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 = e^jθ. 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, …, z_n) in n variables that are holomorphic in the open polydisc Uⁿ = {z:|z₁| < 1, |z₂| < 1, …, |z_n| < 1}, continuous in the closure Ūⁿ = {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 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, …, 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(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.

© 1989 Optical Society of America

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