We consider the Sobolev embedding operator $E_s : H^s(\Omega) \to L_2(\Omega)$ and its role in the solution of inverse problems. In particular, we collect various properties and investigate different characterizations of its adjoint operator $E_s^*$, which is a common component in both iterative and variational regularization methods. These include variational representations and connections to boundary value problems, Fourier and wavelet representations, as well as connections to spatial filters. Moreover, we consider characterizations in terms of Fourier series, singular value decompositions and frame decompositions, as well as representations in finite dimensional settings. While many of these results are already known to researchers from different fields, a detailed and general overview or reference work containing rigorous mathematical proofs is still missing. Hence, in this paper we aim to fill this gap by collecting, introducing and generalizing a large number of characterizations of $E_s^*$ and discuss their use in regularization methods for solving inverse problems. The resulting compilation can serve both as a reference as well as a useful guide for its efficient numerical implementation in practice.

翻译：我们考虑Sobolev嵌入算子$E_s : H^s(\Omega) \to L_2(\Omega)$及其在反问题求解中的作用。特别地，我们收集了其伴随算子$E_s^*$的多种性质，并研究其不同刻画方法——该算子既是迭代正则化方法也是变分正则化方法中的常见组成部分。这些刻画包括变分表示及其与边值问题的关联、傅里叶与小波表示，以及与空间滤波器的联系。此外，我们还考虑了傅里叶级数、奇异值分解、框架分解以及有限维空间中的表示。尽管这些结果中有许多已为不同领域的研究者所知，但尚缺乏一个包含严格数学证明的详细通用综述或参考著作。因此，本文旨在通过收集、引入并推广$E_s^*$的大量刻画，探讨其在求解反问题的正则化方法中的应用，从而填补这一空白。本文的成果既可作为参考手册，也可作为实际高效数值实现的实用指南。