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)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/1467130
- author
- Aleman, Alexandru ^{LU} and Korenblum, Boris
- organization
- publishing date
- 2008
- 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}, }