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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Properties on the unit circle of polynomials with unimodular coefficients

Authors: Donald J. Newman and André Giroux
Journal: Proc. Amer. Math. Soc. 109 (1990), 113-116
MSC: Primary 30C15; Secondary 30C10, 42A16, 42A28
MathSciNet review: 1000163
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Abstract: A concrete explicit construction of a unimodular polynomial with prescribed zeros on the unit circle is given. More precisely a polynomial $ P(z) = {a_0} + {a_1}z + \cdots {a_N}{z^N}$ is produced for which $ \vert{a_i}\vert = 1$ for all $ i = 0,1, \ldots ,N$ and for which $ P({\alpha _j}) = 0$ for a given set of $ {\alpha _j},j = 1,2, \ldots ,n,\vert{\alpha _j}\vert = 1$, and $ P(z) \ne 0$ elsewhere on $ \vert z\vert = 1$. It is further shown how to extend this construction so as to maintain these properties and force the maximum of $ \vert P(z)\vert$ to occur at any given number $ \beta \ne {\alpha _j},j = 1,2, \ldots ,n$ and $ \vert\beta \vert = 1$. The dependence of $ N$ on $ n$ is exponential, but there is rėason to believe that this is actually necessary and not just a weakness of the method.

References [Enhancements On Off] (What's this?)

  • [1] J. S. Byrnes and D. J. Newman, Null steering employing polynomials with restricted coefficients, IEEE Transactions on Antennas and Propagation 36 (1988), 301-303.

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Additional Information

PII: S 0002-9939(1990)1000163-7
Article copyright: © Copyright 1990 American Mathematical Society