Fast Computation of Orthogonal Systems with a <scp>Skew‐Symmetric</scp> Differentiation Matrix

Orthogonal systems in L2(ℝ), once implemented spectral methods, enjoy a number of important advantages if their differentiation matrix is skew-symmetric and highly structured. Such systems, where the skew-symmetric, tridiagonal, irreducible, have been recently fully characterised. In this paper we go step further, imposing extra requirement fast computation: specifically, that first N coefficients expansion can be computed to high accuracy operations. We consider two settings, one approximating function f directly (−∞, ∞) other [f(x) + f(−x)]/2 − separately [0, ∞). each setting prove there single family, parametrised by α, β > 1, orthogonal with irreducible whose as Jacobi polynomial modified function. The four special cases = ± 1/2 are particular interest, since using sine cosine transforms. Banded, Toeplitz-plus-Hankel multiplication operators also possible for representing variable method. Fourier space these related an apparently new generalisation Carlitz polynomials. © 2020 Wiley Periodicals, Inc.