the Weierstrass approximation theorem
/ VY-er-shtrass /
Before you spend effort building a polynomial approximation of a function, you would like a guarantee that the effort can succeed — that polynomials are rich enough to mimic ANY reasonable curve as closely as you wish. The Weierstrass approximation theorem is exactly that promise, the existence guarantee underpinning all of polynomial approximation.
It states: any continuous function f on a closed bounded interval [a, b] can be approximated UNIFORMLY by polynomials to any desired accuracy. In symbols, for every tolerance epsilon greater than 0 there exists a polynomial p with max over x in [a, b] of |f(x) - p(x)| less than epsilon — you can squeeze a polynomial within an epsilon-thin band around the whole curve at once, no matter how wiggly f is (as long as it is continuous and the interval is finite). One classic constructive proof uses the Bernstein polynomials, an explicit averaging of f's values that provably converges uniformly to f as the degree grows; this not only proves existence but exhibits an actual sequence that works. The result generalizes far beyond polynomials in the Stone-Weierstrass theorem.
Weierstrass is the foundation stone: it is what licenses minimax approximation (the BEST polynomial error can be driven to zero, so the question 'how good can degree n be?' has answers that keep improving) and reassures us that approximation as a whole is not a hopeless quest. But read its honesty carefully. It is an EXISTENCE theorem — it promises a good polynomial exists but says nothing about its degree or how to find it, and the degree needed can be enormous. Crucially, it does NOT say that INTERPOLATION at any fixed nodes converges: a degree-n interpolant through equispaced points can still diverge (Runge's phenomenon) even though SOME degree-n polynomial approximates f well. The good approximant Weierstrass guarantees may simply not be the one your nodes produce — which is exactly why node choice (Chebyshev) and splines matter.
Take f(x) = |x| on [-1, 1], which has a sharp corner and is NOT a polynomial. Weierstrass guarantees polynomials approaching it uniformly: a degree-10 best polynomial keeps the error under about 0.04 everywhere, and higher degrees shrink it toward 0 — a smooth polynomial squeezing arbitrarily close to a cornered curve.
Even a cornered function can be squeezed inside any thin polynomial band.
Weierstrass guarantees a good polynomial EXISTS; it does NOT say interpolation finds it. Equispaced interpolation can still diverge (Runge), and the theorem gives no bound on the degree needed. Existence is not a construction.