Perturbations of Roots under Linear Transformations of PolynomialsConstructive Approximation
AbstractLet Pn be the complex vector space of all polynomials of degree at most n. We give several characterizations of the linear operators T:Pn→Pn for which there exists a constant C > 0 such that for all nonconstant f∈Pn there exist a root u of f and a root v of Tf with |u−v|≤C. We prove that such perturbations leave the degree unchanged and, for a suitable pairing of the roots of f and Tf, the roots are never displaced by more than a uniform constant independent on f. We show that such "good" operators T are exactly the invertible elements of the commutative algebra generated by the differentiation operator. We provide upper bounds in terms of T for the relevant constants.
Citation InformationBranko Ćurgus and Vania Mascioni. "Perturbations of Roots under Linear Transformations of Polynomials" Constructive Approximation Vol. 25 Iss. 3 (2007) p. 255 - 277
Available at: http://works.bepress.com/branko_curgus/18/