Document Type

Article

Publication Date

4-2007

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

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.

Included in

Mathematics Commons

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