# Distributional {D}unkl transform and {D}unkl convolution operators

• Volume: 9-B, Issue: 1, page 221-245
• ISSN: 0392-4033

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## Abstract

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In this paper, that is divided in two parts, we study the distributional Dunkl transform on R. In the first part we investigate the Dunkl transform and the Dunkl convolution operators on tempered distributions. We prove that the tempered distributions defining Dunkl convolution operators on the Schwartz space  are the elements of $\mathcal{O}^{\prime}_{c}$, the space of usual convolution operators on $S$. In the second part we define the distributional Dunkl transform by employing the kernel method. We introduce Frechet function spaces containing the kernel of the Dunkl transform. In theproof of the properties of the distributional Dunkl transform, defined on the correspoding dual spaces, certain representations of the elements of the dual spaces will play an important role. These representations allows us to simplify, in contrast with the previous and usual methods (see, for instance [7] and [13]), the mentioned proofs. Our new procedure also applies to other distributional integral transforms that had been investigated by other authors (Hankel transforms ([7] and [13]), amongst others).

## How to cite

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Betancor, Jorge J.. "Distributional {D}unkl transform and {D}unkl convolution operators." Bollettino dell'Unione Matematica Italiana 9-B.1 (2006): 221-245. <http://eudml.org/doc/289605>.

@article{Betancor2006,
abstract = {In this paper, that is divided in two parts, we study the distributional Dunkl transform on R. In the first part we investigate the Dunkl transform and the Dunkl convolution operators on tempered distributions. We prove that the tempered distributions defining Dunkl convolution operators on the Schwartz space  are the elements of $$\mathcal\{O\}'\_c$$, the space of usual convolution operators on $$S$$. In the second part we define the distributional Dunkl transform by employing the kernel method. We introduce Frechet function spaces containing the kernel of the Dunkl transform. In theproof of the properties of the distributional Dunkl transform, defined on the correspoding dual spaces, certain representations of the elements of the dual spaces will play an important role. These representations allows us to simplify, in contrast with the previous and usual methods (see, for instance [7] and [13]), the mentioned proofs. Our new procedure also applies to other distributional integral transforms that had been investigated by other authors (Hankel transforms ([7] and [13]), amongst others).},
author = {Betancor, Jorge J.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {221-245},
publisher = {Unione Matematica Italiana},
title = {Distributional \{D\}unkl transform and \{D\}unkl convolution operators},
url = {http://eudml.org/doc/289605},
volume = {9-B},
year = {2006},
}

TY - JOUR
AU - Betancor, Jorge J.
TI - Distributional {D}unkl transform and {D}unkl convolution operators
JO - Bollettino dell'Unione Matematica Italiana
DA - 2006/2//
PB - Unione Matematica Italiana
VL - 9-B
IS - 1
SP - 221
EP - 245
AB - In this paper, that is divided in two parts, we study the distributional Dunkl transform on R. In the first part we investigate the Dunkl transform and the Dunkl convolution operators on tempered distributions. We prove that the tempered distributions defining Dunkl convolution operators on the Schwartz space  are the elements of $$\mathcal{O}'_c$$, the space of usual convolution operators on $$S$$. In the second part we define the distributional Dunkl transform by employing the kernel method. We introduce Frechet function spaces containing the kernel of the Dunkl transform. In theproof of the properties of the distributional Dunkl transform, defined on the correspoding dual spaces, certain representations of the elements of the dual spaces will play an important role. These representations allows us to simplify, in contrast with the previous and usual methods (see, for instance [7] and [13]), the mentioned proofs. Our new procedure also applies to other distributional integral transforms that had been investigated by other authors (Hankel transforms ([7] and [13]), amongst others).
LA - eng
UR - http://eudml.org/doc/289605
ER -

## References

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