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Derivation-Invariant Subspaces of C∞

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Author

Summary, in English

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.

Publishing year

2008

Language

English

Pages

493-512

Publication/Series

Computational Methods in Function Theory

Volume

8

Issue

1-2

Document type

Journal article

Publisher

Heldermann

Topic

  • Mathematics

Keywords

  • Fourier transform.
  • nearly invariance
  • spectrum
  • Differentiation operator

Status

Published

ISBN/ISSN/Other

  • ISSN: 1617-9447