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We consider the Sobolev embedding operator $E_s : H^s(\Omega) \toL_2(\Omega)$ and its role in the solution of inverse problems. In particular,we collect various properties and investigate different characterizations ofits adjoint operator $E_s^*$, which is a common component in both iterative andvariational regularization methods. These include variational representationsand connections to boundary value problems, Fourier and waveletrepresentations, as well as connections to spatial filters. Moreover, weconsider characterizations in terms of Fourier series, singular valuedecompositions and frame decompositions, as well as representations in finitedimensional settings. While many of these results are already known toresearchers from different fields, a detailed and general overview or referencework containing rigorous mathematical proofs is still missing. Hence, in thispaper we aim to fill this gap by collecting, introducing and generalizing alarge number of characterizations of $E_s^*$ and discuss their use inregularization methods for solving inverse problems. The resulting compilationcan serve both as a reference as well as a useful guide for its efficientnumerical implementation in practice.

Characterizations of Adjoint Sobolev Embedding Operators with Applications in Inverse Problems