Bernstein Polynomial
The polynomials defined by
where
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
positivity
for
and
occurring at 
The envelope
illustrated above for
SEE ALSO: Bernstein Expansion,
Bézier Curve, Spline
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 H?lder 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. Weisstein, Eric W. "Bernstein Polynomial." From MathWorld--A Wolfram Web Resource. http://mathworld./BernsteinPolynomial.html |
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is a 
form a basis for
the 
. The first few
polynomials are
