Hadamard-Finite-Part simultaneous infrared and ultraviolet self-regularization of universal functions in electrodynamics

Considering standard finite-support basis- and testing functions it is rigorously shown that spectral domain Green´s functions in electrodynamics can be simultaneously regularized in infrared (small wavenumber) and ultraviolet (large wavenumber) regions. Thereby, the key insight is that the Hadamard Finite Part regularization emerges “naturally” and the infinities cancel out automatically, thus justifying the notion of self-regularization. The latter property enables a myriad of constructive procedures for renormalizing divergences in computational electrodynamics to any order desired. Three theorems facilitate the discussion.