Generalized Fourier expansions for zero-solutions of surjective convolution operators on $\mathcal D\'(R)$ and $\mathcal D\'_ω(R)$

Abstract

It is well-known that each distribution $\mu$ with compact support can be convolved with an arbitrary distribution and that this defines a convolution operator $S_\mu$, acting on $\mathcal D'(R)$.The surjectivity of $S_\mu$, was characterized by Ehrenpreis [5]. Extending this result,we characterize in the present article the surjectivity of convolution operators on the space $\mathcal D'(R)$ of all w-ultradistributions of Beurling type on R.This is done in two steps. In the first one we show that $kerS_\mu$ has an absolute basis whenever $S_\mu$, admits a fundamental solution $\nu∈ \mathcal D'_w(R)$. The expansion of an element in $kerS_\mu$, with respect to this basis can be regarded as a generalization of the Fourier expansion of periodic ultradistributions.In the second step we use this sequence space representation together with results of Palamodov [15] and Vogt [17], [18] on the projective limit functor to obtain the desired characterization.It turns out that $S_\mu$ is surjective if and only if $S_\mu$ admits a fundamental solution. Hence the elements of $kerS_\mu$ admit a generalized Fourier expansion for each surjective convolution operator $S_\mu$ on $\mathcal D'(R)$.Note that this differs from the behavior of convolution operators on the space $\varepsilon_{{w}}(R)$ of w-ultradifferentiable functions of Roumieu-type, as Braun,Meise and Vogt [4] have shown. Note also that the results of the present article apply to convolution operators on $\mathcal D'(R)$,too.