### 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 = 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(z_{1}, z_{2},…, z_{n}) in n variables that are holomorphic in the open polydisc U^{n} = {z: z_{l} <1, z_{2} <1,…, z_{n} <1}, continuous in the closure U^{n} = [z: z_{1} <1, z_{2} <1,…, z_{n} <1), and free of zeros on its distinguished boundary T^{n} = {z: z_{1} = 1, z_{2} = 1,…, z_{n} = 1}. The collection A(U^{n}) 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(U^{n}) and be free of ze¬ros in P1. For real θ = (θ_{l}, θ_{2},…,θ_{n}), write, in terms of real and imaginary parts, H(exp(jθ_{1}), exp(jθ_{2}),…, exp(jθ_{n})) = H(e^{j} ^{θ}) = u(θ) + jv(θ), and let A(θ) = H(e^{j} ^{θ}); then H(z) is determined uniquely by the functions A(0) and sign u(0), together with its value at one point z_{0} exp(jθ_{0}) ϵ T^{n} for which u(θ_{0}) ≠ 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 language | English (US) |
---|---|

Pages (from-to) | 859-862 |

Number of pages | 4 |

Journal | Journal of the Optical Society of America A: Optics and Image Science, and Vision |

Volume | 6 |

Issue number | 6 |

DOIs | |

State | Published - 1989 |

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### ASJC Scopus subject areas

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

### Cite this

**A uniqueness characterization in terms of signed magnitude for functions in the polydisc algebra A(U ^{n}).** / Youla, Dante C.; Pillai, Unnikrishna.

Research output: Contribution to journal › Article

}

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.

UR - http://www.scopus.com/inward/record.url?scp=84975565189&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84975565189&partnerID=8YFLogxK

U2 - 10.1364/JOSAA.6.000859

DO - 10.1364/JOSAA.6.000859

M3 - Article

AN - SCOPUS:84975565189

VL - 6

SP - 859

EP - 862

JO - Journal of the Optical Society of America A: Optics and Image Science, and Vision

JF - Journal of the Optical Society of America A: Optics and Image Science, and Vision

SN - 0740-3232

IS - 6

ER -