### Generalized Fourier expansions for zero-solutions of surjective convolution operators on and

#### Abstract

It is well-known that each distribution with compact support can be convolved with an arbitrary distribution and that this defines a convolution operator , acting on .The surjectivity of , was characterized by Ehrenpreis [5]. Extending this result,we characterize in the present article the surjectivity of convolution operators on the space of all

*w*-ultradistributions of Beurling type on R.This is done in two steps. In the first one we show that has an absolute basis whenever , admits a fundamental solution . The expansion of an element in , 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 is surjective if and only if admits a fundamental solution. Hence the elements of admit a generalized Fourier expansion for each surjective convolution operator on .Note that this differs from the behavior of convolution operators on the space 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 ,too.DOI Code:
10.1285/i15900932v10supn1p251

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