Editing Fourier analysis (section)

{{Short description|Branch of mathematics}}
{{Use dmy dates|date=June 2020}}
{{Use American English|date = March 2019}}
[[File:Bass Guitar Time Signal of open string A note (55 Hz).png|thumb|upright=1.5| Bass guitar time signal of open string A note (55 Hz).]]
[[File:Fourier Transform of bass guitar time signal.png|thumb|upright=1.5| Fourier transform of bass guitar time signal of open string A note (55 Hz). Fourier analysis reveals the oscillatory components of signals and [[Wave function|functions]].]]
{{Fourier transforms}}

In [[mathematics]], '''Fourier analysis''' ({{IPAc-en|ˈ|f|ʊr|i|eɪ|,_|-|i|ər}})<ref>{{Dictionary.com|Fourier}}</ref> is the study of the way general [[function (mathematics)|functions]] may be represented or approximated by sums of simpler [[trigonometric functions]]. Fourier analysis grew from the study of [[Fourier series]], and is named after [[Joseph Fourier]], who showed that representing a function as a [[Summation|sum]] of trigonometric functions greatly simplifies the study of [[heat transfer]].

The subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, the process of decomposing a function into [[Oscillation|oscillatory]] components is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as '''Fourier synthesis'''. For example, determining what component [[Frequency|frequencies]] are present in a musical note would involve computing the Fourier transform of a sampled musical note. One could then re-synthesize the same sound by including the frequency components as revealed in the Fourier analysis. In mathematics, the term ''Fourier analysis'' often refers to the study of both operations.

The decomposition process itself is called a [[Fourier transformation]]. Its output, the [[Fourier transform]], is often given a more specific name, which depends on the [[Domain of a function|domain]] and other properties of the function being transformed. Moreover, the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, and the general field is often known as [[harmonic analysis]]. Each [[Transform (mathematics)|transform]] used for analysis (see [[list of Fourier-related transforms]]) has a corresponding [[inverse function|inverse]] transform that can be used for synthesis.

To use Fourier analysis, data must be equally spaced. Different approaches have been developed for analyzing unequally spaced data, notably the [[least-squares spectral analysis]] (LSSA) methods that use a [[least squares]] fit of [[Sine wave|sinusoid]]s to data samples, similar to Fourier analysis.<ref>{{cite book | title = Variable Stars As Essential Astrophysical Tools | author = Cafer Ibanoglu | publisher = Springer | year = 2000 | isbn = 0-7923-6084-2 | url = https://books.google.com/books?id=QzGbOiZ3OnkC&q=vanicek+spectral+sinusoids&pg=PA269 }}</ref><ref name=birn>{{cite book | title = Observational Astronomy |author1=D. Scott Birney |author2=David Oesper |author3=Guillermo Gonzalez | publisher = Cambridge University Press | year = 2006 | isbn = 0-521-85370-2 | url = https://books.google.com/books?id=cc9L8QWcZWsC&q=Lomb-Scargle-periodogram&pg=RA3-PA263  }}</ref> Fourier analysis, the most used spectral method in science, generally boosts long-periodic noise in long gapped records; LSSA mitigates such problems.<ref name=pres>{{cite book | url = https://books.google.com/books?id=9GhDHTLzFDEC&q=%22spectral+analysis%22+%22vanicek%22+inauthor:press&pg=PA685 | author = Press | title = Numerical Recipes | edition = 3rd | year = 2007 | publisher = Cambridge University Press | isbn = 978-0-521-88068-8 }}</ref>