In the paper [Stinchcombe and White,1990, APPROXIMATING AND LEARNING UNKNOWN MAPPINGS USING MULTILAYER FEEDFORWARD NETWORKS WITH BOUNDED WEIGHTS]
Theorem 2.8:
If $G$ is super-analytic (analytic but is not polynomial) at $a$ with convergence radius $gamma$, ${G(x+b): |b| \le 1}$ is uniformly dense on compact subsets of $(-gamma, gamma)$ in $span({G^(k)|(-gamma, gamma): k > 0))$ where $G ^(k)$ is the $k$-th derivative of $G$.
If in addition $sp({G(k) |(-gamma, gamma): k >0)) = C(R)|(-r, r)$ then.....
The proof is given in the paper with a differential perspective:
Let $K$ be a compact subset of (-r, r) and let $epsilon > 0$,
$|(G^(k-1)(x+h)-G^(k-1)(x) )/h - G^(k)(x) | \infty$
Then the result always can not stand:
$sp({G(k) |(-gamma, gamma): k >0)) = C(R)|(-r, r)$
I don`t know if I am right.
The original paper is in the following:
If an analytic function G is uniformly dense on compact subsets of (-r, r) in the span of its derivatives G^(k) for k > 0, then it is also uniformly dense on C(R), the set of continuous functions on the real line.
To prove this, we can use the Stone-Weierstrass theorem, which states that any continuous function on a compact interval can be uniformly approximated by polynomials. Here's an outline of the proof:
Therefore, we have shown that if an analytic function G is uniformly dense on compact subsets of (-r, r) in the span of its derivatives G^(k) for k > 0, then it is also uniformly dense on C(R), the set of continuous functions on the real line.
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