Stone-Weierstrass theorem applied to Wiener processes: Does it require always a polynomial of infinite order?

Question

This is a conceptual question: I know beforehand I don't have enough background to understand a detailed answer, so please keep it as simple as possible.

A few days ago I discover in Wikipedia the Stone–Weierstrass theorem, where is explained that

the Weierstrass approximation theorem states that every continuous function defined on a closed interval $[a,b]$ can be uniformly approximated as closely as desired by a polynomial function.

Since a Wiener process is an example of a function that "is continuous everywhere but differentiable nowhere", in principle for some closed interval $[a,b]$ the Stone–Weierstrass theorem should apply.

But since Wiener processes show Self-similarity:

I believe that the only way for a polynomial function to approximate on any closed interval $[a,b]$ as closely as desired a Wiener process if and only if the polynomial is of infinite order, this due the function is as "complicated" as desired also when zooming it on every section of the real line: Does this line of thought make any sense?

Answer 1

Your reading of the theorem is incorrect.

The idea of the Stone-Weierstrass theorem is that for any continuous function $f$ on $[a,b]$ and any specified $\varepsilon > 0$, it is possible to find a polynomial $p$ (of some finite degree) such that $$ |f(x) - p(x)| < \varepsilon$$ for all $x \in [a,b]$. What you should be imagining is "fattening up" the function $f$ to create a kind of "tube" of radius $\varepsilon$. The S-W theorem asserts that a polynomial function can be "threaded through" that fattened-up tube.

Roughly speaking, the picture looks like the image below.

The black curve is fattened up into a tube of radius $\varepsilon$, bounded above and below by the red curves. The blue curve is a polynomial which stays between the red curves. The error between the polynomial and the original black curve is always less than the chosen $\varepsilon$.

Note that the polynomial approximation will not necessarily have all of the same properties of the original function. For example, when approximating a nowhere differentiable function, all of the approximations will be smooth (i.e. infinitely many times differentiable). In the case of a self-similar curve, none of the approximations will be self-similar.

Answer 2

You are misreading the theorem. The desired closeness is specified first - suppose it's some small number $\epsilon$. The theorem says that no matter how wiggly the continuous function, there is a polynomial that's everywhere within $\epsilon$ of the function.

The smaller $\epsilon$ the higher the degree of the polynomial you will need. But you don't get that higher degree polynomial just by adding higher order terms. All the coefficients are in play.