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

Dante C. Youla, Unnikrishna Pillai

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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(9) and the sign of its real part u(0) for z = ej0. 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,…, zn) in n variables that are holomorphic in the open polydisc Un = {z: zl <1, z2 <1,…, zn <1}, continuous in the closure Un = [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 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 = (θl, θ2,…,θn), write, in terms of real and imaginary parts, H(exp(jθ1), exp(jθ2),…, exp(jθn)) = H(ej θ) = u(θ) + jv(θ), and let A(θ) = H(ej θ); 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.

Original languageEnglish (US)
Pages (from-to)859-862
Number of pages4
JournalJournal of the Optical Society of America A: Optics and Image Science, and Vision
Volume6
Issue number6
DOIs
StatePublished - 1989

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uniqueness
Algebra
algebra
polynomials
Polynomials
trucks
rational functions
Rational functions
closures
constrictions
theorems
recovery
Recovery

ASJC Scopus subject areas

  • Atomic and Molecular Physics, and Optics
  • Computer Vision and Pattern Recognition
  • Electronic, Optical and Magnetic Materials

Cite this

@article{80d8da971e444d91b876ed7ff8a7e1e6,
title = "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(9) and the sign of its real part u(0) for z = ej0. 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,…, zn) in n variables that are holomorphic in the open polydisc Un = {z: zl <1, z2 <1,…, zn <1}, continuous in the closure Un = [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 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 = (θl, θ2,…,θn), write, in terms of real and imaginary parts, H(exp(jθ1), exp(jθ2),…, exp(jθn)) = H(ej θ) = u(θ) + jv(θ), and let A(θ) = H(ej θ); 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.",
author = "Youla, {Dante C.} and Unnikrishna Pillai",
year = "1989",
doi = "10.1364/JOSAA.6.000859",
language = "English (US)",
volume = "6",
pages = "859--862",
journal = "Journal of the Optical Society of America A: Optics and Image Science, and Vision",
issn = "0740-3232",
publisher = "The Optical Society",
number = "6",

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TY - JOUR

T1 - A uniqueness characterization in terms of signed magnitude for functions in the polydisc algebra A(Un)

AU - Youla, Dante C.

AU - Pillai, Unnikrishna

PY - 1989

Y1 - 1989

N2 - 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 = ej0. 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,…, zn) in n variables that are holomorphic in the open polydisc Un = {z: zl <1, z2 <1,…, zn <1}, continuous in the closure Un = [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 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 = (θl, θ2,…,θn), write, in terms of real and imaginary parts, H(exp(jθ1), exp(jθ2),…, exp(jθn)) = H(ej θ) = u(θ) + jv(θ), and let A(θ) = H(ej θ); 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.

AB - 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 = ej0. 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,…, zn) in n variables that are holomorphic in the open polydisc Un = {z: zl <1, z2 <1,…, zn <1}, continuous in the closure Un = [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 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 = (θl, θ2,…,θn), write, in terms of real and imaginary parts, H(exp(jθ1), exp(jθ2),…, exp(jθn)) = H(ej θ) = u(θ) + jv(θ), and let A(θ) = H(ej θ); 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.

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