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TASK: given an interval $[a,b]\subset \mathbb R ^n$ and $f \in C^0[a,b]$, show that $f$ can be uniformly approximated by polynomials.

HINT: Exercise is given in the context of PDE (so I guess one isn't asked to copycat Weierstrass polynomial approximation theorem) and the task is given the following hint: use continuation of $f$ and corresponding solution of the heat equation.

IDEA: Let $\Phi$ be heat kernel, i.e.: $$\Phi(t,x) = \frac{1}{\sqrt{4\pi t}} \exp (\frac{-|x|^2}{4t}).$$ Let's extend $f$ by

$$\tilde f(x)=\begin{cases} f(x) &x\in[a,b]\\0 &\text{otherwise}\end{cases}$$

Then we can define $u$ as the solution of the heat equation

$$\begin{cases}\partial_t u - \partial_{xx}u = 0 & t>0\\u=\tilde f &t=0\end{cases}$$

so $u$ is given as the convolution $u(t,x)=\Phi(t)\ast \tilde f$.

Then a natural approach would be to try to approximate $f$ by going backwards, for example:

$$g_n(x) = u\left(\frac 1n, x\right).$$

Since $\tilde f$ is continuous and has compact support, we know that $ g_n \rightarrow \tilde f$ in $C^0[a,b]$.

So now the question is, how do I relate this problem with the polynomials, considering that $g_n$ is not a polynomial? So if I understand correctly, at this point I have to find some mapping $P:g_n\mapsto h_n$ s.t. $g_n$ is like the one above and $h_n$ is polynomial of $n$-th degree. Naturally Taylor series come to mind (notice that it wasn't possible to approximate via Taylor series from the beginning since $f$ is only in $C^0$), so we could take the Taylor series at $0$ up to $n$-th degree. But for each function $h_n$ the polynomials will have variable constants if proceed in a straightforward way, so I don't seem to find the exact way to define the polynomial using the solution of the heat equation. Hence asking you for hints and clues.

Thank you in advance!

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    I think you have the right idea with Taylor approximation. The reason for using $g_n$ is because $g_n$ is smooth, even analytic. If you want to get $\epsilon$ close to $f$, then get $\epsilon/2$ close with some $g_n$ and then use a Taylor remainder bound on some very high degree approx of $g_n$ to get $\epsilon/2$ close to $g_n$ with a polynomial. – 2017-01-27

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Try approximating the heat kernel $\Phi$ by a truncated Taylor series in $x$. Then you have an approximation of $u$ that is a convolution of a polynomial with $f$ (which turns out to be again a polynomial).