A couple things come to mind...
Performing convolutions efficiently as products in the Fourier domain. An example would be training large convolutional neural nets.
For example, see: Fast Training of Convolutional Networks through FFTs (Mathieu et al. 2013)
Another application is sparse signal processing, where the goal is to approximate a signal as a sparse linear combination of basis functions from a 'signal dictionary'. The link here is that the set of sinusoids are, of course, a good dictionary for signals that are sparse in the Fourier domain. If I recall correctly, Fourier dictionaries show up in this literature.
On a related note, you should also be able to find Fourier methods in the compressed sensing literature