On kernels of convolution operators in the Roumieu spaces of ultradifferentiable functions

We consider convolution operators in the Roumieu spaces of ultradiffereniable functions of mean type on the real axis. The famous Gevrey classes are also the Roumieu spaces. As particular cases, convolution operators include the differential equations of infinite order with constant coefficients, difference-differential and integro-differential equations. From recent results for convolution operators in the Beurling spaces of mean type and from the connection between the Roumieu and the Beurling spaces it follows that for the surjectivity of convolution operator it is necessary that the symbol of the operator is slowly decreasing with respect to the weight function. Under this assumption, we obtain the isomorphic description for the kernel of the convolution operator as a sequence space. We also construct an absolute basis in the space of all solutions of the homogeneous convolution equation. These results are of their own interest. On the other hand, they are the necessary step for investigation of the problem of surjectivity of the convolution operator in the Roumieu spaces of mean type.