Bernstein polynomial

Definition

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

is a Bernstein polynomial approximating f(x). These polynomials are used in a constructive proof of the Weierstrass approximation theorem.

A 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.

Proof

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).

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