On mean-periodicity. II

Abstract:

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, \[ \rho :\mathcal {F}\mathcal {M} \backepsilon F \to (F({\lambda _1}),F’({\lambda _1}), \ldots ,{F^{({p_1} - 1)}}({\lambda _1}),F({\lambda _2}), \ldots ,{F^{({p_2} - 1)}}({\lambda _2}), \ldots ),\] 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 $\mathcal {F}\mathcal {M}/\ker \rho$.