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(-∞,∞) 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(-∞,∞), 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 (-∞,∞) 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 dense in L2(-∞,∞), with tridiagonal skew-Hermitian differentiation matrix and whose generalised Fourier coefficients can be computed with a single FFT.

Introductions of the speaker：

Arieh Iserles (born 2 September 1947) is a computational mathematician, currently Professor of the Numerical Analysis of Differential Equations at the University of Cambridge and a member of the Department of Applied Mathematics and Theoretical Physics.In 1999, he was awarded the Onsager Medal, by the Norwegian University of Science and Technology, in 2012 he received the David Crighton medal, presented by the Institute of Mathematics and its Applications and London Mathematical Society "for services to mathematics and the mathematics community" and in 2014 he was awarded by Society for Industrial and Applied Mathematics the SIAM Prize for Distinguished Service to the Profession. In 2012, Professor Iserles was an invited speaker at the 6th European Congress of Mathematics in Kraków, 2–7 July 2012.