On Bessel-Riesz operators
This article deals with certain kind of potential operator defined as convolution with the generalized function Wα (P ± i0,m,n)depending on a complex parameter α and a real non negative one m. The definitory formulae and several properties of the family {W (P ± i m n)} α∈C α 0, , α; have been in...
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Universidad Nacional del Nordeste. Facultad de Ciencias Exactas y Naturales y Agrimensura
2023
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| Acceso en línea: | http://repositorio.unne.edu.ar/handle/123456789/51668 |
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I48-R184-123456789-516682025-03-06T10:59:37Z On Bessel-Riesz operators Cerutti, Rubén Alejandro Bessel-Riesz potentials Fractional derivative Hypersingular integral This article deals with certain kind of potential operator defined as convolution with the generalized function Wα (P ± i0,m,n)depending on a complex parameter α and a real non negative one m. The definitory formulae and several properties of the family {W (P ± i m n)} α∈C α 0, , α; have been introduced and studied by Trione (see [14]) specially the important followings two: a) Wα ∗Wβ =Wα+β , α and β complex numbers, and b) k W −2 is a fundamental solution of the k-times iterated Klein-Gordon operator Writing Wα (P ± i0,m,n) as an infinite linear combination of the ultrahyperbolic Riesz kernel of different orders Rα (P ± i0)which is a causal (anticausal) elementary solution of the ultrahyperbolic differential operator and taking into account its Fourier transform it is possible to evaluate the Fourier transform of the kernel Wα (P ± i0,m,n). We prove the composition formula Wα ∗Wβϕ =Wα+βϕ for a sufficiently good function. The proof of this result is based on the composition formulae presented by Trione in [14], but we also present a different way. Other simple property studied is the one that establish the relationship between the ultrahyperbolic Klein-Gordon operator and the Wα Bessel-Riesz operator. Finally we obtain an expression that will be consider a fractional power of the Klein-Gordon operator. 2023-06-12T12:01:37Z 2023-06-12T12:01:37Z 2007 Artículo Cerutti, Rubén, 2007. On Bessel-Riesz operators. FACENA. Corrientes: Universidad Nacional del Nordeste. Facultad de Ciencias Exactas y Naturales y Agrimensura, vol. 23, p. 17-27. ISSN 1851-507X. 1851-507X http://repositorio.unne.edu.ar/handle/123456789/51668 spa openAccess http://creativecommons.org/licenses/by-nc-nd/2.5/ar/ application/pdf p. 17-27 application/pdf Universidad Nacional del Nordeste. Facultad de Ciencias Exactas y Naturales y Agrimensura FACENA, 2007, vol. 23, p. 17-27. |
| institution |
Universidad Nacional del Nordeste |
| institution_str |
I-48 |
| repository_str |
R-184 |
| collection |
RIUNNE - Repositorio Institucional de la Universidad Nacional del Nordeste (UNNE) |
| language |
Español |
| topic |
Bessel-Riesz potentials Fractional derivative Hypersingular integral |
| spellingShingle |
Bessel-Riesz potentials Fractional derivative Hypersingular integral Cerutti, Rubén Alejandro On Bessel-Riesz operators |
| topic_facet |
Bessel-Riesz potentials Fractional derivative Hypersingular integral |
| description |
This article deals with certain kind of potential operator defined as convolution
with the generalized function Wα (P ± i0,m,n)depending on a complex parameter α and
a real non negative one m.
The definitory formulae and several properties of the family
{W (P ± i m n)} α∈C α 0, , α; have been introduced and studied by Trione (see [14])
specially the important followings two:
a) Wα ∗Wβ =Wα+β , α and β complex numbers, and
b) k W −2 is a fundamental solution of the k-times iterated Klein-Gordon operator
Writing Wα (P ± i0,m,n) as an infinite linear combination of the ultrahyperbolic
Riesz kernel of different orders Rα (P ± i0)which is a causal (anticausal) elementary
solution of the ultrahyperbolic differential operator and taking into account its Fourier
transform it is possible to evaluate the Fourier transform of the kernel Wα (P ± i0,m,n).
We prove the composition formula Wα ∗Wβϕ =Wα+βϕ for a sufficiently good
function. The proof of this result is based on the composition formulae presented by
Trione in [14], but we also present a different way.
Other simple property studied is the one that establish the relationship between the
ultrahyperbolic Klein-Gordon operator and the Wα Bessel-Riesz operator.
Finally we obtain an expression that will be consider a fractional power of the
Klein-Gordon operator. |
| format |
Artículo |
| author |
Cerutti, Rubén Alejandro |
| author_facet |
Cerutti, Rubén Alejandro |
| author_sort |
Cerutti, Rubén Alejandro |
| title |
On Bessel-Riesz operators |
| title_short |
On Bessel-Riesz operators |
| title_full |
On Bessel-Riesz operators |
| title_fullStr |
On Bessel-Riesz operators |
| title_full_unstemmed |
On Bessel-Riesz operators |
| title_sort |
on bessel-riesz operators |
| publisher |
Universidad Nacional del Nordeste. Facultad de Ciencias Exactas y Naturales y Agrimensura |
| publishDate |
2023 |
| url |
http://repositorio.unne.edu.ar/handle/123456789/51668 |
| work_keys_str_mv |
AT ceruttirubenalejandro onbesselrieszoperators |
| _version_ |
1832344392975777792 |