Speaker
Description
We consider a spectral synthesis problem for differentiation-invariant subspaces in a general space $\mathcal E_{\Omega} (a;b)$ of $\Omega$-ultradifferentiable functions, where
$(a;b)\subseteq\mathbb R $ and $\Omega=\{\omega_n\}$ is a sequence of nonquasianalytic weights subjected some standard restrictions of $\Omega$-ultradifferentiable functions theory.
Do there exist differentiation-invariant subspaces $W\subset\mathcal E_{\Omega} (a;b)$ for which weak spectral synthesis fails?
Alexandru Aleman, Anton Baranov and Yurii Belov constructed the first example of differentiation-invariant subspace in $C^{\infty}(a;b)$ which does not admit weak spectral synthesis (2015).
We answer the above question using a dual scheme. Namely, we consider a topological module $P=\mathcal F(\mathcal E'_{\Omega} (a;b))$, where $\mathcal F$ denotes the Fourier-Laplace transform, and find $unlocalisable$ primary submodules $J\subset P$. Then, the differentiation-invariant subspaces in $\mathcal E_{\Omega} (a;b)$ which dual submodules are $J$ do not admit the weak spectral synthesis.
