Unique reconstruction of a band-limited multidimensional signal from its phase or magnitude

The mathematical problem of unique recovery of a band-limited multidimensional signal from its phase or its magnitude is considered. Specifically, we show that any irreducible band-limited function f(s₁, …, s_n), s_i∈ $\mathcal{C}$, i = 1, …, n is uniquely determined, except for trivial associates, from ( 1 ) the phase of f(x₁, …, x_n), x_i∈ ℛ, i = 1, …, n, if not all the zeros of f(s₁, …, s_n) occur in conjugate pairs; or ( 2 ) the magnitude of f(x₁, …, x_n), x_i∈ ℛ, i = …, n.