报告题目：Skew-symmetric Differentiation Matrices and Spectral Methods on the Real Line
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 dense in $L2(-\infty,\infty)$, with tridiagonal skew-Hermitian differentiation matrix and whose generalised Fourier coefficients can be computed with a single FFT.
Arieh Iserles教授是国际著名数值分析学家、英国剑桥大学应用数学和理论物理系终身教授。研究领域为微分方程数值分析和高振荡积分，在微分方程的李群方法、高振荡积分的高效计算等方面做出了杰出贡献，发表国际高水平学术论文140多篇。Arieh Iserles教授1999年获得挪威Lars Onsager奖，2012年获得英国伦敦数学学会授予的David Crighton奖，2012年应邀在第6届欧洲数学大会上作邀请报告。