A journey to approximation theory by Michael Fochler, Department of Mathematical Sciences, Binghamton University April 1, 2014   The subject of approximation theory is the representation of certain functions as the limit of functions of a specific type. If you have taken Calc 2 then you know at least one example: the approximation of infinitely often differentiable functions by Taylor polynomials. A more general setting is addressed by Weierstrass' theorem which states that any continuous function on the unit interval, even if it is not differentiable, can be approximated by polynomials. Korovkin showed in the 1950s the following: Let Ln be any sequence of positive linear operators on the space of all continuous functions defined for 0 ≤ x ≤ 1. If you can prove that Lnf converges uniformly to f just for the following three functions: f(x) = 1, f(x) = x, f(x) = x2 then you know that Lnf converges to f for any continuous function on the unit interval. The first part of this talk will be a review of the tools helpful to understand the formulation and proof of Korovkin's theorem and, as an application, the proof of Weierstrass' theorem. Here are some of them: convergence and uniform convergence, compactness, continuity and uniform continuity, binomial coeffients, Bernstein polynomials, positive and linear operators. If time allows, there will be a quick intro to the following areas of probability theory and statistics: independently executed binomial trials, expected value and variance of a random event, sample mean, law of large numbers and Chebyshev's inequality. This will be followed by a probabilistic proof (due to Bernstein) of Weierstrass' approximation theorem. With so much ground to cover some proofs will be done "by example", others will be skipped entirely, but some will be given in detail.