Analyticity and phase retrieval

Abstract

The general problem of constructing a spectrumg(v) from the knowledge of the magnitude of its Fourier transform ¦ γ(τr) ¦ is considered. The question reduces to locating the zeros of the analytic continuation γ(τ) in the upper half-plane. It is shown that ifg(v) is real, the complex zeros of γ(τ) in the u.h.p. either are on the imaginary axis or occur pairwise in a symmetrical position. If, in addition, g(v)≥0, the zeros on the imaginary axis disappear. The conditiong(v) ≥ 0 also leads to the requirement that γ(τ) must be representable as a convolution of a functionh(τ) with itself. The analytic properties ofh(τ) and the equations to determine it are discussed. Possible ways to obtain the solution of the ensuing nonlinear eigenvalue problem are suggested.