Let $\Omega \subset \mathbb{R}^m$ be an open subset bounded with a smooth boundary.

Problem : Given any bounded continuous function $f:\Omega\to\mathbb{R}$, can we learn it to a given accuracy $\epsilon$? ($\epsilon>0$).

Definition : What do you mean by learning a function to a given accuracy $\epsilon$?

Using samples of $f$, at sufficiently large but finite number of data points that are drawn randomly(iid) from the set $\Omega$ (under a uniform probability distribution), and using a sufficiently large but finite number of registers whose precision is sufficiently large but finite (this finite precision is an important condition), should be able to compute a function $F$ with only a finite number of computations (they could be additions, multiplications, and divisions but performed using registers of finite precision) such that $\|f-F\|_{L^\infty(\Omega)} \le \epsilon$.

compute a function $F$ : Given any query point $x$, one should give out $F(x)$.

Conjecture : There exists a method of learning such that one can derive a bound on required precision $p$ that depends only on $\Omega$ and $epsilon$ and is independent of $f$.

Question: Has anyone formulated this problem before (any reference). Has anyone solved it? If I solve it, what is its market value? (mathematics market)

PS: solving means coming up with a method to learn such functions in the defined way.

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