Let $X$ be a domain in $\mathbb{C}^d$ and let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$. Consider the Banach algebra $H^{\infty}(X)$ consisting of bounded holomorphic functions on $X$ with the supremum norm.
Let $f_1,...,f_n\in H^{\infty}(X)$ be non-constant functions of norm at most $1$, and so they are in fact elements of $H(X,\mathbb{D})$. Let $G(f_1,...,f_n)$ be the smallest subset of $H(X,\mathbb{D})$ that satisfies the following conditions:
- $f_1,...,f_n\in G(f_1,...,f_n)$;
- if $f,g\in G(f_1,...,f_n)$, and $\varphi\in H(\mathbb{D}\times \mathbb{D},\mathbb{D})$, then $\varphi(f,g)\in G(f_1,...,f_n)$.
In particular, $G(f_1,...,f_n)$ is convex balanced and closed under multiplication.
What are the "minimal" sufficient conditions on $f_1,...,f_n$ in order for $G(f_1,...,f_n)$ to be somewhere dense in $H^{\infty}(X)$? Or even more specificly to be dense in $\overline{B}_{H^{\infty}(X)}$?
It is easy to see that $f_1,...,f_n$ have to separate points of $X$, and their gradients must not vanish collectively. I feel another useful condition would be that if $x$ is close to $\partial X$, there should be $k$ such that $|f_k(x)|$ is large.
This question is somewhat reminiscent of the Kolmogorov-Arnold Theorem which does not hold for holomorphic functions.
A similar problem was considered in a paper [1], although the "rules of generating" were quite different: under some rather technical conditions the algebra generated by $\{\psi\circ f_k,~ k=1,...,n,~ \psi\in H^{\infty}(\mathbb{D})\}$ is actually the whole $H^{\infty}(X)$.
[1]: Michael A. Dritschel, Daniel Estévez, Dmitry V. Yakubovich, Traces of analytic uniform algebras on subvarieties and test collections
