I am learning about different interpolation methods in my internship. Today as I was looking this article on Wikipedia to learn about the Runge's Phenomenon exhibited by Polynomial Interpolation. I noticed that Weierstrass Approximation Theorem is contradicted by Runge's Phenomenon:
Weierstrass Theorem: $$\lim_{n \rightarrow \infty} \left( \max_{a \leq x \leq b} | f(x) -P_n(x)| \right) = 0.$$
Runge's Phenomenon for Equidistant partition of interval: $$\lim_{n \rightarrow \infty} \left( \max_{-1 \leq x \leq 1} | f(x) -P_n(x)| \right) = +\infty. $$ This is the first time I am seeing a theorem being contradicted. But I am pretty sure that I have misunderstood something because a theorem, by definition, should always be correct, isn't it? Please clarify my misunderstanding. Thanks .
One can see this as essentially a parsing error (the $P_n$ in the two statements refer to two different things).
Stone-Weierstrass tells us that every continuous function on an interval $[a,b]$ can be approximated arbitrarily closely by some sequence of polynomials.
We contrast this to polynomial interpolation: this is a specific method for generating a sequence of polynomials that will approximate our data set. If our data set was just the outputs of some continuous function, we would hope that the sequence of polynomials would approximate it arbitrarily closely as we generate them of higher and higher degree. The Runge phenomenon says that this is too optimistic: approximating $1/(1+x^2)$ over $[-5,5]$ with equidistant nodes does not converge in the way we want.
Once we've identified that it is possible for polynomial interpolation to fail, we begin to investigate its various failures and successes. The section on wikipedia discusses what happens when we look at other sample point generation schemes.