Banach Spaces of Solenoidal Solutions of the Vector Helmholtz Equation in the Space and Its Associated Reproducing Kernel

In this article we study Banach spaces of solenoidal solutions, that is, vector fields of zero divergence that satisfy the vector Helmholtz equation in R3. As it is known, these solutions are related to the time-harmonic Maxwell equations or Maxwell equations in a simple medium which makes this study important. We define an integral operator given by the Fourier transform of the tangential fields which are square integrable on the unit sphere S2 that will generate pairs of vector fields satisfying the time-harmonic Maxwell equations, which we will call electromagnetic Herglotz pairs. By last, in the context of Hilbert spaces, the space of solutions is studied, seen as a reproducing kernel Hilbert space and its reproducing kernel is calculated. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.

Banach Spaces of Solenoidal Solutions of the Vector Helmholtz Equation in the Space and Its Associated Reproducing Kernel Academic Article in Scopus