  We consider the inverse medium scattering of reconstructing the medium
contrast using Born data, including the full aperture, limited-aperture, and
multi-frequency data. We propose data-driven basis functions for these inverse
problems based on the generalized prolate spheroidal wave functions and related
eigenfunctions. Such data-driven eigenfunctions are eigenfunctions of a Fourier
integral operator; they remarkably extend analytically to the whole space, are
doubly orthogonal, and are complete in the class of band-limited functions. We
first establish a Picard criterion for reconstructing the contrast using the
data-driven basis, where the reconstruction formula can also be understood from
the viewpoint of data processing and analytic extrapolation. Another salient
feature associated with the generalized prolate spheroidal wave functions is
that the data-driven basis for a disk is also a basis for a Sturm-Liouville
differential operator. With the help of Sturm-Liouville theory, we estimate the
$L^2$ approximation error for a spectral cutoff approximation of $H^s$
functions. This yields a spectral cutoff regularization strategy for noisy data
and an explicit stability estimate for contrast in $H^s$ ($0<s<1/2$) in the
full aperture case. In the limited-aperture and multi-frequency cases, we also
obtain spectral cutoff regularization strategies for noisy data and stability
estimates for a class of contrast.
