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On the Approximation Properties of Bernstein Polynomials via Probabilistic Tools

We study two loosely related problems concerning approximation properties of Bernstein polynomials Bnf(x) of some function f on [0,1]: the absence of Gibbs phenomenon at points at which f has jumps, and the convergence of (Bnf) 0 (x) towards f 0 (x). Both results are obtained using classical probabilistic tools. In particular the proof of the second statement relies on the representation of the derivative (Bnf) 0 (x) as the expectation of the functional of a random variable.