Bernstein Polynomial - Approximating Continuous Functions

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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