On semi-separability and differentiation matrices

TL;DR

This paper investigates the structure of differentiation matrices in spectral methods, showing that Jacobi polynomial bases produce semi-separable matrices of rank 2, extending known results for Laguerre and ultraspherical bases.

Contribution

It proves that Jacobi polynomial-based differentiation matrices are semi-separable of rank 2, providing new insights and results on semi-separable matrices in spectral methods.

Findings

Jacobi polynomial bases lead to semi-separable differentiation matrices of rank 2

New results on semi-separable matrices are established

Extends understanding of matrix structures in spectral methods

Abstract

The theory of spectral methods for partial differential equations leads to infinite-dimensional matrices which represent the derivative operator with respect to an underlying orthonormal basis. Favourable properties of such differentiation matrices are crucial in the design of good spectral methods. It is known that bases using Laguerre and ultraspherical polynomials lead to semi-separable differentiation matrices of rank 1. In this paper we consider orthonormal bases constructed from Jacobi polynomials and prove that the underlying differentiation matrices are semi-separable of rank 2. This requires new results on semi-separable matrices which might be interesting in a wider context.