Generalized Fourier expansions for zero-solutions of surjective convolution operators on
and ![\mathcal D\'_ω(R)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/007af8908515c9ecfa71d020f01dfa2e.png)
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.
![\mu](http://siba-ese.unile.it/plugins/generic/latexRender/cache/c9faf6ead2cd2c2187bd943488de1d0a.png)
![S_\mu](http://siba-ese.unile.it/plugins/generic/latexRender/cache/278b8537b692165f66d39794d139dc9a.png)
![\mathcal D'(R)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/f9ad13e26db1f89e79e81913cc751a5a.png)
![S_\mu](http://siba-ese.unile.it/plugins/generic/latexRender/cache/278b8537b692165f66d39794d139dc9a.png)
![\mathcal D'(R)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/f9ad13e26db1f89e79e81913cc751a5a.png)
![kerS_\mu](http://siba-ese.unile.it/plugins/generic/latexRender/cache/1586c2f1f38dcaa19ee9819011212bf7.png)
![S_\mu](http://siba-ese.unile.it/plugins/generic/latexRender/cache/278b8537b692165f66d39794d139dc9a.png)
![\nu∈ \mathcal D'<sub>w</sub>(R)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/dd97bcada970b8e725927b6a6b5e57f8.png)
![kerS_\mu](http://siba-ese.unile.it/plugins/generic/latexRender/cache/1586c2f1f38dcaa19ee9819011212bf7.png)
![S_\mu](http://siba-ese.unile.it/plugins/generic/latexRender/cache/278b8537b692165f66d39794d139dc9a.png)
![S_\mu](http://siba-ese.unile.it/plugins/generic/latexRender/cache/278b8537b692165f66d39794d139dc9a.png)
![kerS_\mu](http://siba-ese.unile.it/plugins/generic/latexRender/cache/1586c2f1f38dcaa19ee9819011212bf7.png)
![S_\mu](http://siba-ese.unile.it/plugins/generic/latexRender/cache/278b8537b692165f66d39794d139dc9a.png)
![\mathcal D'(R)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/f9ad13e26db1f89e79e81913cc751a5a.png)
![\varepsilon_{{w}}(R)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/8150e45fdc2ca16877ce5ca3debab910.png)
![\mathcal D'(R)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/f9ad13e26db1f89e79e81913cc751a5a.png)
DOI Code:
10.1285/i15900932v10supn1p251
Full Text: PDF