A uniqueness characterization in terms of signed magnitude for functions in the polydisc algebra A(Un)

Abstract

In an excellent paper written in 1983 Van Hove et al. [ IEEE Trans. Acoust. Speech Signal Process.ASSP-31, 1286 ( 1983)] proved that a real polynomial P(z), free of zeros in |z| = 1, is determined uniquely by its magnitude A(θ) and the sign of its real part u(θ) for z = ejθ. Van Hove et al. also indicated a possible extension to real polynomials in more than one variable and suggested an iterative procedure for the recovery of P(z). The restriction to polynomials (or rational functions), real or otherwise, is not crucial. In fact, on due reflection, it is seen that the problem continues to make sense for functions H(z) = H(z1, z2, …, zn) in n variables that are holomorphic in the open polydisc Un = {z:|z1| < 1, |z2| < 1, …, |zn| < 1}, continuous in the closure Ūn = {z:|z1| ≤ 1, |z2| ≤ 1, …, |zn| ≤ 1}, and free of zeros on its distinguished boundary Tn = {z:|z1| = 1, |z2| = 1, …, |zn| = 1}. The collection A(Un) of functions holomorphic in Un and continuous in Ūn is Rudin’s polydisc algebra [ W. Rudin , Function Theory in Polydiscs ( Benjamin, NewYork, 1969)]. Our main result can be stated as follows. Let H(z) belong to A(Un) and be free of zeros in Tn. For real θ = (θ1, θ2, …, θn), write, in terms of real and imaginary parts, H(exp(jθ1), exp(jθ2), …, exp(jθn)) = H(ejθ) = u(θ) + jv(θ), and let A(θ) = |H(ejθ)|; then H(z) is determined uniquely by the functions A(θ) and sign u(θ), together with its value at one point z0 exp(jθ0) ∈ Tn for which u(θ0) ≠ 0. We also clarify the meaning and the scope of this theorem by the use of specific examples, but reconstruction techniques are not discussed.