Probabilistic Perspective of Weierstrass Approximation Theorem via Bernstein Polynomial

In view of the importance of Weierstrass approximation theorem, this paper presents a probabilistic approach to establishing the Weierstrass approximation theorem via Bernstein polynomials, emphasizing the interplay between probability theory and mathematical analysis. By interpreting Bernstein polynomials through the lens of Bernoulli trials, the author associates the Bernstein polynomial function with the expectationE[f(Sn/n)], whereSnfollows a binomial distribution with parameterx. Leveraging the law of large numbers, the author demonstrates the uniform convergence of the Bernstein polynomial function to the original function on[0,1]. The key technical contributions involve bounding the approximation error through variance analysis and employing piecewise estimation to address non-uniform continuity. This approach not only circumvents traditional analytical complexities but also highlights how probabilistic tools, such as particularly moment calculations and distributional convergence, naturally unify discrete approximations with continuous function spaces. The methodology simplifies computational procedures while maintaining rigorous convergence guarantees, offering an instructive example of cross-domain synergy between probability and classical analysis.