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 (−∞, ∞) and the other approximatingIn 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.

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

“…The motivation for this paper is the numerical solution of time-dependent partial differential equations on the real line. It continues an ongoing project of the present authors, begun in (Iserles & Webb 2019b), which studied orthonormal systems Φ = {ϕ n } n∈Z in L 2 (R) which satisfy the differential-difference relation,…”

“…once ϕ 0 is known. The obvious idea is to compute explicitly the derivatives of ϕ 0 and form their linear combination (6), but equally useful is a generalisation of an approach originating in (Iserles & Webb 2019b). Thus, Fourier-transforming (6),φ…”

“…where {b n } n∈Z+ ⊂ C and {c n } n∈Z+ ⊂ R. While the substantive theory underlying the characterisation of orthonormal systems in L 2 (R) with skew-Hermitian, tridiagonal, irreducible differentiation matrices is a fairly straightforward extension of (Iserles & Webb 2019b), its ramifications are new and, we believe, important. In Section 2 we establish this theory, characterising Φ as Fourier transforms of weighted orthogonal polynomials with respect to some absolutely-continuous Borel measure dµ.…”

“…Adding rigorous but reasonable assumptions to these requirements, we prove that, modulo a simple generalisation, the MT system is the only system which bears all three. 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.…”

“…The difference is that (2) is replaced by (1), dµ must be even, g must have even real part and odd imaginary part, and θ n is chosen so that e iθn = (−i) n . We will prove sufficiency because it is elementary but enlightening, and leave necessity and uniqueness for the reader to prove by modifying the proof in (Iserles & Webb 2019b). That part of the proof depends on Favard's theorem and properties of the Fourier transform, and we wish to avoid it for the sake of brevity.…”