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Localisation des zéros de polynômes intervenant en théorie du signal. (French) Zbl 0703.30004
Fifty years of polynomials, Proc. Conf. in Honour of Alain Durand, Paris/Fr. 1988, Lect. Notes Math. 1415, 167-179 (1990).
[For the entire collection see Zbl 0683.00011.]
The authors consider two families of polynomials, namely $f_ n(z):=z^{n+1}-(n+1)z+n$ and $A_{m+2}(z):=z^{m+2}- 2\frac{m+2}{m}z^{m+1}+\frac{(m+2)(m+1)}{m(m-1)}z^ m-2\frac{m+2}{m(m- 1)}z+\frac{2}{m}$ which are related to spectral analysis of signal processing. J.-L. Lacoume, C. Hanna and J.-L. Nicholas [Ann. Télécommun. 36, 579-584 (1981; Zbl 0474.94005)] noticed that the roots $$\zeta_ k:=\rho_ k \exp (i\theta_ k)$$ of $$f_ n$$ can be denoted such that $$\zeta_ n=1$$ and each argument $$\theta_ k$$ verifying $| \theta_ k-(2k\pi /n)| \leq \pi /(n+1),\quad 1\leq k\leq n.$ The authors show that $$\rho_ k<\rho_{k+1}$$ for $$1\leq k<n/2$$ (Theorem 1). Moreover, there exists a constant A (say $$A=10^{40})$$ such that $| \zeta_ j-r_ k \exp (i\pi (4k+1)/(2n+1))| \leq 3/n$ for $$A<k<n+1-A$$ (Theorem 2). Here the values $$r_ k$$ are explicitly given. The asymptotic expansion $\theta_ 1=b/n-b/2n^ 2+a_ 3/n^ 3+a_ 4/n^ 4+a_ 5/n^ 5+{\mathcal O}(1/n^ 6)$ is carefully computed. As a consequence, $$\rho_ k\geq 1+(2/n)$$ for n large enough. Analogous results are given for the zeros $$\alpha\neq 1$$ of $$A_{m+2}$$. In particular, it is shown that $$| \alpha | <1-(2/5m)$$ for all m large enough. Proofs are elementary.
Reviewer: P.Liardet

##### MSC:
 30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) 26C10 Real polynomials: location of zeros 94A12 Signal theory (characterization, reconstruction, filtering, etc.)