# Whitney approximation theorem proof using a convolution

I am asked to prove the Whitney's approximation theorem:

Any continuous map $$F\colon M \rightarrow N$$ is continuously homotopic to a smooth map.

By somehow using this convolution: $$(\xi \ast h_t)(x) = \int_\mathbb{R} h_t(x-y)\xi(y)dy$$ where: \begin{align} h_t(x) &\colon = \frac{g_t(x)}{\int_{-1/t}^{1/t}g_t(s)ds}\\\\ g_t(x) &\colon = f(1+t^{-1}x) f(1-t^{-1} x) \text{ for } t>0\\\\ f(t) &\colon = \begin{cases} e^{-1/t} &\text{ if } t>0\\\\ 0 &\text{ if } t\leq 0 \end{cases} \end{align}

The convolution has good properties, i.e.:

1. Given $$\xi\colon\mathbb{R}\rightarrow \mathbb{R}$$ continuous, $$\xi\ast h_t$$ is smooth for every $$t$$.
2. $$\xi\ast h_t \to \xi$$ as $$t \to 0$$ in the $$C^0$$ norm.

MY TRY

My objective is to somehow define $$F$$ in terms of $$\mathbb{R}\rightarrow \mathbb{R}$$ continuous maps. Then I would be able to convolute them by $$h_t$$.

Given a point $$p\in M$$, there exists a chart $$(U,\varphi\colon U\rightarrow \mathbb{R}^m)$$ s.t. $$p\in U$$. Define $$\tilde{p}\colon = \varphi(p)$$. Then: $$F(p) = (F\circ \varphi^{-1})(\tilde{p}) = (F\circ \varphi^{-1})(\tilde{p}_1,\dots,\tilde{p}_m)$$

Now we have maps $$F\circ \varphi^{-1}$$ that go from an open set $$\varphi(U)$$ of $$\mathbb{R}^m$$ to $$N$$.

Let's proceed similarly for $$N$$. There exists a chart $$(V,\psi\colon N\rightarrow \mathbb{R}^n)$$ in $$N$$ s.t. $$F(p)\in V$$. Let me define the open set $$V'\colon=V\cap F(U)$$. Then the following map makes sense: $$\psi^{-1}\circ \psi \circ F \circ \varphi^{-1}\colon \varphi(U) \longrightarrow V'$$ and it coincides with $$F$$ where it is defined. So, following the previous: $$F(p) = (\psi^{-1}\circ \psi \circ F \circ \varphi^{-1})(\tilde{p}_1,\dots,\tilde{p}_m) = \psi^{-1}((\psi \circ F \circ \varphi^{-1})(\tilde{p}_1,\dots,\tilde{p}_m))$$

Now, the map $$\psi \circ F \circ \varphi^{-1}$$ goes from $$\varphi(U)\subset \mathbb{R}^m$$ to $$\psi(V')\subset \mathbb{R}^n$$. And I'm lost right here.

I have a feeling that we can "split" the continuous map $$\psi \circ F \circ \varphi^{-1}$$ in terms of continuous maps from $$\mathbb{R}$$ to $$\mathbb{R}$$, but I don't know how.

Any help will be appreciated. Thanks a lot!

• Do you have a compactness assumption on, say, $N$? If $N$ is embedded it becomes quite easier. Commented Nov 21, 2019 at 11:49
• @Mindlack The only assumption is that both $M$ and $N$ are smooth manifolds... Commented Nov 21, 2019 at 18:58