Abstract:

A most welcome feature of orthogonal bases employed in spectral methods is that their differentiationmatrix is skew

symmetric, since this makes energy conservation automatic in conservative time-evolvingproblems. A familiar example

is given by Hermite functions, which are dense in L(-∞,∞) and give raise toa 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 havea differentiation matrix which is skew-

symmetric, tridiagonal and irreducible. We also present a constructivealgorithm 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.