In order to study the effects of aging on plutonium, a new nonparametric super-resolution method of Fourier analysis is used to transform X-ray absorption fine-structure data of aged and newly-processed plutonium. This method of Fourier analysis, called LIFT, creates a sparse expansion of an underlying sum of complex exponentials by minimizing the norm of the expansion coefficients. Each complex exponential in the sum corresponds to the position of an atom in the crystal. LIFT increases inter-atomic spatial resolution by a factor of 64 over that delivered by a windowed Fourier transform. The anomalous site in newly-processed plutonium observed by Conradson using a windowed Fourier transform is also observed with LIFT, but with greatly improved resolution, and at a larger site separation. The newly processed plutonium is a slightly modified form of plutonium. The anomalous site is not present in aged plutonium. This site appears to disappears as plutonium ages. However, to establish changes in plutonium crystal structure with age, repeated EXFS measurements must be made on the same sample as it ages.