Generalised Hermite functions and their applications in Spectral Approximations

Abstract

In 1939, G. Szeg\"{o} first introduced a family of generalised Hermite polynomials (GHPs) as a generalisation of usual Hermite polynomials, which are orthogonal with respect to the weight function \(|x|^{2\mu} \e^{-x^2},\mu>-\frac 12\) on the whole line. Since then, there have been a few works on the study of their properties, but no any on their applications to numerical solutions of partial differential equations (PDEs). The main purposes of this paper are twofold. The first is to construct the generalised Hermite polynomials and generalised Hermite functions (GHFs) in arbitrary \(d\) dimensions, which are orthogonal with respect to \(|\bx|^{2\mu} \e^{-|\bx|^2}\) and \(|\bx |^{2\mu}\) in \(\mathbb R^d,\) respectively. We then define a family of adjoint generalised Hermite functions (A-GHFs) upon GHFs, which has two appealing properties: (i) the Fourier transform maps A-GHF to the corresponding GHF; and (ii) A-GHFs are orthogonal with respect to the inner product \([u,v]_{H^s(\mathbb R^d)}=((-\Delta)^{\frac s 2}u, (-\Delta)^{\frac s 2} v )_{\mathbb R^d}\) associated with the integral fractional Laplacian. The second purpose is to explore their applications in spectral approximations of PDEs. As a remarkable consequence of the fractional Sobolev-type orthogonality, the spectral-Galerkin method using A-GHFs as basis functions leads to an identity stiffness matrix for the integral fractional Laplacian operator \((-\Delta)^s,\) which is known to be notoriously difficult and expensive to discretise. ....