Properties on the unit circle of polynomials with unimodular coefficients
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- by Donald J. Newman and André Giroux PDF
- Proc. Amer. Math. Soc. 109 (1990), 113-116 Request permission
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 $|{a_i}| = 1$ for all $i = 0,1, \ldots ,N$ and for which $P({\alpha _j}) = 0$ for a given set of ${\alpha _j},j = 1,2, \ldots ,n,|{\alpha _j}| = 1$, and $P(z) \ne 0$ elsewhere on $|z| = 1$. It is further shown how to extend this construction so as to maintain these properties and force the maximum of $|P(z)|$ to occur at any given number $\beta \ne {\alpha _j},j = 1,2, \ldots ,n$ and $|\beta | = 1$. 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
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J. S. Byrnes and D. J. Newman, Null steering employing polynomials with restricted coefficients, IEEE Transactions on Antennas and Propagation 36 (1988), 301-303.
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 109 (1990), 113-116
- MSC: Primary 30C15; Secondary 30C10, 42A16, 42A28
- DOI: https://doi.org/10.1090/S0002-9939-1990-1000163-7
- MathSciNet review: 1000163