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Reproducing kernel for elastic Herglotz functions

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Abstract

We study the elastic Herglotz wave functions, which are entire solutions of the spectral Navier equation appearing in linearized elasticity theory with \(L^2\)-far-field patterns. We characterize in three-dimensions the set of these functions \({\varvec{\mathcal {W}}},\) as a closed subspace of a Hilbert space \({\varvec{\mathcal {H}}}\) of vector-valued functions such that they and their spherical gradients belong to a certain weighted \(L^2\) space. This allows us to prove that \({\varvec{\mathcal {W}}}\) is a reproducing kernel Hilbert space and to calculate the reproducing kernel. Finally, we outline the proof for the two-dimensional case and give the corresponding reproducing kernel.

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Notes

  1. We have found a misprint on identity (6) of [8]. There, the factor |x| has to be removed.

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Authors and Affiliations

  1. Departamento de Análisis Matemático y Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense de Madrid, Madrid, Spain

    T. Luque

  2. Departamento de Matemática e Informática aplicadas a las Ingenierías Civil y Naval, Universidad Politécnica de Madrid, Madrid, Spain

    M. C. Vilela

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  1. T. Luque
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  2. M. C. Vilela
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The first author was supported by Spanish Grant MTM2017-82160-C2-1-P, and the second by Spanish Grant MTM2014-57769-C3-1-P and MTM2017-85934-C3-3-P.

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Luque, T., Vilela, M.C. Reproducing kernel for elastic Herglotz functions. Rev Mat Complut 33, 495–525 (2020). https://doi.org/10.1007/s13163-019-00327-w

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