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Saito's method has been applied successfully for measuring potentials with compact support in three dimensions. Also potentials have been reconstructed in the sense of distributions using a weak version of the method. Saito's method does not depend on the decay of the boundary value of the resolvent operator, but instead on certain Reimann-Lebesgue type properties of convolutions of the kernel of the unperturbed resolvent. In this paper these properties are extended from three to higher dimensions. We also provide an important application to inverse scattering by extending reconstruction results to measure potentials with unbounded support.