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

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AbstractOrthogonal systems in L2(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 operations. 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.