Closed-form Dirichlet integral harmonic interpolation-fits for real n-dimensional and complex half-space: DIDACKS III

Abstract

This article gives a ``fundamental solution'' based energy-norm harmonic interpolation approach for two half-space settings of interest: the upper-half $\mathbb{R}^n$ plane, where fundamental solutions satisfy Laplace's equation, and the upper-half complex plane, where simple poles are of interest. This approach can handle higher-order pole fits, as well as logarithmic source fits, in the complex setting and it can handle higher-order multipole fits in the general real $\mathbb{R}^n$ setting. Higher-order multipoles in the real $\mathbb{R}^n$ half-space setting are of particular interest since fits based on a commonly used type of radial-basis function (inverse multiquadrics) can be reinterpreted as multipole based interpolations that minimize energy forms.