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
Famous quotes containing the words continuous and/or functions:
“There was a continuous movement now, from Zone Five to Zone Four. And from Zone Four to Zone Three, and from us, up the pass. There was a lightness, a freshness, and an enquiry and a remaking and an inspiration where there had been only stagnation. And closed frontiers. For this is how we all see it now.”
—Doris Lessing (b. 1919)
“The English masses are lovable: they are kind, decent, tolerant, practical and not stupid. The tragedy is that there are too many of them, and that they are aimless, having outgrown the servile functions for which they were encouraged to multiply. One day these huge crowds will have to seize power because there will be nothing else for them to do, and yet they neither demand power nor are ready to make use of it; they will learn only to be bored in a new way.”
—Cyril Connolly (1903–1974)