10.8: Multi-Dimensional Fourier Transforms

When studying problems such as wave propagation, we often deal with Fourier transforms of several variables. This is conceptually straightforward. For a function \(f(x_1, x_2, \dots, x_d)\) which depends on \(d\) independent spatial coordinates \(x_1, x_2, \dots x_d\), we can Fourier transform each coordinate individually: \[F(k_1, k_2, \dots, k_d) = \int_{-\infty}^\infty dx_1\; e^{-ik_1x_1}\; \int_{-\infty}^\infty dx_2\; e^{-ik_2x_2}\,\cdots\, \int_{-\infty}^\infty dx_d\; e^{-ik_d x_d}\, f(x_1,x_2, \dots,x_N)\] Each coordinate gets Fourier-transformed into its own independent \(k\) variable, so the result is also a function of \(d\) independent variables.

We can express the multi-dimensional Fourier transform more compactly using vector notation. If \(\vec{x}\) is a \(d\)-dimensional coordinate vector, the Fourier-transformed coordinates can be written as \(\vec{k}\), and the Fourier transform is \[F(\vec{k}) = \int d^d x \; \exp\left(-i\,\vec{k}\cdot\vec{x}\right) \, f\big(\vec{x}\big),\] where \(\int d^d x\) denotes an integral over the entire \(d\)-dimensional space, and \(\vec{k}\cdot\vec{x}\) is the usual dot product of two vectors. The inverse Fourier transform is \[f(\vec{x}) = \int \frac{d^dk}{(2\pi)^d}\; \exp\left(i\,\vec{k}\cdot\vec{x}\right)\, F\big(\vec{k}\big).\] The delta function, which we introduced in Section 10.7, can also be defined in \(d\)-dimensional space, as the Fourier transform of a plane wave: \[\delta^d(\vec{x}-\vec{x}') = \int \frac{d^dk}{(2\pi)^d} \, \exp\left[i\vec{k} \cdot \left(\vec{x}-\vec{x}'\right)\right].\] Note that \(\delta^d\) has the dimensions of \([x]^{-d}\). The multi-dimensional delta function has a “filtering” property similar to the one-dimensional delta function. For any \(f(x_1,\dots,x_d)\), \[\int d^dx \; \delta^d(\vec{x}-\vec{x}') \, f(\vec{x}) = f(\vec{x}').\]