The Journal of Online Mathematics and Its Applications, Volume 6 (2006)
The Chebyshev Equioscillation Theorem, Robert Mayans

## 2. The Weierstrass Approximation Theorem

Throughout this article, f will be a continuous real-valued function on an interval [a, b], and p will be a real polynomial that approximates f on [a, b].

The first step is to show that polynomial approximations exist to arbitrary accuracy. This is the conclusion of the famous Weierstrass approximation theorem, named for Karl Weierstrass.

#### Theorem 1 (Weierstrass)

Let f be a continuous real-valued function on the closed interval [a, b]. Then there is a sequence of real polynomials p1, p2, ... that converge uniformly to f.

The proof of the Weierstrass theorem by Sergi Bernstein is constructive: it defines explicitly a sequence of polynomials that converge to f. Suppose that f is a continuous real-valued function defined on [0, 1] (there is no loss of generality in restricting the interval in this way). Define the Bernstein polynomials on f by:

#### Theorem 2 (Bernstein)

The Bernstein polynomials Bn(f; x) converge uniformly to f on [0, 1].