Reproducing Kernel for the Herglotz Functions in \(\mathbb {R}^n\) and Solutions of the Helmholtz Equation

The purpose of this article is to extend to (mathbb {R}^{n}) known results in dimension 2 concerning the structure of a Hilbert space with reproducing kernel of the space of Herglotz wave functions. These functions are the solutions of Helmholtz equation in (mathbb {R} ^{n}) that are the Fourier transform of measures supported in the unit sphere with density in (L^{2}(mathbb {S}^{n-1})). As a natural extension of this, we define Banach spaces of solutions of the Helmholtz equation in (mathbb {R}^{n}) belonging to weighted Sobolev type spaces (mathcal {H}^{p}) having in a non local norm that involves radial derivatives and spherical gradients. We calculate the reproducing kernel of the Herglotz wave functions and study in (mathcal {H}^{p}) and in mixed norm spaces, the continuity of the orthogonal projection (mathcal {P}) of (mathcal {H}^{2}) onto the Herglotz wave functions.