Orthogonal systems with a skew-symmetric differentiation matrix

In this paper we explore orthogonal systems in L2(R) which give rise to a real skew-symmetric, tridiagonal, irreducible differentiation matrix. Such systems are important since they are stable by design and, as necessary, preserve Euclidean energy for a variety of time-dependent partial differential equations. We prove that there is a one-to-one correspondence between such an or- thonormal system {φn}n∈Z+ and a sequence of polynomials {pn}n∈Z+ orthonor- mal with respect to a symmetric probability measure dμ(ξ) = w(ξ)dξ. If dμ is supported by the real line this system is dense in L2(R), otherwise it is dense in a Paley–Wiener space of band-limited functions. The path leading from dμ to {φn}n∈Z+ is constructive and we provide detailed algorithms to this end. We also prove that the only such orthogonal system consisting of a polynomial sequence multiplied by a weight function is the Hermite functions. The paper is accompanied by a number of examples illustrating our argument.status: accepte