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Fast Computation of Orthogonal Systems with a Skew‐Symmetric Differentiation Matrix
Author(s) -
Iserles Arieh,
Webb Marcus
Publication year - 2021
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21973
Subject(s) - mathematics , tridiagonal matrix , skew symmetric matrix , toeplitz matrix , orthogonal polynomials , matrix (chemical analysis) , symmetric function , computation , skew , pure mathematics , symmetric matrix , algorithm , square matrix , eigenvalues and eigenvectors , physics , materials science , quantum mechanics , astronomy , composite material
Orthogonal systems in L 2 (ℝ) , once implemented in spectral methods, enjoy a number of important advantages if their differentiation matrix is skew‐symmetric and highly structured. Such systems, where the differentiation matrix is skew‐symmetric, tridiagonal, and irreducible, have been recently fully characterised. In this paper we go a step further, imposing the extra requirement of fast computation: specifically, that the first N coefficients of the expansion can be computed to high accuracy in O N log 2 Noperations. We consider two settings, one approximating a function f directly in (−∞, ∞) and the other approximating [ f ( x ) + f (− x )]/2 and [ f ( x ) − f (− x )]/2 separately in [0, ∞) . In each setting we prove that there is a single family, parametrised by α , β > − 1 , of orthogonal systems with a skew‐symmetric, tridiagonal, irreducible differentiation matrix and whose coefficients can be computed as Jacobi polynomial coefficients of a modified function. The four special cases where α , β = ± 1/2 are of particular interest, since coefficients can be computed using fast sine and cosine transforms. Banded, Toeplitz‐plus‐Hankel multiplication operators are also possible for representing variable coefficients in a spectral method. In Fourier space these orthogonal systems are related to an apparently new generalisation of the Carlitz polynomials. © 2020 Wiley Periodicals, Inc.