Characterizations of Adjoint Sobolev Embedding Operators for Inverse Problems
share
We consider the Sobolev embedding operator E_s : H^s(Ω) → L_2(Ω) 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^*. The resulting compilation can serve both as a reference as well as a useful guide for its efficient numerical implementation in practice.
READ FULL TEXT