Bernstein Expansion
The nth order Bernstein expansion of a function f(x) in terms of a variable x is given by
(Gzyl and Palacios 1997, Mathé 1999), where (n; k) is a binomial coefficient and
is a Bernstein polynomial.
Letting f(x)=x gives the identity
| B_n(x,x)=x |
(3)
|
for n in Z and n>=0.
See also
Bernstein PolynomialExplore with Wolfram|Alpha
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References
Gzyl, H. and Palacios, J. L. "The Weierstrass Approximation Theorem and Large Deviations." Amer. Math. Monthly 104, 650-653, 1997.Mathé, P. "Approximation of Hölder Continuous Functions by Bernstein Polynomials." Amer. Math. Monthly 106, 568-574, 1999.Referenced on Wolfram|Alpha
Bernstein ExpansionCite this as:
Weisstein, Eric W. "Bernstein Expansion." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BernsteinExpansion.html