About approximation by Haar polynomials

Question

I'm reading about Haar functions, and I found the statement of a theorem which says that if $f$ is a continuous function on $\mathbb{T}$ and $\varepsilon >0$, then there exists a Haar polynomial of degree $n(\varepsilon)$, and $S_n(f)=\sum_{k=0}^{n-1}\alpha_kh_k$ ($\alpha_k\in\mathbb{C}$) such that

$$\|f-S_n(f)\|_{L^{\infty}(\mathbb{T})}<\varepsilon.$$

That is to say, that Haar polynomials can approximate continuous functions on the torus.

However, I can't find a proof of this result. Can anyone tell me a book or article where this theorem is proved?

Thanks a lot!

Answer 1

This is basically the Stone-Weierstrass theorem applied to Haar polynomials. Just show that haar polynomials seperate points and form a sub-algebra.