Subspaces of C-infinity invariant under the differentiation

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Abstract

Let L be a proper differentiation invariant subspace of C-infinity (a, b) such that the restriction operator d/dx vertical bar L has a discrete spectrum Lambda (counting with multiplicities). We prove that L is spanned by functions vanishing outside some closed interval I subset of (a, b) and monomial exponentials x(k)e(lambda x) corresponding to Lambda if its density is strictly less than the critical value vertical bar I vertical bar/2 pi, and moreover, we show that the result is not necessarily true when the density of Lambda 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. (C) 2015 Elsevier Inc. All rights reserved.

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Subject classification (UKÄ) – MANDATORY

  • Mathematics

Keywords

  • Spectral synthesis, Entire functions, Paley-Wiener spaces, Invariant, subspaces
Original languageEnglish
Pages (from-to)2421-2439
JournalJournal of Functional Analysis
Volume268
Issue number8
Publication statusPublished - 2015
Publication categoryResearch
Peer-reviewedYes