Sobolev-Orthogonal Systems with Tridiagonal Skew-Hermitian Differentiation Matrices

TL;DR

This paper develops a theory of Sobolev orthogonal systems with tridiagonal skew-Hermitian differentiation matrices, linking them to Fourier transforms of orthogonal polynomials and providing efficient computational methods.

Contribution

It introduces a complete characterization of Sobolev-orthogonal systems with skew-Hermitian matrices and demonstrates computational efficiency for coefficient calculation.

Findings

Complete characterization of Sobolev-orthogonal systems as Fourier transforms of orthogonal polynomials.

Existence of a Sobolev-orthogonal system with coefficients computable in O(N log N) operations.

New concepts and analysis for Sobolev orthogonality with skew-Hermitian differentiation matrices.

Abstract

We introduce and develop a theory of orthogonality with respect to Sobolev inner products on the real line for sequences of functions with a tridiagonal, skew-Hermitian differentiation matrix. While a theory of such L2-orthogonal systems is well established, Sobolev orthogonality requires new concepts and their analysis. We characterise such systems completely as appropriately weighed Fourier transforms of orthogonal polynomials and present a number of illustrative examples, inclusive of a Sobolev-orthogonal system whose leading N coefficients can be computed in $\mathcal{O}(N \log N)$ operations.