Given a family of real-valued curved $f(x;\theta)$, say for example, $$ f(x,\theta) = e ^ {\theta x}, x \in [-1, 1], $$
the Stone-Weierstrass approximation theorem says that any member in the family of curve can be uniformly approximated by a polynomial $p$ $$ \| f(x;\theta) - p(x;\theta) \|_\infty < \epsilon. $$
To formulate the question assume that the degree of the polynomial is $n$, so that the polynomial can be written as $$ p(x;\theta) = \sum_{i=0}^n a_{\theta, i} x ^ i, $$ So the approximation operator $A$ maps a curve to a polynomial $$ A: f(x;\theta) \mapsto p(x;\theta) $$
Can you point to some reference that discuss the continuity of the approximation operator? In the sense that a small change in $\theta$, should produce a small change in the coefficient $a_{\theta,i}$ of the approximating polynomial.
I known that by using the proof of the Weierstrass approximation that relies on Bernstein polynomial, the continuity of the operator can be prooved. I guess this is routinely prooved in approximation theory book, but I couldn't find a reference that treat this.