Geodesic
Associative algebra of functionals containing $\delta(x)$ and $r^n$
Teoretičeskaâ i matematičeskaâ fizika, Tome 52 (1982) no. 2, pp. 327-331 
V. A. Smirnov. Associative algebra of functionals containing $\delta(x)$ and $r^n$. Teoretičeskaâ i matematičeskaâ fizika, Tome 52 (1982) no. 2, pp. 327-331. http://geodesic.mathdoc.fr/item/TMF_1982_52_2_a17/
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     title = {Associative algebra of functionals containing $\delta(x)$ and~$r^n$},
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Voir la notice de l'article provenant de la source Math-Net.Ru

Shirokov's results [1, 2] are generalized to the case of arbitrary dimension. This leads to the construction of an associative algebra with differentiation containing the elements $\delta(\mathbf x)$ and $r^n$ ($\mathbf x=(x_1,\dots,x_d)$, $r=|\mathbf x|$, $n=0,\pm1,\pm2,\dots$). The algebra is realized on a subset of functionals defined on the space of functions which can be represented in the form $\varphi=r^{-2n_1}\varphi_1+r^{-2n_2{-1}}\varphi_2$, $\varphi_{1,2}\in S(\mathrm R^d)$.

[1] Shirokov Yu. M., TMF, 39:3 (1979), 291–301 | MR | Zbl

[2] Shirokov Yu. M., TMF, 40:3 (1979), 348–354 | MR | Zbl

[3] Shirokov Yu. M., TMF, 41:3 (1979), 291–302 | MR | Zbl

[4] Shirokov Yu. M., TMF, 42:1 (1980), 45–49 | MR

[5] Gelfand I. M., Shilov G. E., Obobschennye funktsii, vyp. 1, Fizmatgiz, M., 1958 | MR

[6] Shirokov Yu. M., TMF, 15:1 (1973), 20–42 | MR