A family of orthogonal rational functions and other orthogonal systems with a skew-Hermitian differentiation matrix

Arieh Iserles, Marcus Webb

Research output: Contribution to journalArticlepeer-review

38 Downloads (Pure)

Abstract

In this paper we explore orthogonal systems in L2(R) which give rise to a skew-Hermitian, tridiagonal differentiation matrix. Surprisingly, allowing the differentiation matrix to be complex leads to a particular family of rational orthogonal functions with favourable properties: they form an orthonormal basis for L2(R), have a simple explicit formulæ as rational functions, can be manipulated easily and the expansion coefficients are equal to classical Fourier coefficients of a modified function, hence can be calculated rapidly. We show that this family of functions is essentially the only orthonormal basis possessing a differentiation matrix of the above form and whose coefficients are equal to classical Fourier coefficients of a modified function though a monotone, differentiable change of variables. Examples of other orthogonal bases with skew-Hermitian, tridiagonal differentiation matrices are discussed as well.
Original languageEnglish
JournalJournal of Fourier Analysis and Applications
DOIs
Publication statusPublished - 23 Jan 2020

Keywords

  • Orthogonal systems
  • orthogonal rational functions
  • spectral methods
  • Fast Fourier Transform
  • Malmquist–Takenaka system

Fingerprint

Dive into the research topics of 'A family of orthogonal rational functions and other orthogonal systems with a skew-Hermitian differentiation matrix'. Together they form a unique fingerprint.

Cite this