  Slepian functions are orthogonal function systems that live on subdomains
(for example, geographical regions on the Earth's surface, or bandlimited
portions of the entire spectrum). They have been firmly established as a useful
tool for the synthesis and analysis of localized (concentrated or confined)
signals, and for the modeling and inversion of noise-contaminated data that are
only regionally available or only of regional interest. In this paper, we
consider a general abstract setup for inverse problems represented by a linear
and compact operator between Hilbert spaces with a known singular-value
decomposition (svd). In practice, such an svd is often only given for the case
of a global expansion of the data (e.g. on the whole sphere) but not for
regional 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-posed
regional inverse problem. Moreover, we describe an algorithm for constructing
the Slepian basis via an algebraic eigenvalue problem. The obtained Slepian
functions can be used to derive an svd for the combination of the regionalizing
projection and the compact operator. As a result, standard regularization
techniques relying on a known svd become applicable also to those inverse
problems where the data are regionally given only. In particular, wavelet-based
multiscale techniques can be used. An example for the latter case is elaborated
theoretically and tested on two synthetic numerical examples.
