The [Stone-Weierstrass theorem][1] has an analog for the algebras of smooth functions, called  

> **Naсhbin's theorem**: An involutive subalgebra $A$ in the algebra ${\mathcal C}^\infty(M)$ of smooth functions on a smooth manifold $M$ is dense in ${\mathcal C}^\infty(M)$ if and only if $A$ separates the points and the tangent vectors of $M$.

See details in: "L.Nachbin. Sur les algèbres denses de fonctions diffèrentiables sur une variètè, C.R. Acad. Sci. Paris 228 (1949) 1549-1551", or in  [J.G.Llavona's monograph][2], or [here][3]. 

This is strange, I can't find an analog for the algebras of holomorphic functions (on complex manifolds). Did anybody think about this?

> **Question**: let $A$ be a subalgebra in the algebra ${\mathcal O}(M)$ of holomorphic functions on a complex manifold $M$ (as a first approximation, we can think that $M$ is just an open subset in ${\mathbb C}^n$). Which conditions should $A$ satisfy for being dense in ${\mathcal O}(M)$? 


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**Remark.** By topology on ${\mathcal O}(M)$ I mean the usual topology of uniform convergence on compact sets in $M$. The algebra ${\mathcal C}^\infty(M)$ is also endowed with its usual topology, which can be described, for example, as follows. 

1. For each function $f\in {\mathcal C}^\infty(M)$ let us define its *support* as the closure of the set of the points where $f$ does not vanish:
$$
\text{supp}f=\overline{\{x\in M:\ f(x)\ne 0\}}.
$$
An equivalent definition: $\text{supp}f$ is the set of the points in $M$ where $f$ has non-zero [germs][4]:
$$
\text{supp}f=\{x\in M:\ f\not\equiv 0\ (\text{mod}\ x)\}.
$$

2. Let us define *differential operators* (see e.g. [S.Helgason's book][5]) on $M$ as linear mappings $D:{\mathcal C}^\infty(M)\to {\mathcal C}^\infty(M)$ which do not extend the support of functions:
$$
\text{supp}Df\subseteq \text{supp}f,\quad f\in{\mathcal C}^\infty(M).
$$
Equivalently, $D$ is *local*, i.e. the value of $Df$ in a point $x\in M$ depends only on the germ of $f$ in $x$: 
$$
\forall f,g\in{\mathcal C}^\infty(M)\quad \forall x\in M\qquad f\equiv g\ (\text{mod}\ x)\quad\Longrightarrow\quad Df(x)=Dg(x).
$$

3. Then we say that a sequence of functions $f_n$ converges to a function $f$ in ${\mathcal C}^\infty(M)$
$$
f_n\overset{{\mathcal C}^\infty(M)}{\underset{n\to\infty}{\longrightarrow}}f
$$
if and only if for each differential operator $D:{\mathcal C}^\infty(M)\to {\mathcal C}^\infty(M)$ the sequence of functions $Df_n$ converges to $Df$ in the space ${\mathcal C}(M)$ of continuous functions with the usual *topology of uniform convergence on compact sets* in $M$:
$$
Df_n\overset{{\mathcal C}(M)}{\underset{n\to\infty}{\longrightarrow}}Df
$$

Of course, this is equivalent to the convergence in ${\mathcal C}^\infty(U)$  for each smooth local chart $\varphi:U\to V$, $U\subseteq\mathbb{R}^m$, $V\subseteq M$. This is also  equivalent to what Alex M. writes about vector fields:
$$
f_n\overset{{\mathcal C}^\infty(M)}{\underset{n\to\infty}{\longrightarrow}}f
\quad\Longleftrightarrow\quad \forall k\ \forall X_1,...,X_k\in{\mathcal X}(M) \quad
X_1...X_kf_n\overset{{\mathcal C}(M)}{\underset{n\to\infty}{\longrightarrow}}X_1...X_kf.
$$

  [1]: https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem
  [2]: https://books.google.ru/books/about/Approximation_of_Continuously_Differenti.html?id=C7auJPRK5nAC&redir_esc=y
  [3]: http://arxiv.org/abs/1303.2424
  [4]: https://en.wikipedia.org/wiki/Germ_(mathematics)
  [5]: http://www.amazon.com/Geometric-Invariant-Differential-Operators-Spherical/dp/0821826735