Homework - Showing any continuous functions on a compact subset of $\mathbb{R}^3$ can be approximated by a polynomial. $ X = \left\{(x, y, z) | \frac{x^2}{3} + \frac{y^2}{5} + \frac{z^2}{7} \le 1 \right\} $
Prove:
If f(x,y,z) is continuous on X, then for any ϵ > 0, there exists a polynomial p(x,y,z) such that |f−p|< ϵ on X.
My approach:
I'm familiar with Weierstrass Approximation Theorem. In class, we proved a similar problem but we showed the property held for a closed interval $[a, b]$. To do this, we first proved two things:
$\forall \epsilon > 0, |(1-t)^{\frac{1}{2}} - P_n(t)| < \epsilon$ for $t \in [0, 1]$ and $P_n(t)$ being a polynomial in t.
We then showed that $\forall \epsilon > 0 ||s| - Q_n(s)| < \epsilon$ for $s \in [-1, 1]$. The proof was finished through a few more easier lemmas such as showing linear and multiplicative combinations of functions that can be approximated by polynomials still maintain that property.
Where I'm stuck:
If I can somehow tie the first two properties we proved to this case in $\mathbb{R}^3$ then I'm fairly confident I can finish the proof. I'm guessing that instead of proving it for $[0, 1]$ we prove it for $[0, 1]^3$ and then generalize from there. If anybody has any tips for approaching the problem then I would greatly appreciate it.
 A: One approach is to use the   more general Stone-Weierstrass theorem which applies to every compact space, and is not much harder to prove than the Weierstrass theorem for the interval. But if you don't have this result at your disposal, the following is one possible strategy: 
Let's say a function $g$ is radial if it is of the form $g(x,y,z)=\varphi( x^2+y^2+z^2 )$ for some continuous $\varphi$. Since $\varphi$ can be uniformly approximated by polynomials on any interval $[0,R]$, it follows that $g$ can be uniformly approximated by polynomials on any closed disk.  
Also, the shifted function $g(x-x_0,y-y_0,z-z_0)$  can be approximated by polynomials. 
It remains to approximate $f$ by a linear combination of radial functions with shifts. One way to do this is to show 
$$f(x,y,z) = \lim_{r\to 0} \frac{c}{r^{3}} \int_X f(u,v,w) \exp\left(-\frac{(x-u)^2+(y-v)^2+(z-w)^2}{r^2}\right)\,du\,dv\,dw $$ 
and then approximate the integral by a Riemann sum. You may have used this convolution integral in the proof of one-dimensional result, possibly with a different kernel.
