Fast Computation of Orthogonal Systems with a Skew‐Symmetric Differentiation Matrix

Abstract

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 skewsymmetric, 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 N / operations. We consider two settings, one approximating a function f directly in . I; I/ and the other approximating f .x/ g f . x/=2 and f .x/ f . x/=2 separately in 0; I/. In each setting we prove that there is a single family, parametrised by ; > 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 ; h ¦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.

A Family of Orthogonal Rational Functions and Other Orthogonal Systems with a skew-Hermitian Differentiation Matrix

“…The FFT, however, is not the only route toward 'fast' computation of coefficients in the context of orthonormal systems on L 2 (R) with skew-Hermitian or skew-symmetric differentiation matrices. In (Iserles & Webb 2019a) we characterised all such real systems (thus, with a skew-symmetric differentiation matrix) whose coefficients can be computed with either Fast Cosine Transform, Fast Sine Transform or a combination of the two, again incurring an O (N log 2 N ) cost. We prove there that there exist exactly four systems of this kind.…”

“…The jury is out on which is the 'best' orthonormal L 2 (R) system with a skew-Hermitian (or skew-symmetric) tridiagonal differentiation matrix and whose first N coefficients can be computed in O (N log 2 N ) operations. While some considerations have been highlighted in (Iserles & Webb 2019a), probably the most important factor is the speed of convergence. Approximation theory in L 2 (R) is poorly understood and much remains to be done to single out optimal orthonormal systems for different types of functions.…”

“…Either course of action is restricted by the limitations on our knowledge of explicit fomulae of orthogonal polynomials for absolutely continuous measures (thereby excluding, for example, Charlier and Lommel polynomials, as well as the Askey-Wilson hierarchy). Thus Hermite polynomials and their generalisations (Iserles & Webb 2019b), Jacobi and Konoplev polynomials (Iserles & Webb 2019b), Carlitz polynomials (Iserles & Webb 2019a) and, in the next section, Laguerre polynomials. Herewith we present another example which, albeit of no apparent practical use, by its very simplicity helps to illustrate our narrative.…”

“…We wish to draw attention to (Iserles & Webb 2019a), a companion paper to this one. While operating there within the original framework of (Iserles & Webb 2019b) -skew-symmetry rather than skew-Hermicity -we seek therein to characterise orthonormal systems in L 2 (R) whose first N coefficients can be computed in O (N log 2 N ) operations by fast expansion in orthogonal polynomials.…”

Approximation of wave packets on the real line

“…Discovered simultaneously by Malmquist and Takenaka in 1926, they have since been studied in fields ranging from signal processing to spectral methods. The MT functions were recently rediscovered by Iserles and Webb in [14] based on their ongoing work classifying bases with banded skew-Hermitian differentiation matrices [13,15,16]. It was the work of Boyd in [1] and Weideman in [22,23] that highlighted the interesting approximation properties of MT functions.…”