I'm working on a problem of estimating a function $y=f(x): \mathbb{R}^d \rightarrow \mathbb{R}$. Namely, I have an unknown function $f(x)$ (like a black box), what I can do is to input $x^{(i)}$ to it, and obtain $y^{(i)}$ ($i=1,2,\cdots, N$). Then I get a dataset $(x^{(i)}, y^{(i)})$ and am able to fit a function on it.
The simplest/dumbest way is to use a uniform grid of $x$, namely sampling $m$ points on each dimension ($x_i, i=1,2,\cdots, d$) and form a grid with $m^d$ points. The number of samples would explode with high dimension (very large $d$).
A better choice might be using Latin Hypercube or some low discrepancy sequences.
I'm wondering are there any literature with rigorous analysis on using those sampling methods to estimate functions? How can I use as less samples as possible (i.e. small $N$) to get a good estimate of $f(x)$.
For simplicity, we could assume $f(x)$ is infinitely differentiable ($f \in C^\infty$).