Linear operators and the distribution of zeros of entire funtions

Abstract

Let π _n denote the vector space (over or ) of all polynomials of degree ≤ n. For (where S is an appropriate set of interest), let π _n (S) denote the class of all polynomials of degree ≤ n, all of whose zeros lie in S. Old and new problems related to the following open problem will be presented. Characterize all linear transformations (operators) T:π_n(S)→ π_n(S), where deg T[p] ≤ deg p for p ∈ π _n (S).

Keywords:

• Complex zero decreasing sequences
• Linear operators
• Hurwitz polynomials

Keywords:

• AMS 2000 Mathematics Subject Classifications: 30D15
• 30D35