Bernstein Polynomial
The polynomials defined by
 |
(1)
|
where 
is a binomial coefficient. The Bernstein polynomials of degree 
form a basis for the power polynomials of degree 
. The first few polynomials are
The Bernstein polynomials are implemented in the Wolfram Language as BernsteinBasis[n, i, t].
The Bernstein polynomials have a number of useful properties (Farin 1993). They satisfy symmetry
 |
(12)
|
positivity
 |
(13)
|
for 
, normalization
 |
(14)
|
and 
with 
has a single unique local maximum of
 |
(15)
|
occurring at 
.
The envelope 
of the Bernstein polynomials 
for 
, 1, ..., 
(Mabry 2003) is given by
 |
(16)
|
illustrated above for 
.
REFERENCES:
Bernstein, S. "Démonstration du théorème de Weierstrass fondée sur le calcul des probabilities." Comm. Soc. Math. Kharkov 13, 1-2, 1912.
Farin, G. Curves and Surfaces for Computer Aided Geometric Design. San Diego: Academic Press, 1993.
Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, p. 222, 1971.
Kac, M. "Une remarque sur les polynomes de M. S. Bernstein." Studia Math. 7, 49-51, 1938.
Kac, M. "Reconnaissance de priorité relative à ma note, 'Une remarque sur les polynomes de M. S. Bernstein.' " Studia Math. 8, 170, 1939.
Lorentz, G. G. Bernstein Polynomials. Toronto: University of Toronto Press, 1953.
Mabry, R. "Problem 10990." Amer. Math. Monthly 110, 59, 2003.
Mathé, P. "Approximation of Hölder Continuous Functions by Bernstein Polynomials." Amer. Math. Monthly 106, 568-574, 1999.
Widder, D. V. The Laplace Transform. Princeton, NJ: Princeton University Press, p. 101, 1941.
