Table of contents |
2 A theorem 3 Proof |
Suppose f is a continuous real-valued function on the interval [0, 1]. The nth-degree polynomial
Definition
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
A theorem
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
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
Proof
Because f, being continuous on a closed bounded interval, must be uniformly continuous on that interval, we can infer a statement of the form