Convergence theory for the two-dimensional modified-Bessel lightning ansatz

Establish a rigorous convergence theory for the full two-dimensional modified-Bessel ansatz used to approximate modified Helmholtz solutions, including the effects of boundary sampling and numerical rank truncation, and thereby justify its observed root-exponential convergence.

Background

The paper develops a lightning approximation for the homogeneous two-dimensional modified Helmholtz equation using modified Bessel functions of the second kind centered at tapered, exponentially clustered exterior poles to represent corner singularities, together with modified Bessel functions of the first kind centered inside the domain to represent the regular component. Numerical experiments show approximately root-exponential decay of the boundary residual, and a one-dimensional Bessel prototype exhibits slopes consistent with established lightning rational-approximation theory.

However, the authors emphasize that existing rational-approximation results do not constitute a convergence proof for the full two-dimensional Bessel-based representation. A complete theory would need to address not only approximation by the ansatz itself but also practical issues arising from discrete boundary sampling and numerical rank truncation in the least-squares computation.