Document Type

Article

Publication Date

4-2007

Keywords

roots of polynomials

Abstract

Let 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.

Publication Title

Constructive Approximation

Volume

25

Issue

3

First Page

255

Last Page

277

DOI

http://dx.doi.org/10.1007/s00365-006-0649-0

Required Publisher's Statement

© Springer International Publishing AG, Part of Springer Science+Business Media

The final publication is available at Springer via http://dx.doi.org/10.1007/s00365-006-0649-0.

Subjects - Topical (LCSH)

Polynomials; Linear operators

Genre/Form

articles

Type

Text

Rights

Copying of this document in whole or in part is allowable only for scholarly purposes. It is understood, however, that any copying or publication of this document for commercial purposes, or for financial gain, shall not be allowed without the author’s written permission.

Language

English

Format

application/pdf

Download

Included in

Mathematics Commons

COinS