On mean-periodicity. II

This paper is devoted to the problem of representing all solutions of certain homogeneous convolution equations through series of exponential polynomials. This representation is sought in the dual space $\mathcal{M}'$ of a function space $\mathcal{M}$, the latter consisting of entire functions satisfying growth conditions in horizontal directions. The space $\mathcal{M}$ is a Fréchet space, which fact permits a simpler and more thorough treatment than that given in the paper [1]. The technique used here is based upon a method developed by L. Ehrenpreis [5] and V. P. Palamodov [3] in the theory of differential equations with constant coefficients. We map the Fourier transform space $\mathcal{F}\mathcal{M}$ into a space of sequences,

$$\rho :\mathcal{F}\mathcal{M} \backepsilon F \to (F({\lambda _1}),... ...lambda _1}),F({\lambda _2}), \ldots ,{F^{({p_2} - 1)}}({\lambda _2}), \ldots ),$$

where $\{ {\lambda _k}\}$ is the spectrum with multiplicity of a mean-periodic element of the dual space $\mathcal{M}'$. The crucial point is to identify the quotient space $\mathcal{F}\mathcal{M}/\ker \rho$.