Approximating Continuous Functions
Let ƒ be a continuous function on the interval . Consider the Bernstein polynomial
It can be shown that
uniformly on the interval . 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 that every real-valued continuous function on a real interval can be uniformly approximated by polynomial functions over R.
A more general statement for a function with continuous kth derivative is
where additionally
is an eigenvalue of Bn; the corresponding eigenfunction is a polynomial of degree k.
Read more about this topic: Bernstein Polynomial
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