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Slepian functions are orthogonal function systems that live on subdomains(for example, geographical regions on the Earth's surface, or bandlimitedportions of the entire spectrum). They have been firmly established as a usefultool for the synthesis and analysis of localized (concentrated or confined)signals, and for the modeling and inversion of noise-contaminated data that areonly regionally available or only of regional interest. In this paper, weconsider a general abstract setup for inverse problems represented by a linearand compact operator between Hilbert spaces with a known singular-valuedecomposition (svd). In practice, such an svd is often only given for the caseof a global expansion of the data (e.g. on the whole sphere) but not forregional data distributions. We show that, in either case, Slepian functions(associated to an arbitrarily prescribed region and the given compact operator)can be determined and applied to construct a regularization for the ill-posedregional inverse problem. Moreover, we describe an algorithm for constructingthe Slepian basis via an algebraic eigenvalue problem. The obtained Slepianfunctions can be used to derive an svd for the combination of the regionalizingprojection and the compact operator. As a result, standard regularizationtechniques relying on a known svd become applicable also to those inverseproblems where the data are regionally given only. In particular, wavelet-basedmultiscale techniques can be used. An example for the latter case is elaboratedtheoretically and tested on two synthetic numerical examples.

A General Approach to Regularizing Inverse Problems with Regional Data using Slepian Wavelets