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

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(9) and the sign of its real part u(0) for z = e^j0. 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 polynomi¬als (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,…, z_n) in n variables that are holomorphic in the open polydisc Uⁿ = {z: z_l < 1, z₂ < 1,…, z_n < 1}, continuous in the closure Uⁿ = [z: z₁ < 1, z₂ < 1,…, z_n < 1), and free of zeros on its distinguished boundary Tⁿ = {z: z₁ = 1, z₂ = 1,…, z_n = 1}. The collection A(Uⁿ) of functions holomorphic in Un and continuous in Un is Rudin’s polydisc algebra [W. Rudin, Function Theory in Polydiscs (Benjamin, New York, 1969)]. Our main result can be stated as follows. Let if (z) belong to A(Uⁿ) and be free of ze¬ros in P1. For real θ = (θ_l, θ₂,…,θ_n), write, in terms of real and imaginary parts, H(exp(jθ₁), exp(jθ₂),…, exp(jθ_n)) = H(e^jθ) = u(θ) + jv(θ), and let A(θ) = H(e^jθ); then H(z) is determined uniquely by the functions A(0) and sign u(0), together with its value at one point z₀ exp(jθ₀) ϵ 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.