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Minimizing and maximizing the Euclidean norm of the product of two polynomials
Publikationstyp
Journal Article
Date Issued
2010-06-09
Sprache
English
Author(s)
Institut
TORE-URI
Volume
35
Issue
2
Start Page
193
End Page
215
Citation
Advances in Computational Mathematics 2 (35): 193-215 (2011-11-01)
Publisher DOI
Scopus ID
Publisher
Baltzer Science Publ.
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.
Subjects
Eigenvalues and eigenvectors of autocorrelation Toeplitz matrices
Inequalities of polynomial products
DDC Class
004: Informatik
510: Mathematik