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

Abstract

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.