# Bernstein polynomial

Suppose *f* is a continuous real-valued function on the interval [0, 1]. The *n*th-degree polynomial

is a

**Bernstein polynomial** approximating

*f*(

*x*). These polynomials are used in a constructive proof of the

Weierstrass approximation theorem.

It can be shown that

**uniformly on the interval [0, 1]**. This is a stronger statement than the proposition that the limit holds for each value of

*x* separately; that would be

pointwise convergence rather than

uniform convergence. specifically, the word

*uniformly* signifies that

Bernstein polynomials thus afford one way to prove the

Weierstrass approximation theorem (named in honor of

Karl Weierstrass) that every continuous function on a closed bounded interval can be uniformly approximated by polynomial functions.

Suppose *K* is a random variable distributed as the number of successes in *n* independent Bernoulli trials with probability *x* of success on each trial; in other words, *K* has a binomial distribution with parameters *n* and *x*. Then we have the expected value E(*K/n*) = *x*.

Then the weak law of large numbers of probability theory tells us that

Because

*f*, being continuous on a closed bounded interval, must be

uniformly continuous on that interval, we can infer a statement of the form

Consequently

And so the second probability above approaches 0 as

*n* grows. But the second probability is either 0 or 1, since the only thing that is random is

*K*, and that appears

*within the scope of the expectation operator E*. Finally, observe that E(

*f*(

*K/n*)) is just the Bernstein polynomial

*B*_{n}(

*f*,

*x*).