Editing Fourier analysis (section)

==History==
{{See also|Fourier series#Historical development}}

An early form of harmonic series dates back to ancient [[Babylonian mathematics]], where they were used to compute [[ephemerides]] (tables of astronomical positions).<ref name=Prestini/><ref name=Rota/><ref name=Neugebauer/><ref name=Brack/>

The Classical Greek concepts of [[deferent and epicycle]] in the [[Ptolemaic system]] of astronomy were related to Fourier series (see {{slink|Deferent and epicycle|Mathematical formalism}}).

In modern times, variants of the discrete Fourier transform were used by [[Alexis Clairaut]] in 1754 to compute an orbit,<ref name=Terras/>
which has been described as the first formula for the DFT,<ref name=thedft/>
and in 1759 by [[Joseph Louis Lagrange]], in computing the coefficients of a trigonometric series for a vibrating string.<ref name=thedft/> Technically, Clairaut's work was a cosine-only series (a form of [[discrete cosine transform]]), while Lagrange's work was a sine-only series (a form of [[discrete sine transform]]); a true cosine+sine DFT was used by [[Carl Friedrich Gauss|Gauss]] in 1805 for [[trigonometric interpolation]] of [[asteroid]] orbits.<ref name=Heideman84/>
Euler and Lagrange both discretized the vibrating string problem, using what would today be called samples.<ref name=thedft/>

An early modern development toward Fourier analysis was the 1770 paper ''[[Réflexions sur la résolution algébrique des équations]]'' by Lagrange, which in the method of [[Lagrange resolvents]] used a complex Fourier decomposition to study the solution of a cubic''':'''<ref name=Knapp/>
Lagrange transformed the roots <math>x_1,</math> <math>x_2,</math> <math>x_3</math> into the resolvents''':'''
<!-- equation to clarify connection; instantly recognizable if familiar with DFT matrix -->
:<math>\begin{align}
r_1 &= x_1 + x_2 + x_3\\
r_2 &= x_1 + \zeta x_2 + \zeta^2 x_3\\
r_3 &= x_1 + \zeta^2 x_2 + \zeta x_3
\end{align}</math>
where {{mvar|ζ}} is a cubic [[root of unity]], which is the DFT of order 3.

A number of authors, notably [[Jean le Rond d'Alembert]], and [[Carl Friedrich Gauss]] used [[trigonometric series]] to study the [[heat equation]],<ref name=Narasimhan/> but the breakthrough development was the 1807 paper ''[[Mémoire sur la propagation de la chaleur dans les corps solides]]'' by [[Joseph Fourier]], whose crucial insight was to model ''all'' functions by trigonometric series, introducing the Fourier series. Independently of Fourier, astronomer [[Friedrich Wilhelm Bessel]] also introduced Fourier series to solve [[Kepler's equation]]. His work was published in 1819, unaware of Fourier's work which remained unpublished until 1822.<ref>{{cite journal |last1=Dutka |first1=Jacques |date=1995 |title=On the early history of Bessel functions |journal=Archive for History of Exact Sciences |volume=49 |issue=2 |pages=105–134 |doi=10.1007/BF00376544}}</ref>

Historians are divided as to how much to credit Lagrange and others for the development of Fourier theory''':''' [[Daniel Bernoulli]] and [[Leonhard Euler]] had introduced trigonometric representations of functions, and Lagrange had given the Fourier series solution to the wave equation, so Fourier's contribution was mainly the bold claim that an arbitrary function could be represented by a Fourier series.<ref name=thedft/> 

The subsequent development of the field is known as [[harmonic analysis]], and is also an early instance of [[representation theory]].

The first fast Fourier transform (FFT) algorithm for the DFT was discovered around 1805 by [[Carl Friedrich Gauss]] when interpolating measurements of the orbit of the asteroids [[3 Juno|Juno]] and [[2 Pallas|Pallas]], although that particular FFT algorithm is more often attributed to its modern rediscoverers [[Cooley–Tukey FFT algorithm|Cooley and Tukey]].<ref name=Heideman84/><ref name=Terras/>