The original motivation for this work has been spectral methods for dispersive PDEs, whereby the underlying spatial domain is the whole real line. Seeking 'good' orthonormal systems on the real line, Marcus Webb and I have developed an overarching theory relating such systems which, in addition, have a tridiagonal differentiation matrix, to Fourier transforms of appropriately weighed orthogonal polynomials. Thus, for every Borel measure we obtain an orthonormal system in a Paley-Wiener space, which is dense in $L_2(\mathbb{R})$ iff the measure is supported on all of $\mathbb{R}$. In this talk I will introduce this theory, illustrate it by examples (in particular, Hermite and Malmquist-Takenaka functions) and, time allowing, take it in some of the following directions:

- Characterisation of all such orthonormal systems whose first $n$
coefficients can be computed in $O(n \log n)$ operations;
- Sobolev-orthogonal systems of this kind;
- The emerging approximation theory on the real line.
Speaker(s): Arieh Iserles (University of Cambridge)