Bernstein Polynomial
The polynomials defined by
where (n; k) is a binomial coefficient. The Bernstein polynomials of degree n form a basis for the power polynomials of degree n. 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
| B_(i,n)(t)=B_(n-i,n)(1-t), |
(12)
|
positivity
| B_(i,n)(t)>=0 |
(13)
|
for 0<=t<=1, normalization
| [画像: sum_(i=0)^nB_(i,n)(t)=1, ] |
(14)
|
and B_(i,n) with i!=0,n has a single unique local maximum of
| [画像: i^in^(-n)(n-i)^(n-i)(n; i) ] |
(15)
|
occurring at t=i/n.
The envelope f_n(x) of the Bernstein polynomials B_(i,n)(x) for i=0, 1, ..., n (Mabry 2003) is given by
| [画像: f_n(x)=1/(sqrt(2pinx(1-x))), ] |
(16)
|
illustrated above for n=20.
See also
Bernstein Expansion, Bézier Curve, SplineExplore with Wolfram|Alpha
More things to try:
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.Referenced on Wolfram|Alpha
Bernstein PolynomialCite this as:
Weisstein, Eric W. "Bernstein Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BernsteinPolynomial.html