Integration of functions in a space with complex number of dimensions

A study is made of analytic continuation with respect to the dimension of integrals of isotropic functions, $I(\nu)=\int f(x_1,\dots,x_n)d^\nu x_1\dots d^\nu x_n$, i.e., of functions such that $f(Ux_1,\dots,Ux_n)=f(x_1,\dots,x_n)$ for any orthogonal transformation $U\in O(\nu)$. The main result of the paper is the proof that if $f$ is a $C^\infty$ rapidly decreasing function, $f \in \mathscr S$, then $I(\nu)$ is an entire function of $\nu$. Its order is estimated as a generalized function over the space for $\mathscr S$ different complex values of $\nu$. A uniqueness theorem for the analytic continuation of $I(\nu)$ is established. Similar results are proved for an operator of integration with respect to some of the variables. The analytic continuation with respect to the dimension of the operator of Fourier transformation is considered.