Comment by ogogmad
5 days ago
Chebyshev polynomials cos(n arcos(x)) provide one of the proofs that every continuous function f:[0,1]->R can be uniformly approximated by polynomial functions. Bernstein polynomials provide a shorter proof, but perhaps not the best numerical method: https://en.wikipedia.org/wiki/Bernstein_polynomial#See_also
Those don't guarantee that that they can be well approximated by a polynomial of degree N though, like we have here. You can apply Jackson's inequality to calculate a maximum error bound, but the epsilon for degree 5 is pretty atrocious.