# Fast approximation on the real line

GCS - Geometry, compatibility and structure preservation in computational differential equations

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, commencing from the familiar Hermite functions and moving to recent results  characterising 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 generalised Malmquist–Takenaka system. The talk concludes with a (too!) long list of open problems and challenges.

This talk is part of the Isaac Newton Institute Seminar Series series.