Suppose I have a continuous differentiable surface such as the following:
$$f(x,z)=\sin(e^{2\pi+x^2+z^2}), \space\space\space x,z\in\mathbb{R}$$
As $x$ and $z$ become large, the function oscillates faster and faster. You can see that in the corners of this image:
Now suppose I only evaluate $f$ at some grid of points instead of the whole plane. In other words, at:
$$x,z\in{\bigcup_{k=0,\space 2n}\{r(k-n)\}}\space\space n\in\mathbb{N}\space\space r\in(0,\infty)$$
for some given $n$ and $r$. These points are evenly spaced apart on the $xz$ plane, but as $f$ oscillates more and more, the height of the points becomes less predictable (visually).
Now suppose I fit a smooth function $g$ to those points.
Here's an animation showing how as I increase $n$, $g$ changes. You can see how sensitive it is near the edges: https://youtu.be/HtH3YCqR-Pw
Is there a name for the set of functions that $g$ belongs to (a set of functions that smoothly interpolate through a set of points)? I'm not really sure what I would call it. Would $g$ be "a smooth continuous interpolation of a uniform discrete sampling of $f$?" I don't think that sounds right since it implies the points $f$ was evaluated at are random. But I'm not really sure what else to call it.