Editing Stone–Weierstrass theorem (section)

=== Degree of approximation ===
For differentiable functions, [[Jackson's inequality]] bounds the error of approximations by polynomials of a given degree: if <math>f</math> has a continuous k-th derivative, then for every <math>n\in\mathbb N</math> there exists a polynomial <math>p_n</math> of degree at most <math>n</math> such that <math>\lVert f-p_n\rVert \leq \frac\pi 2\frac 1{(n+1)^k} \lVert f^{(k)}\rVert</math>.<ref>{{Cite book |last=Cheney |first=Elliott W. |title=Introduction to approximation theory |date=2000 |publisher=AMS Chelsea Publ |isbn=978-0-8218-1374-4 |edition=2. ed., repr |location=Providence, RI}}</ref>

However, if <math>f</math> is merely continuous, the convergence of the approximations can be arbitrarily slow in the following sense: for any sequence of positive real numbers <math>(a_n)_{n\in\mathbb N}</math> decreasing to 0 there exists a function <math>f</math> such that <math>\lVert f-p\rVert > a_n</math> for every polynomial <math>p</math> of degree at most <math>n</math>.<ref>{{Cite journal |last=de la Cerda |first=Sofia |date=2023-08-09 |title=Polynomial Approximations to Continuous Functions |url=https://www.tandfonline.com/doi/full/10.1080/00029890.2023.2206324 |journal=The American Mathematical Monthly |language=en |volume=130 |issue=7 |pages=655 |doi=10.1080/00029890.2023.2206324 |issn=0002-9890|url-access=subscription }}</ref>