I'm a simple guy. I see anything related to Fourier series/transform, me like. To me, it's like solution to at least 16% of all the problems I have. My only gripe is that I can't thank Fourier personally for the genius he had.
There's a good list in Wikipedia for FFT (though it's a bit more specific than just FT, but still application of it):
- fast large-integer and polynomial multiplication,
- efficient matrix–vector multiplication for Toeplitz,- circulant and other structured matrices,
- filtering algorithms (see overlap–add and overlap–save methods),
- fast algorithms for discrete cosine or sine transforms (e.g. fast DCT used for JPEG and MPEG/MP3 encoding and decoding),
- fast Chebyshev approximation,
- solving difference equations,
- computation of isotopic distributions.[47]
- modulation and demodulation of complex data symbols using orthogonal frequency division multiplexing (OFDM) for 5G, LTE, Wi-Fi, DSL, and other modern communication systems.
And naturally for any time-based signal about half a gazillion applications due to it's ability to detect "traits" which can be stored/read efficiently due to time => freq transform
It's not so much a stand-in as the possibility to compute convolutions (and auto-correlation and cross-correlation) really easily and cheaply in Fourier space.
