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.

中文翻译：

---

复制内核以实现弹性Herglotz功能

我们研究了弹性Herglotz波函数，它是线性弹性理论中以\（L ^ 2 \） -远场模式出现的频谱Navier方程的完整解。我们将这些函数\（{\ varvec {\ mathcal {W}}}，\）的三个维度描述为希尔伯特空间\（{\ varvec {\ mathcal {H}}} \）的封闭子空间向量值函数的集合，使得它们及其球形渐变属于某个加权\（L ^ 2 \）空间。这使我们能够证明\（{\ varvec {\ mathcal {W}}} \）是一个再生内核Hilbert空间，并计算出该再生内核。最后，我们概述了二维情况的证明，并给出了相应的复制内核。