Speaker: Charles Epstein

Abstract: Interpolation, which entails using samples of a function to define function values at intermediate arguments, and extrapolation, which entails extending a function beyond its domain of definition, are two basic operations of analysis and numerical analysis. I will first discuss the easier problem of interpolation, explaining various surprising phenomena that arise and the central role played by Chebyshev polynomials in the essentially optimal solution of these problems.  Most of this material is classical. I will then turn to the much harder problem of extrapolation, and describe a new approach, developed by Shidong Jiang and myself, with superior accuracy and stability properties. It again relies heavily on remarkable properties of Chebyshev polynomials.