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Bernstein Polynomial

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BernsteinPolynomial

The polynomials defined by

B_(i,n)(t)=(n; i)t^i(1-t)^(n-i),
(1)

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

B_(0,0)(t)=1
(2)
B_(0,1)(t)=1-t
(3)
B_(1,1)(t)=t
(4)
B_(0,2)(t)=(1-t)^2
(5)
B_(1,2)(t)=2(1-t)t
(6)
B_(2,2)(t)=t^2
(7)
B_(0,3)(t)=(1-t)^3
(8)
B_(1,3)(t)=3(1-t)^2t
(9)
B_(2,3)(t)=3(1-t)t^2
(10)
B_(3,3)(t)=t^3.
(11)

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

sum_(i=0)^nB_(i,n)(t)=1,
(14)

and B_(i,n) with i!=0,n has a single unique local maximum of

i^in^(-n)(n-i)^(n-i)(n; i)
(15)

occurring at t=i/n.

BernsteinPolynomialEnvelope

The envelope f_n(x) of the Bernstein polynomials B_(i,n)(x) for i=0, 1, ..., n (Mabry 2003) is given by

f_n(x)=1/(sqrt(2pinx(1-x))),
(16)

illustrated above for n=20.

See also

Bernstein Expansion, Bézier Curve, Spline

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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.

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Bernstein Polynomial

Cite this as:

Weisstein, Eric W. "Bernstein Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BernsteinPolynomial.html

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