Academy of Mathematics and Systems Science, CAS Colloquia & Seminars
Speaker:
Prof. Arieh Iserles, University of Cambridge
Inviter:
Title:
Skew-symmetric differentiation matrices and spectral methods on the real line
Time & Venue:
2018.10.10 16:00-17:00 N222
Abstract:
A most welcome feature of orthogonal bases employed in spectral methods is that their differentiation matrix is skew symmetric, since this makes energy conservation automatic in conservative time-evolving problems. A familiar example is given by Hermite functions, which are dense in $L(-\infty,\infty)$ and give raise to a skew-symmetric, tridiagonal differentiation matrix. In this talk, describing joint work with Marcus Webb (KU Leuven), we present full characterisation of all orthogonal systems acting on $L2(-\infty,\infty)$, dense either there or in a Paley-Wiener space, and that have a differentiation matrix which is skew-symmetric, tridiagonal and irreducible. We also present a constructive algorithm for their generation -essentially,given any symmetric Borel measure on $(-\infty,\infty)$ or on $(-a,a)$ for some $a>0$, there exists a unique (up to rescaling) basis of this kind and it can be generated constructively. We conclude with a number of examples, related to Konoplev, Carlitz and Freud measures. Finally, we address the more general question of skew-Hermitian differentiation matrices. This brings us to very recent work on a variant of Malmquist-Takenaka basis, which appears to tick every desirable box: an orthonormal system densein $L2(-\infty,\infty)$, with tridiagonal skew-Hermitian differentiation matrix and whose generalised Fourier coefficients can be computed with a single FFT.