Subspaces of C^∞ invariant under the differentiation

2013/09/26 by Alexandru Aleman, Aleman, Alexandru, Anton Baranov +3
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #math.CV

paper · pdf · doi:10.48550/arxiv.1309.6968

17 pages

arxiv created 2013/12/29 · arxiv updated 2013/12/31

Abstract

Let L be a proper differentiation invariant subspace of C^∞(a,b) such that the restriction operator (d)/(dx)|_L has a discrete spectrum Λ (counting with multiplicities). We prove that L is spanned by functions vanishing outside some closed interval I⊂(a,b) and monomial exponentials x^ke^λx corresponding to Λ if its density does not exceed the critical value (|I|)/(2π), and moreover, we show that the result is not necessarily true when the density of Λ equals the critical value. This answers a question posed by the first author and B. Korenblum. Finally, if the residual part of L is trivial, then L is spanned by the monomial exponentials it contains.

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