Computational Methods and Function Theory 8 (2008), No. 2, 493--512
Copyright Heldermann Verlag 2008
Alexandru Aleman
aleman@maths.lth.se , Lund University, Department of Mathematics, P.O. Box 118, S-221 00 Lund, Sweden.
Boris Korenblum
borisko@csc.albany.edu , SUNY at Albany, Department of Mathematics, Albany, NY 12222-0001, U.S.A.
Let $C^\infty(a,b)$ be the Fr\'echet space of all complex-valued infinitely differentiable functions on a (finite or infinite) interval $(a,b)\subset\mathbb{R}$. Let ${L\subset C^\infty(a,b)}$ be a closed subspace such that $DL\subset L$, where $D=\frac{d}{dx}$. Then the spectrum $\sigma_L$ of $D$ on $L$ is either the whole complex plane, or a discrete possibly void set of eigenvalues $\lambda$, each one with some finite multiplicity $m_\lambda\in \mathbb{N}$ such that the monomial exponentials $e_{\lambda,j}(x)=x^j\exp(\lambda x)$, $0\leq j\leq m_\lambda-1$ belong to $L$. If the spectrum is void there is a relatively closed interval $I\subset (a,b)$ such that $L$ consists of those functions from $C^\infty(a,b)$ which vanish identically on $I$. The interval may reduce to a point in which case $L$ consists of the functions that vanish together with all their derivatives at that point.
Keywords: Differentiation operator, spectrum, nearly invariance, Fourier transform.
MSC 2000: 46E10, 42B35, 34Axx, 30H05.
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