Editing Fast Fourier transform (section)

==Applications==
The FFT is used in digital recording, sampling, [[additive synthesis]] and [[pitch correction]] software.<ref>{{cite book |last1=Burgess |first1=Richard James |title=The History of Music Production |date=2014 |publisher= Oxford University Press |isbn=978-0199357178 |url=https://books.google.com/books?id=qMKiAwAAQBAJ |access-date=1 August 2019}}</ref>

The FFT's importance derives from the fact that it has made working in the frequency domain equally computationally feasible as working in the temporal or spatial domain. Some of the important applications of the FFT include:<ref name="Rockmore_2000"/><ref name="Chu_George_1999"/>

* fast large-integer [[multiplication algorithm]]s and polynomial multiplication,
* efficient matrix–vector multiplication for [[Toeplitz matrix|Toeplitz]], [[circulant]] and other structured matrices,
* filtering algorithms (see [[overlap–add method|overlap–add]] and [[overlap–save method|overlap–save]] methods),
* fast algorithms for [[discrete cosine transform|discrete cosine]] or [[discrete sine transform|sine transforms]] (e.g. [[Discrete cosine transform|fast DCT]] used for [[JPEG]] and [[MPEG]]/[[MP3]] encoding and decoding),
* fast [[Chebyshev approximation]],
* solving [[Recurrence relation|difference equation]]s,
* computation of [[Mass spectrometry|isotopic distributions]].<ref name="Fernandez-de-Cossio_2012"/>
* modulation and demodulation of complex data symbols using [[orthogonal frequency-division multiplexing]] (OFDM) for 5G, LTE, Wi-Fi, DSL, and other modern communication systems.

An original application of the FFT in [[finance]] particularly in the [[Valuation of options]] was developed by Marcello Minenna.<ref name="MinennaJBF">{{cite journal |title=A revisited and stable Fourier transform method for affine jump diffusion models |author-first1=Marcello |author-last1=Minenna |date=October 2008 |journal=Journal of Banking and Finance |doi=10.1016/j.jbankfin.2007.05.019|volume=32 |issue=10 |pages=2064–2075}}</ref>