# Lund University Publications

## LUND UNIVERSITY LIBRARIES

### Derivation-Invariant Subspaces of C∞

(2008) In Computational Methods in Function Theory 8(1-2). p.493-512
Abstract
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... (More)
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. (Less)
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Fourier transform., nearly invariance, spectrum, Differentiation operator
in
Computational Methods in Function Theory
volume
8
issue
1-2
pages
493 - 512
publisher
Heldermann
ISSN
1617-9447
language
English
LU publication?
yes
id
557fd37e-b1e1-4b8d-8dc0-5f18416f7557 (old id 1467130)
alternative location
http://www.heldermann-verlag.de/cmf/cmf08/cmf08037.pdf
2009-09-07 14:55:11
date last changed
2016-04-16 01:29:53
@article{557fd37e-b1e1-4b8d-8dc0-5f18416f7557,
abstract     = {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.},
author       = {Aleman, Alexandru and Korenblum, Boris},
issn         = {1617-9447},
keyword      = {Fourier transform.,nearly invariance,spectrum,Differentiation operator},
language     = {eng},
number       = {1-2},
pages        = {493--512},
publisher    = {Heldermann},
series       = {Computational Methods in Function Theory},
title        = {Derivation-Invariant Subspaces of C∞},
volume       = {8},
year         = {2008},
}