Invariant Subspaces in Non-Quasianalytic Spaces of \( \Omega \)-Ultradifferentiable Functions on an Interval
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Keywords:
\( \Omega \)-ultradifferentiable function, \( \Omega \)-ultradistribution, Fourier-Laplace transform, invariant subspace, spectral synthesisAbstract
We consider and solve a weakened version of the classical spectral synthesis problem for differentiation operator in non-quasianalytic spaces of ultradifferentiable functions (UDF). Moreover, we deal with the widest class of UDF among all known ones. Namely, we study the spaces of \( \Omega \)-ultradifferentiable functions introduced by Alexander Abanin in 2007–08. For subspaces of these spaces which are invariant under the differentiation operator we establish general conditions of weak spectral synthesis.