<p>We consider convolution operators in the Roumieu spaces ofultradifferentiable functions of mean type on the real axis.The well-known Gevrey classes are also Roumieu spaces.The convolution operators includethe differential equations of infinite order with constant coefficients,difference-differential and integro-differential operatorsas particular cases.The recent results on convolution operators in the Beurling spacesof mean type and the connection between the Roumieu andBeurling spaces imply that the surjectivity ofthe convolution operator requires that the symbol ofthe operator be slowly decreasing with respect tothe weight function defining the space.Under this assumption, we obtainthe isomorphic description of the kernel of the convolution operatoras some space of sequences of functionals and in the form ofsome space of numeric sequences. Using the isomorphic description theorems,we construct an absolute basis for the solution space ofthe homogeneous convolution equation. These results are not only of interest in their own rightbut also form the necessary step in studying the problem of surjectivity of the convolution operatorin the Roumieu spaces of mean type which remains unsettled by now.</p>