This procedure returns the nth degree Bernstein polynomial in x approximating the function f(x) on the interval $\left[0\,1\right]$.  Note that f must be a function of one variable specified as a procedure or operator.

Bernstein polynomials arise in the Stone-Weierstrass approximation theorem of analysis that says any continuous function (R->R) can be uniformly approximated on a closed interval by a sequence of polynomials.  The Bernstein polynomials are one such set for doing this.

Given $p≔\left(n\,i\,x\right)\mapsto \left(\genfrac{}{}{0ex}{}{n}{i}\right)\cdot x^i\cdot {\left(1-x\right)}^{n-i}$ Bernstein is defined to be

$\mathrm{bernstein}\left(3\,x\mapsto \frac{1}{x+1}\,z\right)$

$-\frac{1}{20}{z}^3+\frac{3}{10}{z}^2-\frac{3}{4}z+1$

f := proc(t) if t < 1/2 then 4*t^2 else 2 - 4*t^2 end if end proc:

$\mathrm{bernstein}\left(2\&comma;f\&comma;x\right)$

$-4x^2+2x$