Phase reconstruction from magnitude of band-limited multidimensional signals

In this paper, the problem of phase reconstruction from magnitude of multidimensional band-limited functions is considered. It is shown that any irreducible band-limited function f(z₁...,z_n), z_i ε{lunate} C, i=1, ..., n, is uniquely determined from the magnitude of f(x₁...,x_n): | f(x₁...,x_n)|, x_i ε{lunate} R, i=1,..., n, except for (1) linear shifts: ^{i(α1z1+...+α_n2n+β}), β, α_iε{lunate}R, i=1,..., n; and (2) conjugation: f^*(z₁^*,...,z_n ^*).