On a particular solution of a nonhomogeneous convolution equation in spaces of ultradifferentiable functions

We consider the Beurling spaces of ultradifferentiable functions of mean type on the real axis determined by special weight functions. These spaces are the general projective analogs of the well-known Gevrey classes. In these spaces we investigate a nonhomogeneous convolution equation (differential equation of infinite order with constant coefficients) generated by the symbol which has only simple zeros and satisfies some natural growth estimates. Given the zeros of a symbol, a symmetric sequence of real numbers is explicitly constructed, in each of which the module of the symbol has a suitable lower estimate. This sequence determines a system of exponentials with imaginary indexes which is absolutely representing in the corresponding space. This allows us to represent the right-hand side of the equation as an absolutely convergent series with respect to this system. Then we establish a particular solution of the equation under considering as an absolutely convergent series with respect to this system, too. The coefficients of the series are naturally determined by the right-hand side of the equation. The proof is essentially based on the analogous results which were earlier obtained in the case of spaces on finite interval. We also use the stability property of weakly sufficient sets and absolutely representing systems. Some concrete examples of constructing the desired sequences are also given in the paper.