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...
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Artículo |
| Lenguaje: | Español |
| Publicado: |
Universidad Nacional del Nordeste. Facultad de Ciencias Exactas y Naturales y Agrimensura
2023
|
| Materias: | |
| Acceso en línea: | http://repositorio.unne.edu.ar/handle/123456789/51668 |
| Aporte de: |
| Sumario: | 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. |
|---|