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

Understand

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.