Subspaces of C-infinity invariant under the differentiation

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Subspaces of C-infinity invariant under the differentiation. / Aleman, Alexandru; Baranov, Anton; Belov, Yurii.

In: Journal of Functional Analysis, Vol. 268, No. 8, 2015, p. 2421-2439.

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Aleman, Alexandru ; Baranov, Anton ; Belov, Yurii. / Subspaces of C-infinity invariant under the differentiation. In: Journal of Functional Analysis. 2015 ; Vol. 268, No. 8. pp. 2421-2439.

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TY - JOUR

T1 - Subspaces of C-infinity invariant under the differentiation

AU - Aleman, Alexandru

AU - Baranov, Anton

AU - Belov, Yurii

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

KW - Spectral synthesis

KW - Entire functions

KW - Paley-Wiener spaces

KW - Invariant

KW - subspaces

U2 - 10.1016/j.jfa.2015.01.002

DO - 10.1016/j.jfa.2015.01.002

M3 - Article

VL - 268

SP - 2421

EP - 2439

JO - Journal of Functional Analysis

T2 - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 8

ER -