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

作者：Buenger F^*

来源：Advances in Computational Mathematics, 2011, 35(2-4): 193-215.

DOI：10.1007/s10444-010-9158-z

• Inequalities of polynomial products
• Eigenvalues and eigenvectors of autocorrelation Toeplitz matrices

We consider the problem of minimizing or maximizing the quotient
fm,n(p,q) := parallel to pq parallel to/parallel to p parallel to parallel to q parallel to,
where p = p(0) + p(1)x +...+ p(m)x(m), q = q(0) + q(1)x + ... + q(n)x(n) is an element of K[x],
K is an element of {R, C}, are non-zero real or complex polynomials of maximum degree m, n is an element of respectively and parallel to p parallel to := (vertical bar p(0)vertical bar(2) + ... + vertical bar p(m)vertical bar(2))(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 K = C and maximizers p(max), q(max) for arbitrary 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 K = 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 fm, n for real polynomials which are slightly better than the known ones and inclusions for max f(m,n).

• 出版日期2011-11