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

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 f _m,n is bounded and assumes its maximum and minimum values min f _m,n = f _m,n(p _min, q _min) and max f _m,n = f(p _max, q _max). We prove that minimizers p _min, q _min for ${\mathbb K}={\mathbb C}$ and maximizers p _max, q _max for arbitrary ${\mathbb K}$ fulfill $\deg(p_{\min})=m=\deg(p_{\max})$, $\deg(q_{\min})=n=\deg(q_{\max})$ and all roots of p _min, q _min, p _max, q _max have modulus one and are simple. For ${\mathbb K}={\mathbb R}$ we can only prove the existence of minimizers p _min, q _min 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 f _m,n for real polynomials which are slightly better than the known ones and inclusions for max f _m,n.