we are allowed to evaluate/sample $f$ at any $K$ points we desire.  How many samples $K$ do we require to completely reconstruct $f$?
Unfortunately, $f$ is not bandlimited in the Fourier domain otherwise we could have used Nyquist criteria.  However, because of the Gaussian tail, it is almost bandlimited, plus we have additional information about its structure (i.e., we know $N$).  So, therefore, I am curious if $K<\infty$ is enough.
This question must have been asked before but I couldn't find any reference and don't really know for what keywords to search.
Some thoughts
One can set this up as a linear algebra problem. First, choose $K$ samples $x_i$ at which we will evaluate $f$ then and let
\begin{align}
F=[f(x_1),\ldots, f(x_k) ]^T
\end{align}
also define a matrix $E$ such that
\begin{align}
E_{ij}= e^{-\frac{(x_i-s_j)^2}{2}}\big/ \sqrt{2\pi}
\end{align}
and let
\begin{align}
P=[p(s_1),...p(s_N)]
\end{align}
Therefore, our problem can be written as
\begin{align}
E P=F
\end{align}
we are interested in recovering $E$ and $P$ based on the knowledge of $F$.

Since
$$\sum_{i=1}^K p(s_i)e^{-(x_j-s_i)^2/2}=e^{-x_j^2/2}\sum_{i=1}^K p(s_i)e^{x_js_i-s_i^2/2}=e^{-x_j^2/2}\sum_{i=1}^K \underbrace{p(s_i)e^{-s_i^2/2}}_{q_i}e^{x_js_i}$$
it is equivalent to consider the system of equations
$$\require{cancel}\cancel{e^{x_j^2/2}}\cancel{e^{-x_j^2/2}}\sum_{i=1}^K q_i\left[e^{s_i}\right]^{x_j}=\alpha_je^{x_j^2/2}\qquad j=1,\ldots,N$$
which should be solvable with $N=2K$ interpolation points $x_j$ taken at random in $(-C,C)$ (without considering the further constraints).
