In this article we study Banach spaces of solenoidal solutions, that is, vector fields of zero divergence that satisfy the vector Helmholtz equation in \(\mathbb {R}^3\) . 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 \(\mathbb {S}^2\) 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.
