Classifications of linear operators preserving elliptic, positive and non-negative polynomials

Details Classifications of linear operators preserving elliptic, positive and non-negative polynomials Classifications of linear operators preserving elliptic, positive and non-negative polynomials

2008-11-26

arxiv.org/abs/0811.4374v3

Mathematics

Classical Analysis and ODEs

Final version, to appear in J. Reine Angew. Math. (Crelle); 12 pages, no figures, LaTeX2e

Scientific paper

We characterize all linear operators on finite or infinite-dimensional spaces of univariate real polynomials preserving the sets of elliptic, positive, and non-negative polynomials, respectively. This is done by means of Fischer-Fock dualities, Hankel forms, and convolutions with non-negative measures. We also establish higher-dimensional analogs of these results. In particular, our classification theorems solve the questions raised in [9] originating from entire function theory and the literature pertaining to Hilbert's 17th problem.

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