Weierstrass Approximation Theorem

Statement

Let $f : [a, b] \to \mathbb{R}$ be continuous. For every $\varepsilon > 0$ there exists a polynomial $p \in \mathbb{R}[x]$ such that

Equivalently, the polynomials are dense in $C([a,b])$ with the uniform norm $\|f\|_\infty = \sup_{[a,b]} |f|$.

Visualization

Bernstein polynomials $B_n f$ approximate $f(x) = |x|$ on $[-1, 1]$ uniformly. Explicitly, reparameterizing to $[0,1]$ with $g(t) = |2t-1|$:

| $x$ | $f(x) = |x|$ | $B_4$ | $B_8$ | $B_{16}$ | |-----|------------|-------|-------|--------| | 0.0 | 0.000 | 0.375 | 0.273 | 0.196 | | 0.2 | 0.200 | 0.261 | 0.231 | 0.214 | | 0.5 | 0.500 | 0.500 | 0.500 | 0.500 | | 0.8 | 0.800 | 0.761 | 0.781 | 0.791 | | 1.0 | 1.000 | 1.000 | 1.000 | 1.000 |

(Values computed for $g(t) = |2t-1|$ on $[0,1]$, so $t = (x+1)/2$.)

The uniform error $\|B_n g - g\|_\infty = O(n^{-1/2})$ for Lipschitz $g$; higher-degree polynomials converge to $|x|$ everywhere.

  Approximation of |x| by polynomials on [-1,1]

  1 |*         *   ← |x|
    | *\     /*
    |  *\   /* ← B₄ (smoothed corners)
    |   *\ /* ← B₈
  0 |----*------   ← x=0, f(0)=0, B_n(0)→0
    |   /* \*
    |  /    \* ← converging to |x|
   -1                   1

Proof Sketch

1. Bernstein polynomials: For $f \in C([0,1])$ define $B_n f(x) = \sum_{k=0}^{n} f(k/n) \binom{n}{k} x^k (1-x)^{n-k}$.
2. Probabilistic identity: $B_n f(x) = \mathbb{E}[f(S_n/n)]$ where $S_n \sim \mathrm{Binomial}(n, x)$. By the law of large numbers $S_n/n \to x$, so $B_n f(x) \to f(x)$.
3. Uniform convergence: Continuity on the compact set $[0,1]$ implies uniform continuity. The estimate $|B_n f(x) - f(x)| \leq \omega_f(\delta) + 2\|f\|_\infty / (n\delta^2)$ (where $\omega_f$ is the modulus of continuity) gives uniform convergence by choosing $\delta = n^{-1/4}$.
4. General $[a,b]$: Apply the $[0,1]$ result after the affine reparameterization $t = (x-a)/(b-a)$.

Connections

• Extreme Value TheoremExtreme Value Theorem$$f\in C(K),\,K\text{ compact}\implies\exists\,x_*,x^*\in K:\;f(x_*)=\min f=f(x^*)\leq\max f$$A continuous function on a compact set attains its maximum and minimum valuesRead more → — $f$ is uniformly continuous on $[a,b]$ (since it is compact) — this is the key regularity that makes the Bernstein approximation converge
• Heine-Borel Theorem — the compact domain $[a,b]$ is essential; the theorem fails for all of $\mathbb{R}$ (polynomials cannot uniformly approximate $e^x$)
• Intermediate Value TheoremIntermediate Value Theorem$$f(a) \le y \le f(b) \Rightarrow \exists\, c \in [a,b]:\; f(c) = y$$A continuous function on a closed interval hits every value between its endpointsRead more → — the density of polynomials in $C[a,b]$ is the analytic analogue of the IVT: every continuous function is a limit of explicit elementary ones
• Stone–Weierstrass TheoremStone–Weierstrass Theorem$$A \subseteq C(X) \text{ sep. pts, contains 1} \Rightarrow \overline{A} = C(X)$$A subalgebra of continuous functions on a compact space that separates points is dense in the sup-normRead more → — the functional-analysis generalization: any subalgebra of $C(K)$ that separates points and contains constants is dense; Weierstrass is the case where the subalgebra is $\mathbb{R}[x]$ on $[a,b]$

import Mathlib.Topology.ContinuousMap.Weierstrass

/-- Weierstrass Approximation Theorem: polynomials are dense in C([a,b]).
    Wraps Mathlib's `polynomialFunctions_closure_eq_top`. -/
theorem weierstrass_approximation (a b : ℝ) :
    (polynomialFunctions (Set.Icc a b)).topologicalClosure = ⊤ :=
  polynomialFunctions_closure_eq_top a b