The method of sparse spectrum estimation developed by Chen and Donoho for real-valued one-dimensional signals has been extended to complex-valued signals, and is used here in two widely different applications: to denoise and superresolve ISAR images, and to transform extended X-ray absorption fine-structure (EXAFS) data of the elements to aid in the determination of their detailed crystal structure. This extension of the Chen-Donoho algorithm, which we call the l1-FFT, incorporates the a priori information that the spectrum is sparse by minimizing the l1 norm of the coefficients of the expansion functions. The l1-FFT is applied to stepped-frequency ISAR imaging where it increases resolution by factors of 4 and 64 over that of the windowed Fourier transform, for the real and simulated data presented here. In the second application, to determine the effects of aging on the crystal structure of plutonium, the l1-FFT is used to transform EXAFS plutonium data. The l1-FFT increases inter-atomic spatial resolution by a factor of 64 over that delivered by a windowed Fourier transform.