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:
“For good and evil, man is a free creative spirit. This produces the very queer world we live in, a world in continuous creation and therefore continuous change and insecurity.”
—Joyce Cary (1888–1957)
“Nobody is so constituted as to be able to live everywhere and anywhere; and he who has great duties to perform, which lay claim to all his strength, has, in this respect, a very limited choice. The influence of climate upon the bodily functions ... extends so far, that a blunder in the choice of locality and climate is able not only to alienate a man from his actual duty, but also to withhold it from him altogether, so that he never even comes face to face with it.”
—Friedrich Nietzsche (1844–1900)