Weierstrass Approximation Theorem

The Weierstrass Approximation Theorem states that every continuous function on a closed interval can be uniformly approximated by polynomials. This profound result shows that polynomials are "dense" in the space of continuous functions and is the foundation for numerical analysis and approximation theory.

Statement

Theorem7.1Weierstrass Approximation Theorem

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

$|f(x) - p(x)| < \epsilon \quad \text{for all } x \in [a, b].$

RemarkUniform approximation

The theorem guarantees uniform approximation: the polynomial approximates $f$ to within $\epsilon$ on the entire interval simultaneously.

Applications

ExampleBernstein polynomials

For $f : [0, 1] \to \mathbb{R}$ continuous, the Bernstein polynomials

$B_n(f)(x) = \sum_{k=0}^n f(k/n) \binom{n}{k} x^k (1-x)^{n-k}$

converge uniformly to $f$ as $n \to \infty$ . This gives a constructive proof of the Weierstrass theorem.

ExampleApproximating e^x

The exponential $e^x$ on $[-1, 1]$ can be approximated by $p_n(x) = 1 + x + x^2/2! + \cdots + x^n/n!$ to arbitrary precision as $n \to \infty$ .

Summary

The Weierstrass Approximation Theorem shows polynomials are dense in $C([a, b])$ :

Every continuous function can be uniformly approximated by polynomials.
Proof via Bernstein polynomials or convolution.
Foundation for numerical methods and functional analysis.

See Uniform Convergence and Stone-Weierstrass.