I have an interpolation method, which takes function $f$ values at any given finite number $N$ of points in the domain and interpolate to get a function $f_{int}$.

I want to do some analysis on how well this function interpolates, by analyzing what happens if we keep increasing the number of given points to infinity and see if $f_{int}$ converges to $f$.

For that If I make $N \to \infty$, its not sufficient and fair, as the all the points still might come from any particular open subset of the domain and no point may come from the rest of the domain, $f_{int}$ will surely not converge to $f$ in an open set from where no points are coming.

So blindly making $N \to \infty$ is not fair and doesn't make sense. I already know that underlying function $f$ is smooth. So what extra assumptions I can make, part from $N\to \infty$, to make a fair analysis of whether $f_{int}$ converges to $f$, for evaluating the interpolation method?