The integral represenatation of the riesz kernel Wα(u,m)
Titel:
The integral represenatation of the riesz kernel Wα(u,m)
Auteur:
Trione, Susana Elena
Verschenen in:
Integral transforms and special functions
Paginering:
Jaargang 7 (1998) nr. 1-2 pagina's 171-174
Jaar:
1998-06
Inhoud:
The Riesz kernel [image omitted] where n is the dimension of the space (cf. form. (1)) can be expressed by means of a Riemann-Liouville integral (cf. form. (13)). The very interesting particular case α=2k expressed by form. (15). The kernel Wα(u,m) has analogous properties to the Riesz kernel (cf. [1]). That is,putting p=1 in (1) we obtain the kernel introduced by Riesz (cf. [1], p.17,[2], p.89, [3], p.179 and [4], p.72). Therefore, we know that [image omitted] ,where [image omitted] is the n-dimensional ultrahyperbolic Klein-Gordon operator, (cf. [5], p.9, form. (2.29));[image omitted] (cf. [5], p.19, form (IV.9)), here [image omitted] is the n-dimensional ultrahyperbolic Klein-Gordon operator iterated K-times W0(u,m)=δ (cf. [5], p.20, form (VI)). OtherwiseWα(u,m) can be expressed as an infinite, linear combination of Rα(u) of different orders (cf. [5], p.15, form. (3.26)) where Rα(u) is the ultrahyperbolic Riesz (cf. [6], p.72) and Wα(u,m)=Rα(u) (cf. [5], p.22, form. (V2.5). We also study the particular case W0(u,m) (see Note).
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