In a given problem, we known that the function
$$ y = ae^x + be^{2x} + ce^{3x}$$
should interpolate a set of data points given by $(x_1, y_1), · · · ,(x_n, y_n)$. However, data is noisy and we have measurement error. Due this fact the interpolation was not made possible. Is it possible to show the mathematical formulation associated with find the correct values for a, b, c such as the interpolation works correctly?
Let us say that, locally, your model is $$y=a z + b z^2+c z^3 \qquad \text{with} \qquad z_i=e^{x_i}$$ and you want to interpolate (not curve fit) between $x_k$ and $x_{k+1}$.
The simplest could be to consider for points $\big[x_{k-1},x_k, x_{k+1},x_{k+2}\big]$ and compute the $(a,b,c)$ using a polynomial least-square fit and use the result to predict the value you need.