I am thinking of if there is high dimensional extension to the well known Stone-Weirstrass theorem.
Wikipedia says it is possible to extend the 1D theorem to 2D, i.e.
If f is a continuous real-valued function defined on the set [a, b] × [c, d] and ε > 0, then there exists a polynomial function p in two variables such that | f (x, y) − p(x, y) | < ε for all x in [a, b] and y in [c, d].
Is it free to say it also works on 3D or multi-D? Or some additional conditions are required?
Best Answer
In fact, Wikipedia says that the Stone-Weierstrass theorem can be extended to continuous real-valued functions over an arbitrary Hausdorff and compact topological space $X$.
It particular: this means that if $f$ is a continuous real-valued function defined on any closed and bounded subset $X \subset \mathbb{R}^n$ for any $n\in \mathbb N$, then for any $\epsilon > 0$, there exists a polynomial $p(x_1,\dots,x_n)$ such that $$ |f(x_1,\dots,x_n) - p(x_1,\dots,x_n)| < \epsilon $$ For any $(x_1,x_2,\dots,x_n) \in X$.
This certainly includes the $n$-dimensional rectangles $X = [a_1,b_1]\times \cdots \times [a_n ,b_n]$.