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

Reproducing Kernel for the Herglotz Functions in $$\mathbb {R}^n$$ R n and Solutions of... The purpose of this article is to extend to $$\mathbb {R}^{n}$$ 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 $$\mathbb {R} ^{n}$$ R n that are the Fourier transform of measures supported in the unit sphere with density in $$L^{2}(\mathbb {S}^{n-1})$$ L 2 ( S n - 1 ) . As a natural extension of this, we define Banach spaces of solutions of the Helmholtz equation in $$\mathbb {R}^{n}$$ R n belonging to weighted Sobolev type spaces $$\mathcal {H}^{p}$$ 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 $$\mathcal {H}^{p}$$ H p and in mixed norm spaces, the continuity of the orthogonal projection $$\mathcal {P}$$ P of $$\mathcal {H}^{2}$$ H 2 onto the Herglotz wave functions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Fourier Analysis and Applications Springer Journals

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

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Publisher
Springer US
Copyright
Copyright © 2016 by Springer Science+Business Media New York
Subject
Mathematics; Fourier Analysis; Signal,Image and Speech Processing; Abstract Harmonic Analysis; Approximations and Expansions; Partial Differential Equations; Mathematical Methods in Physics
ISSN
1069-5869
eISSN
1531-5851
D.O.I.
10.1007/s00041-016-9492-8
Publisher site
See Article on Publisher Site

Abstract

The purpose of this article is to extend to $$\mathbb {R}^{n}$$ 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 $$\mathbb {R} ^{n}$$ R n that are the Fourier transform of measures supported in the unit sphere with density in $$L^{2}(\mathbb {S}^{n-1})$$ L 2 ( S n - 1 ) . As a natural extension of this, we define Banach spaces of solutions of the Helmholtz equation in $$\mathbb {R}^{n}$$ R n belonging to weighted Sobolev type spaces $$\mathcal {H}^{p}$$ 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 $$\mathcal {H}^{p}$$ H p and in mixed norm spaces, the continuity of the orthogonal projection $$\mathcal {P}$$ P of $$\mathcal {H}^{2}$$ H 2 onto the Herglotz wave functions.

Journal

Journal of Fourier Analysis and ApplicationsSpringer Journals

Published: Jul 27, 2016

References

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