Asymptotic error in the eigenfunction expansion for the Green's function of a Sturm-Liouville problem

TL;DR

This paper analyzes the asymptotic error in approximating Green's functions of Sturm-Liouville problems via eigenfunction expansion, providing formulas and scaling insights for various classical orthogonal polynomials.

Contribution

It derives explicit asymptotic error expressions and scaling exponents for eigenfunction truncations of Green's functions, including new formulas for classical orthogonal polynomials.

Findings

Asymptotic error on the diagonal expressed in terms of differential equation coefficients

Scaling exponents depend on eigenvalue asymptotics for non-degenerate limits

Established Christoffel-Darboux type formula for classical orthogonal polynomials

Abstract

We study the asymptotic error arising when approximating the Green's function of a Sturm-Liouville problem through a truncation of its eigenfunction expansion, both for the Green's function of a regular Sturm-Liouville problem and for the Green's function associated with the Hermite polynomials, the associated Laguerre polynomials, and the Jacobi polynomials, respectively. We prove that the asymptotic error obtained on the diagonal can be expressed in terms of the coefficients of the related second-order Sturm-Liouville differential equation, and that the suitable scaling exponent which yields a non-degenerate limit on the diagonal depends on the asymptotic behaviour of the corresponding eigenvalues. We further consider the asymptotic error away from the diagonal and analyse which scaling exponents ensure that it remains at zero. For the Hermite polynomials, the associated Laguerre polynomials, and the Jacobi polynomials, a Christoffel-Darboux type formula, which we establish for all classical orthogonal polynomial systems, allows us to obtain a better control away from the diagonal than what a sole application of known asymptotic formulae gives. As a consequence of our study for regular Sturm-Liouville problems, we identify the fluctuations for the Karhunen-Lo\`eve expansion of Brownian motion.