Bernstein Polynomial and Weierstrass Approximation Theorem
The Weierstrass Approximation Theorem tells us that any continuous function $f$ on a closed interval can be uniformly approximated by polynomial functions. In 1912, Bernstein, based on probability theory, constructed Bernstein polynomials and proved the Weierstrass Approximation Theorem. This paper will introduce the essence of how Bernstein polynomials can approximate continuous functions from both the probabilistic and pure analytic perspectives. First, we present the Weierstrass Approximation Theorem:
Weierstrass Approximation Theorem
Theorem 1 (Weierstrass Approximation Theorem): Let $f$ be a continuous function on the closed interval $[a,b]$. For any $\varepsilon > 0$, there exists a polynomial $p$ such that
$$
\max_{x \in [a,b]} |p(x) - f(x)| < \varepsilon
$$