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Derivation-Invariant Subspaces of C∞

Aleman, Alexandru LU and Korenblum, Boris (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)
Please use this url to cite or link to this publication:
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
date added to LUP
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},
}