Orthogonal systems with a skew-symmetric differentiation matrix

Arieh Iserles, Marcus Webb

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper we explore orthogonal systems in L2(R) which give rise to a real
skew-symmetric, tridiagonal, irreducible differentiation matrix. Such systems are
important since they are stable by design and, as necessary, preserve Euclidean
energy for a variety of time-dependent partial differential equations.
We prove that there is a one-to-one correspondence between such an or-
thonormal system {f'n}nεZ+ and a sequence of polynomials {pn}nεZ+ orthonor-
mal with respect to a symmetric probability measure dμ(ξ) = w(ξ)dξ. If dμ is
supported by the real line this system is dense in L2(R), otherwise it is dense in
a Paley{Wiener space of band-limited functions. The path leading from dµ to
{f'n}nεZ+ is constructive and we provide detailed algorithms to this end.
We also prove that the only such orthogonal system consisting of a polynomial
sequence multiplied by a weight function is the Hermite functions.
The paper is accompanied by a number of examples illustrating our argument.
Original languageEnglish
JournalFoundations of Computational Mathematics
Early online date10 Oct 2019
DOIs
Publication statusPublished - 2019

Keywords

  • Differentiation matrix
  • special functions
  • orthogonal polynomials
  • energy preserving numerical methods

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