Publisher DOI: 10.1007/s10444-010-9158-z
Title: Minimizing and maximizing the Euclidean norm of the product of two polynomials
Language: English
Authors: Bünger, Florian 
Keywords: Eigenvalues and eigenvectors of autocorrelation Toeplitz matrices; Inequalities of polynomial products
Issue Date: 9-Jun-2010
Publisher: Baltzer Science Publ.
Source: Advances in Computational Mathematics 2 (35): 193-215 (2011-11-01)
Abstract (english): 
We consider the problem of minimizing or maximizing the quotient, where p = p0 + p1x + ... + pmxm, q = q0 + q1x + ... + qnxn ∈ K[x], K ∈ R, C, are non-zero real or complex polynomials of maximum degree m, n ∈ ℕ respectively and double pipepdouble pipe := (pipep0pipe2 + ... + pipepmpipe2)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 K = ¢ and maximizers pmax, qmax for arbitrary K fulfill deg(pmin) = m = deg(pmax), deg(qmin) = n = deg(qmax) and all roots of pmin, qmin, pmax, qmax have modulus one and are simple. For K = 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. © 2010 Springer Science+Business Media, LLC.
ISSN: 1019-7168
Journal: Advances in computational mathematics 
Institute: Zuverlässiges Rechnen E-19 
Document Type: Article
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