Advances in Computational Mathematics

, Volume 35, Issue 2–4, pp 193–215 | Cite as

Minimizing and maximizing the Euclidean norm of the product of two polynomials

Article

Abstract

We consider the problem of minimizing or maximizing the quotient
$$f_{m,n}(p,q):=\frac{\|{pq}\|}{\|{p}\|\|{q}\|} \ ,$$
where \(p=p_0+p_1x+\dots+p_mx^m\), \(q=q_0+q_1x+\dots+q_nx^n\in{\mathbb K}[x]\), \({\mathbb K}\in\{{\mathbb R},{\mathbb C}\}\), are non-zero real or complex polynomials of maximum degree \(m,n\in{\mathbb N}\) respectively and \(\|{p}\|:=(|p_0|^2+\dots+|p_m|^2)^{\frac{1}{2}}\) is simply the Euclidean norm of the polynomial coefficients. Clearly fm,n is bounded and assumes its maximum and minimum values min fm,n = fm,n(pmin, qmin) and max fm,n = f(pmax, qmax). We prove that minimizers pmin, qmin for \({\mathbb K}={\mathbb C}\) and maximizers pmax, qmax for arbitrary \({\mathbb K}\) fulfill \(\deg(p_{\min})=m=\deg(p_{\max})\), \(\deg(q_{\min})=n=\deg(q_{\max})\) and all roots of pmin, qmin, pmax, qmax have modulus one and are simple. For \({\mathbb K}={\mathbb R}\) we can only prove the existence of minimizers pmin, qmin of full degree m and n respectively having roots of modulus one. These results are obtained by transferring the optimization problem to that of determining extremal eigenvalues and corresponding eigenvectors of autocorrelation Toeplitz matrices. By the way we give lower bounds for min fm,n for real polynomials which are slightly better than the known ones and inclusions for max fm,n.

Keywords

Inequalities of polynomial products Eigenvalues and eigenvectors of autocorrelation Toeplitz matrices 

Mathematics Subject Classifications (2010)

15A42 26D05 

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Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  1. 1.Institute for Reliable ComputingHamburg University of TechnologyHamburgGermany

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