Polya-Schur master theorems for circular domains and their boundaries

• 成果类型: Article
• 署名作者: Borcea, Julius; Branden, Petter
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2009.170.465
• 发表日期: 2009
• 页码: 465-492
• 关键词: transformed polynomials; linear transformations; multiplier sequences; algebraic equations; zeros; Operators

摘要

We characterize all linear operators on finite or infinite-dimensional polynomial spaces that preserve the property of having the zero set inside a prescribed region Omega subset of C for arbitrary closed circular domains Omega (i.e., images of the closed unit disk under a Mobius transformation) and their boundaries. This provides a natural framework for dealing with several long-standing fundamental problems, which we solve in a unified way. In particular, for Omega = R our results settle open questions that go back to Laguerre and Polya-Schur.