Fast approximation on the real line

Speaker: Prof. Dr. Arieh Iserles
Affiliation: University of Cambridge, UK
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Abstract: While approximation theory in an interval is thoroughly understood, the real line represents something of a mystery. In this talk we review the state of the art in this area. Motivating our interest by the design of spectral methods for dispersive PDEs, we commence from familiar Hermite functions and move to recent results characterizing all orthonormal sets on $L_2(-infty,infty)$ that have a skew-symmetric (or skew-Hermitian) tridiagonal differentiation matrix and such that their first $n$ expansion coefficients can be calculated in O(n log n) operations. In particular, we describe the generalized Malmquist-Takenaka system. The talk concludes with a (too!) long list
of open problems and challenges. This is joint work with Marcus Webb.