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  "body": "\n## Definition\n\nLet $f \\colon [0, 1] \\to \\mathbb{R}$ be a function and let $n \\ge 1$ be an integer.\nThe $n$-th Bernstein polynomial of $f$ is\n$$B_n(f)(x) = \\sum_{k=0}^{n} f\\Big( \\frac{k}{n} \\Big) \\binom{n}{k} x^k (1 - x)^{n-k}.$$\nThe polynomials $b_{k,n}(x) = \\binom{n}{k} x^k (1 - x)^{n-k}$ are the Bernstein basis polynomials.\n\n## Remarks\n\nIf $f$ is continuous, then $B_n(f) \\to f$ uniformly on $[0, 1]$. This proves the [[Weierstrass approximation theorem|0JQE130DHW]].\nThe basis polynomials are nonnegative on $[0, 1]$, and they sum to $1$ by the binomial theorem.\nSergei Bernstein gave this proof in 1912, with an argument from probability. Mathlib defines the basis polynomials as `bernsteinPolynomial`.\n\n## Explanations\n\n### L1 human\n\nThe Bernstein polynomial samples $f$ at the points $k/n$ and blends the values with smooth weights.\nNear $x$, the weights concentrate on the samples with $k/n$ close to $x$.\n",
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