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.
