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


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.)