  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.
