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We consider the inverse medium scattering of reconstructing the mediumcontrast using Born data, including the full aperture, limited-aperture, andmulti-frequency data. We propose data-driven basis functions for these inverseproblems based on the generalized prolate spheroidal wave functions and relatedeigenfunctions. Such data-driven eigenfunctions are eigenfunctions of a Fourierintegral operator; they remarkably extend analytically to the whole space, aredoubly orthogonal, and are complete in the class of band-limited functions. Wefirst establish a Picard criterion for reconstructing the contrast using thedata-driven basis, where the reconstruction formula can also be understood fromthe viewpoint of data processing and analytic extrapolation. Another salientfeature associated with the generalized prolate spheroidal wave functions isthat the data-driven basis for a disk is also a basis for a Sturm-Liouvilledifferential 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 dataand an explicit stability estimate for contrast in $H^s$ ($0<s<1/2$) in thefull aperture case. In the limited-aperture and multi-frequency cases, we alsoobtain spectral cutoff regularization strategies for noisy data and stabilityestimates for a class of contrast.

Data-driven basis for reconstructing the contrast in inverse scattering: Picard criterion, regularity, regularization, and stability