Reproducing Kernel for the Herglotz Functions in $$\mathbb {R}^n$$ and Solutions of the Helmholtz Equation |
| |
Authors: | Salvador?Pérez-Esteva Email author" target="_blank">Salvador?Valenzuela-DíazEmail author |
| |
Institution: | 1.Instituto de Matemáticas-Unidad Cuernavaca,Universidad Nacional Autónoma de México,Mexico,Mexico |
| |
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. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |