Mathematical folklore states that polynomial extrapolation of a function is hopeless, especially when only equally-spaced function samples are known. In this talk, we explain how a more precise statement carries an interesting nuance. Provided a standard oversampling condition is satisfied, we give a recipe for constructing an asymptotically best extrapolant as a piecewise polynomial. Along the way, we derive explicit bounds for the quality of least squares approximations from equally-spaced samples, robustness to perturbed samples, and a faster direct algorithm for constructing least squares approximants. This is joint work with Laurent Demanet (MIT).