Invariant subspaces in nonquasianalytic spaces of $\Omega$-ultradifferentiable functions on an interval

In this paper we consider a weakened version of the spectral synthesis for the differentiation operator in nonquasianalytic spaces of ultradifferentiable functions. We deal with the widest possible class of spaces of ultradifferentiable functions among all known ones. Namely, these are spaces of $\Omega$-ultradifferentiable functions which have been recently introduced and explored by A.V. Abanin. For differentiation invariant subspaces in these spaces, we establlish conditions of weak spectral synthesis. As an application, we prove that a kernel of a local convolution operator admits weak spectral synthesis. We also show that a conjunction of kernels of convolution operators admits weak spectral synthesis if all generating ultradistributions have the same support equaled to $\{0\}$ and there exists one generated by an ultradistribution which characteristic function is a multiplier in the corresponding space of entire functions.