Equiconvergence for perturbed Jacobi polynomial expansions

TL;DR

This paper establishes asymptotic expansions for eigenfunctions of perturbed Jacobi operators and demonstrates equiconvergence with cosine basis expansions, leading to new pointwise convergence results.

Contribution

It introduces asymptotic eigenfunction expansions for perturbed Jacobi operators and proves their equiconvergence with cosine expansions, a novel connection in spectral theory.

Findings

Eigenfunctions have asymptotic expansions

Equiconvergence between Jacobi and cosine expansions

New pointwise convergence results

Abstract

We show asymptotic expansions of the eigenfunctions of certain perturbations of the Jacobi operator in a bounded interval, deducing equiconvergence results between expansions with respect to the associated orthonormal basis and expansions with respect to the cosine basis. Several results for pointwise convergence then follow.