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Bernstein Expansion

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The nth order Bernstein expansion of a function f(x) in terms of a variable x is given by

B_n(f,x)=sum_(j=0)^n(n; j)x^j(1-x)^(n-j)f(j/n),
(1)

(Gzyl and Palacios 1997, Mathé 1999), where (n; k) is a binomial coefficient and

B_(j,n)(x)=(n; j)x^j(1-x)^(n-j)
(2)

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 Polynomial

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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 Expansion

Cite this as:

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

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