Everything about Bernstein Polynomial totally explained
» For the Bernstein polynomial in D-module theory, see Bernstein-Sato polynomial.
In the
mathematical subfield of
numerical analysis, a
Bernstein polynomial, named after
Sergei Natanovich Bernstein, is a
polynomial in the
Bernstein form, that's a
linear combination of
Bernstein basis polynomials.
A
numerically stable way to evaluate polynomials in Bernstein form is
de Casteljau's algorithm.
Polynomials in Bernstein form were first used by Bernstein in a constructive proof for the
Stone-Weierstrass approximation theorem. With the advent of computer graphics, Bernstein polynomials, restricted to the interval
x ∈ [0, 1], became important in the form of
Bézier curves.
Definition
The
n + 1
Bernstein basis polynomials of degree
n are defined as
»
And so the second probability above approaches 0 as
n grows. But the second probability is either 0 or 1, since the only thing that's random is
K, and that appears
within the scope of the expectation operator E. Finally, observe that E(
f(
K/n)) is just the Bernstein polynomial
Bn(
f,
x).
Further Information
Get more info on 'Bernstein Polynomial'.
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