Editing Fourier series (section)

=== Beginnings ===
Joseph Fourier wrote<ref>{{cite book |title=Oeuvres de Fourier|date= 1890| pages= 218–219|last= Fourier|first= Jean-Baptiste-Joseph|authorlink= Jean-Baptiste-Joseph Fourier |editor=Gaston Darboux|chapter=Mémoire sur la propagation de la chaleur dans les corps solides, présenté le 21 Décembre 1807 à l'Institut national| publisher=Gauthier-Villars et Fils|location=Paris |language=fr |isbn=9781139568159|volume=2|trans-title=The Works of Fourier |doi=10.1017/CBO9781139568159.009}} {{pb}} Whilst the cited article does list the author as Fourier, a footnote on page 215 indicates that the article was actually written by [[Siméon_Denis_Poisson|Poisson]] and that it is, "for reasons of historical interest", presented as though it were Fourier's original memoire.</ref>

{{blockquote|<math>\varphi(y)=a_0\cos\frac{\pi y}{2}+a_1\cos 3\frac{\pi y}{2}+a_2\cos5\frac{\pi y}{2}+\cdots.</math>

Multiplying both sides by <math>\cos(2k+1)\frac{\pi y}{2}</math>, and then integrating from <math>y=-1</math> to <math>y=+1</math> yields:

<math>a_k=\int_{-1}^1\varphi(y)\cos(2k+1)\frac{\pi y}{2}\,dy.</math>
|Joseph Fourier|[[Mémoire sur la propagation de la chaleur dans les corps solides]] (1807).}}

This immediately gives any coefficient ''a<sub>k</sub>'' of the [[trigonometric series]] for φ(''y'') for any function which has such an expansion. It works because if φ has such an expansion, then (under suitable convergence assumptions) the integral
<math display="block">\begin{align}
&\int_{-1}^1\varphi(y)\cos(2k+1)\frac{\pi y}{2}\,dy \\
&= \int_{-1}^1\left(a\cos\frac{\pi y}{2}\cos(2k+1)\frac{\pi y}{2}+a'\cos 3\frac{\pi y}{2}\cos(2k+1)\frac{\pi y}{2}+\cdots\right)\,dy
\end{align}</math>
can be carried out term-by-term. But all terms involving <math>\cos(2j+1)\frac{\pi y}{2} \cos(2k+1)\frac{\pi y}{2}</math> for {{nowrap|''j'' &ne; ''k''}} vanish when integrated from −1 to 1, leaving only the <math>k^{\text{th}}</math> term, which is ''1''.

In these few lines, which are close to the modern [[Formalism (mathematics)|formalism]] used in Fourier series, Fourier revolutionized both mathematics and physics. Although similar trigonometric series were previously used by [[Euler]], [[Jean le Rond d'Alembert|d'Alembert]], [[Daniel Bernoulli]] and [[Carl Friedrich Gauss|Gauss]], Fourier believed that such trigonometric series could represent any arbitrary function. In what sense that is actually true is a somewhat subtle issue and the attempts over many years to clarify this idea have led to important discoveries in the theories of [[Convergent series|convergence]], [[function space]]s, and [[harmonic analysis]].

When Fourier submitted a later competition essay in 1811, the committee (which included [[Joseph Louis Lagrange|Lagrange]], [[Laplace]], [[Étienne-Louis Malus|Malus]] and [[Adrien-Marie Legendre|Legendre]], among others) concluded: ''...the manner in which the author arrives at these equations is not exempt of difficulties and...his analysis to integrate them still leaves something to be desired on the score of generality and even [[Mathematical rigour|rigour]]''.<ref>{{cite book |title=Oeuvres de Fourier|date= 1888|last= Fourier|first= Jean-Baptiste-Joseph|authorlink= Jean-Baptiste-Joseph Fourier |editor=Gaston Darboux|chapter=Avant-propos des oevres de Fourier| publisher=Gauthier-Villars et Fils|location=Paris |language=fr |volume=1|trans-title=The Works of Fourier  | isbn=978-1-108-05938-1 | doi=10.1017/cbo9781139568081.001 |pages=VII-VIII}}</ref>