Multidimensional transform

Remarks

The transform is separable when $\Omega = \Omega_1 \times \cdots \times \Omega_n$, $U = U_1 \times \cdots \times U_n$ and $K(u, x) = K_1(u_1, x_1) \cdots K_n(u_n, x_n)$. Then, by Fubini's theorem, $T$ is the composition of the one-variable transforms with kernels $K_j$ applied in each variable.

Example: $\Omega = U = \R^n$ and $K(\xi, x) = e^{-i \xi \cdot x}$ give the $n$-dimensional Fourier transform, a separable transform since $e^{-i \xi \cdot x} = \prod_j e^{-i \xi_j x_j}$.

Non-example: the Radon transform sends a function on $\R^n$ to its integrals over hyperplanes. It is not of the form above with a function $K$, since each value is an integral over a set of measure zero.