Let $n_1, n_2 \in \mathbb{N}$ be known constants such that $n_1 + n_2 = n$. Moreover, let $0 \leq p_1, p_2 \leq 1-1/n$. Assume an exponential model $f(k) = n_1 p_1^k + n_2 p_2^k,\ k\in \mathbb{N}$.
I have access to points $y_k$ such that $|y_k - f(k)| \leq \epsilon$.
I would like to take the minimum number of points $y$ in order to construct an approximation $\hat{f}$ of $f$ such that $| \hat{f}(k) - f(k) | \leq \epsilon$, for every $k \in \mathbb{N}$.
We just need to approximate the first $O(n/\epsilon)$ points of $f$ because for $k \in \omega(n/\epsilon)$, $f(k)$ is approximately $0$ since $(1-\frac1n)^{n/\epsilon} \approx (1/e)^{1/\epsilon} \approx 0$.
The problem is of course trivial if we obtain $O(n)$ approximations: $y_1\approx f(1),\ y_2 \approx f(2),\ \ldots,\ y_n \approx f(n), \ldots, y_{c n} \approx f(c n)$.
Since $f$ is an exponential decreasing function I am thinking of obtaining $\log(n)$ points \begin{equation*} y_1 \approx f(1),\ y_2 \approx f(2),\ y_4 \approx f(4),\ y_8 \approx f(8),\ \ldots\ ,\ y_{2^i} \approx f(2^i) \end{equation*} because $f$ decreases rapidly in the beggining and slower for greater values of $k$. Are there any references for such cases of function approximation ?