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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On mean-periodicity. II

Author: Edwin J. Akutowicz
Journal: Trans. Amer. Math. Soc. 157 (1971), 449-457
MSC: Primary 42.30
MathSciNet review: 0284765
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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,

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

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 $.

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Keywords: Spectral analysis and synthesis, mean-periodic distributions, homogeneous convolution equations, linear topological space of entire functions and its dual, Fourier transform, Fantappiè indicator, interpolation
Article copyright: © Copyright 1971 American Mathematical Society

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