Stabilized inversion for limited angle tomography

Many problems in applied mathematics involve recovering a function f from measurements of Lf, where L is a known operator. We study the recovery of a function from limited knowledge of its Fourier transform. The inversion of a compact operator L (for limited data) is considered from a discrete signal processing perspective. In this context, the continuous operator L is naturally viewed as the limiting case of a series of discrete operators. Since L will generally be compact and self-adjoint, its spectrum can be analyzed via standard techniques. Real-world problems are generated from discretely sampled data sets. Therefore, we believe that it is more informative to study the spectra of the discrete approximations to L rather than the spectrum of L itself. Our main tool for analyzing the spectra of these discrete approximations to L is the theory of finite Toeplitz forms, originally introduced by Szego (1915). We show that the study of these finite Toeplitz forms give us some clues concerning the construction of an accurate, stable inversion for L, even when the continuous spectra of L suggests that it is not invertible