Fast Computation Of Orthogonal Systems With A Skew-Symmetric Differentiation Matrix

Orthogonal systems in L-2(Double-struck capital 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 ONlog2N operations. We consider two settings, one approximating a function f directly in (-infinity, infinity) and the other approximating [f(x) + f(-x)]/2 and [f(x) - f(-x)]/2 separately in [0, infinity). In each setting we prove that there is a single family, parametrised by alpha, beta > - 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 alpha, beta = +/- 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. (c) 2020 Wiley Periodicals, Inc.