Associative algebra of functionals containing $\delta(x)$ and $r^n$

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 $\varphi=r^{-2n_1}\varphi_1+r^{-2n_2{-1}}\varphi_2$, $\varphi_{1,2}\in S(\mathrm R^d)$.