We study the space of tempered ultradistributions whose convolutions with test functions are all contained in a given translation-modulation invariant Banach space of ultradistributions. Our main ...result will be the first structural theorem for the aforementioned space. As an application we consider several extensions of convolution.
The paper aims to introduce and study an algebra of asymptotically almost periodic generalized ultradistributions. These generalized ultradistributions contain asymptotically almost periodic ...ultradistributions and asymptotically almost periodic generalized functions. The definition and main properties of these generalized ultradistributions are studied. An application to difference differential systems is given.
We develop a convolution theory for quasianalytic ultradistributions of Gelfand–Shilov type. We also construct a special class of ultrapolynomials, and use it as a base for the parametrix method in ...the study of new topological and structural properties of several quasianalytic spaces of functions and ultradistributions. In particular, our results apply to Fourier hyperfunctions and Fourier ultra-hyperfunctions.
On dévelope une théorie de convolution pour les ultradistributions quasianalytiques de type Gelfand–Shilov. On construit aussi une classe spéciale des ultrapolynômes et on l'utilise comme base pour la méthode de parametrix dans l'étude des nouvelles propriétés topologiques et structurelles de quelques espaces quasianalytiques fonctionnels et ultradistributionnels. En particulier, nos résultats appellent aux hyperfonctions et ultrahyperfonctions de Fourier.
We introduce and study a new class of translation–modulation invariant Banach spaces of ultradistributions. These spaces show stability under Fourier transform and tensor products; furthermore, they ...have a natural Banach convolution module structure over a certain associated Beurling algebra, as well as a Banach multiplication module structure over an associated Wiener–Beurling algebra. We also investigate a new class of modulation spaces, the Banach spaces of ultradistributions
M
F
on
R
d
, associated to translation–modulation invariant Banach spaces of ultradistributions
F
on
R
2
d
.
In this paper we give a global characterisation of classes of ultradifferentiable functions and corresponding ultradistributions on a compact manifold X. The characterisation is given in terms of the ...eigenfunction expansion of an elliptic operator on X. This extends the result for analytic functions on compact manifolds by Seeley in 1969, and the characterisation of Gevrey functions and Gevrey ultradistributions on compact Lie groups and homogeneous spaces by the authors (2014).
The global Kotake-Narasimhan theorem Hoepfner, G.; Rampazo, P.
Proceedings of the American Mathematical Society,
03/2022, Volume:
150, Issue:
3
Journal Article
Peer reviewed
Open access
In this paper we introduce the notion of global ultradifferentiable functions with respect to weight functions and include a discussion of its functional analytic theory and prove a characterization ...in terms of certain exponential decay of the Fourier-Broz-Iagolnitzer transform—a Paley-Wiener type theorem. As an application we investigate the regularity of global ultradifferentiable vectors proving a version of the Kotake-Narasimhan theorem in this setting.
We first introduce an algebra of almost periodic generalized
ultradistributions containing classical almost periodic ultradistributions
as well as the algebra of almost periodic generalized ...functions, and then we
study the fundamental properties of this algebra.
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We define Wiener amalgam spaces of (quasi)analytic ultradistributions whose local components belong to a general class of translation and modulation invariant Banach spaces of ultradistributions and ...their global components are either weighted Lp or weighted C0 spaces. We provide a discrete characterisation via so called uniformly concentrated partitions of unity. Finally, we study the complex interpolation method and we identify the strong duals for most of these Wiener amalgam spaces.
We study a class of semilinear elliptic equations on spaces of tempered ultradistributions of Beurling and Roumieu type. Assuming that the linear part of the equation is a pseudodifferential operator ...of infinite order satisfying a suitable ellipticity condition we prove a regularity result in the functional setting above for weak Sobolev type solutions.
We consider several general sequential conditions for convolvability of two Roumieu ultradistributions on
R
d
in the space
D
′
{
M
p
}
and prove that they are equivalent to the convolvability of ...these ultradistributions in the sense of Pilipović and Prangoski. The discussed conditions, based on two classes
U
{
M
p
}
and
U
¯
{
M
p
}
of approximate units and corresponding sequential conditions of integrability of Roumieu ultradistributions, are analogous to the known convolvability conditions in the space
D
′
of distributions and in the space
D
′
(
M
p
)
of ultradistributions of Beurling type. Moreover, the following property of the convolution and ultradifferential operator
P
(
D
) of class
{
M
p
}
is proved: if
S
,
T
∈
D
′
{
M
p
}
(
R
d
)
are convolvable, then
P
(
D
)
(
S
∗
T
)
=
(
P
(
D
)
S
)
∗
T
=
S
∗
(
P
(
D
)
T
)
.
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