Probability and Korovkin's Theorem by Michael Fochler, Department of Mathematical Sciences, Binghamton University Thursday, April 10, 2014         A classical theorem of Weierstrass states that any continuous function on the unit interval can be approximated in the norm of uniform convergence by polynomials. We shall briefly review Bernstein's probabilistic proof which is based on the law of large numbers for functions f(X̄) of the sample mean of an iid sequence of Bernoulli trials: The n-th Bernstein polynomial Bnf(p) is the expected value Epf(X̄) if we denote by p the success probability of any one of the Bernoulli trials and Bernstein showed that this sequence converges uniformly in p to f(p) for 0 ≤ p ≤ 1 if f is continuous.       Korovkin showed in the 1950s the following: Let H0 be the set of three test functions h(x) = 1, h(x) = x, h(x) = x2. Let Ln be any sequence of positive linear operators on the space C[0,1] of all continuous functions on the unit interval. If you can prove that Lnh converges uniformly to h for h ∈ H0 then this uniform convergence extends to all f ∈ C[0,1]. Pre-calculus math is sufficient to compute Bnh(x) for h ∈ H0 and it is immediate that Bnh(x) → h uniformly. Korovkin's theorem hence gives yet another proof of Weierstrass' approximation theorem.       We shall look at Korovkin's theorem for continuous functions on compact metric spaces along the lines of an article of Heinz Bauer. Here the task is to find as small a set H0 of continuous test functions as possible which satisfy the following, just as in the classical case: Extension of uniform convergence Lnh → h for all h ∈ H0 to all continuous functions on E. Let us say that such a test set has the Korovkin property. A very geometric criterion based on envelopes is given and the role of representing measures and the Choquet boundary for a given set of test functions is discussed.       We shall then look at Weba's extension of Korovkin's theorem to positive linear operators on stochastic processes.       Finally we shall go back to the function case and examine Markov processes that leave all functions in H0 invariant. It turns out that a test set has the Korovkin property if and only if only the Markov process associated with the identity semigroup (the constant paths Markov process) leaves it invariant.