# Analytic approximations of smooth vector fields

Let $$M$$ be the set of smooth divergence-free vector fields $$u$$ on $$\mathbb{R}^3$$ with $$|\partial_x^{\alpha} u(x)| \leq C_{\alpha K}(1+|x|)^{-K}$$ on $$\mathbb{R}^3$$ for any $$\alpha,K$$. Further, we consider the subset of analytic functions $$M_0 \subset M$$. The question is now, whether for each $$u \in M$$ and $$\varepsilon >0$$, there is an $$v \in M_0$$, such that $$|u(x)-v(x)| < \varepsilon$$ for each $$x \in \mathbb{R}^3$$.

As a second question consider $$M$$ to be the set of smooth divergence-free vector fields $$u$$ on $$\mathbb{R}^3$$ with $$u(x+e_j) = u(x)$$ for all $$x \in \mathbb{R}^3$$ and $$1 \leq j \leq 3$$. Again, consider the subset of analytic functions $$M_0 \subset M$$. The question here is now again, whether for each $$u \in M$$ and $$\varepsilon >0$$, there is an $$v \in M_0$$, such that $$|u(x)-v(x)| < \varepsilon$$ for each $$x \in \mathbb{R}^3$$.

These questions are fairly similar to the Stone-Weierstrass theorem, but it seems it is not possible to derive it directly from it.

I believe the most natural approach to this particular question is via Fourier analysis. In the periodic case we have the series $$u(x)=\sum_{k\in\mathbb{Z}^3}u_k e^{2\pi i (k,x)},$$ and the condition $$\nabla\cdot u=0$$ simply means $$(u_k,k)=0$$. Taking $$v(x)=\sum_{|k| for sufficiently large $$K>0$$ we can approximate $$u$$ as close as we want because the Fourier series of a smooth function converges uniformly and very quickly. Obviously, a polynomial $$v(x)$$ is analytic.

The same trick works in the first case too except the Fourier series has to be replaced by the Fourier integral, and the fact that $$v$$ is analytic may be a little less obvious (but still true).

• In fact in problem1, one can also approximate by convolution with the Weierstrass kernel $C\exp(-||x||^2/\epsilon)$ Aug 12, 2020 at 13:00

Here is how to derive both results from the Stone-Weierstras theorem. As you say, it's not direct, but not a long way either. Recall these simple applications of the S-W theorem, to be used in PB1 resp. in PB2

• The algebra $$A$$ of rapidly decreasing real analytic functions on $$\mathbb{R}^3$$ is uniformly dense in the space of continuous functions vanishing at infinity on $$\mathbb{R}^3$$. This follows from the S-W theorem applied to the one-point compactification $$\mathbb{R}^3\cup\{\infty\}$$, the $$3$$-sphere. To check that the above algebra separates points it is sufficient to consider the function $$\exp(-\|x\|^2)$$ and its translates.

• The algebra $$A$$ of $$\mathbb{Z}^3$$-periodic real analytic functions on $$\mathbb{R}^3$$ is uniformly dense in the space of $$\mathbb{Z}^3$$-periodic continuous functions on $$\mathbb{R}^3$$. This follows from the S-W theorem applied to the quotient $$\mathbb{R}^3/\mathbb{Z}^3$$, the $$3$$-torus. To check that the above algebra separates points it is sufficient to consider the functions $$\sin(2\pi x_1)$$, $$\sin(2\pi x_2)$$, $$\sin(2\pi x_3)$$ and their translates.

Now, given $$u\in M$$ and $$\epsilon>0$$, we find $$w_i\in A$$, such that $$\|u_i-w_i\|_{\infty}\le\epsilon$$ (for $$i=1,..,3$$). To define a divergence-free approximation $$v=(v_1,v_2,v_3)\in M_0$$, we may then take $$v_1:=w_1$$, $$v_2=w_2$$ and for all $$(x,y,z)\in\mathbb{R}^3$$ $$v_3(x,y,z):=w_3(x,y,0)-\int_0^z\big\{\partial_1w_1(x,y,s)+\partial_2w_2(x,y,s)\big\}ds.$$ It is easy to check that, both in problem 1 and 2, $$v_3\in A$$, that $${\rm div\, } v=0$$, and that $$v$$ is still uniformly close to $$u$$.