 Bernstein polynomial

For $f\in C([0,1{]}^d)$, its tensor-product Bernstein polynomial is the expectation of $f(X_1/n,\dots ,X_d/n)$ for independent binomial variables $X_j\sim \mathrm{Bin}(n,x_j)$. Uniform continuity and concentration of the binomial variables prove uniform convergence to $f$.

The Bernstein polynomial is a crucial concept in approximation theory and mathematical analysis, particularly in the context of polynomial interpolation and approximation of continuous functions. The Bernstein polynomials are defined to approximate a continuous function on a closed interval [0, 1] by a weighted sum of polynomials.