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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?

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  • $\begingroup$ You could insist that the sequence of points be uniformly distributed. $\endgroup$
    – Gerry Myerson
    Commented Oct 30, 2019 at 11:39
  • $\begingroup$ @GerryMyerson : you mean define a probability distribution for the points? $\endgroup$
    – Rajesh D
    Commented Oct 30, 2019 at 11:43
  • $\begingroup$ There's a concept of a uniformly distributed sequence of points. It means that for every interval the difference between the proportion of the domain contained in the interval and the proportion of points contained in the interval approaches zero. $\endgroup$
    – Gerry Myerson
    Commented Oct 30, 2019 at 11:45
  • $\begingroup$ @GerryMyerson : Thats is a much stronger condition? I am not interested in uniform distribution, but, something like, given any open subset, the sequence contains atleast one point from it, if we make $N$ sufficiently large. I think this would work, but I don't know if it is known by any name. $\endgroup$
    – Rajesh D
    Commented Oct 30, 2019 at 14:15
  • $\begingroup$ Yes, that might be good enough for your purposes, just asking for the points to be dense in the domain. It's only if you want the average of the approximation to converge to the integral of the function that you need the stronger condition of uniform distribution. $\endgroup$
    – Gerry Myerson
    Commented Oct 30, 2019 at 20:53

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