View source for Fourier series

{{Short description|Decomposition of periodic functions into sums of simpler sinusoidal forms}}
A '''Fourier series''' ({{IPAc-en|ˈ|f|ʊr|i|eɪ|,_|-|i|ər}}<ref>{{Dictionary.com|Fourier}}</ref>) is an [[Series expansion|expansion]] of a [[Periodic function|periodic function]] into a sum of [[Trigonometric functions|trigonometric functions]].  The Fourier series is an example of a [[Trigonometric series|trigonometric series]], but not all trigonometric series are Fourier series.<ref>{{cite book | last1 = Zygmund| first1 = A. | title = Trigonometric Series | publisher =Cambridge University Press | edition =3nd | date =2002 | location =Cambridge, UK | isbn =0-521-89053-5 }}</ref> By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by [[Biography:Joseph Fourier|Joseph Fourier]] to find solutions to the [[Heat equation|heat equation]]. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always [[Convergent series|converge]]. Well-behaved functions, for example [[Smoothness|smooth]] functions, have Fourier series that converge to the original function.  The coefficients of the Fourier series are determined by [[Integral|integral]]s of the function multiplied by trigonometric functions, described in [[Fourier series#Common forms of the Fourier series|Common forms of the Fourier series]] below.

The study of the [[Convergence of Fourier series|convergence of Fourier series]] focus on the behaviors of the ''partial sums'', which means studying the behavior of the sum as more and more terms from the series are summed. The figures below illustrate some partial Fourier series results for the components of a [[Square wave|square wave]].

<gallery widths="224" heights="224">
File:SquareWaveFourierArrows,rotated,nocaption 20fps.gif|A square wave (represented as the blue dot) is approximated by its sixth partial sum (represented as the purple dot), formed by summing the first six terms (represented as arrows) of the square wave's Fourier series. Each arrow starts at the vertical sum of all the arrows to its left (i.e. the previous partial sum).
File:Fourier Series.svg|The first four partial sums of the Fourier series for a [[Square wave|square wave]]. As more harmonics are added, the partial sums ''converge to'' (become more and more like) the square wave.
File:Fourier series and transform.gif|Function <math>s_6(x)</math> (in red) is a Fourier series sum of 6 harmonically related sine waves (in blue). Its Fourier transform <math>S(f)</math> is a frequency-domain representation that reveals the amplitudes of the summed sine waves.
</gallery>

Fourier series are closely related to the [[Fourier transform]], which can be used to find the frequency information for functions that are not periodic. Periodic functions can be identified with functions on a circle, for this reason Fourier series are the subject of [[Fourier analysis]] on a circle, usually denoted as <math>\mathbb{T}</math> or <math>S_1</math>. The Fourier transform is also part of [[Fourier analysis]], but is defined for functions on <math>\mathbb{R}^n</math>.

Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for [[Real number|real]]-valued functions of real arguments, and used the [[Sine and cosine|sine and cosine functions]] in the decomposition. Many other [[List of Fourier-related transforms|Fourier-related transforms]] have since been defined, extending his initial idea to many applications and birthing an [[Areas of mathematics|area of mathematics]] called [[Fourier analysis]].

==Common forms of the Fourier series==
A Fourier series is a continuous, [[Periodic function|periodic function]] created by a summation of harmonically related sinusoidal functions.  It has several different, but equivalent, forms, shown here as partial sums.  But in theory <math>N \rightarrow \infty.</math>  The subscripted symbols, called ''coefficients'', and the period, <math>P,</math> determine the function <math>s_{\scriptscriptstyle N}(x)</math> as follows''':'''
[[File:Fourier_series_illustration.svg|right|thumb|400x400px|Fig 1. The top graph shows a non-periodic function ''s''(''x'') in blue defined only over the red interval from ''0'' to ''P''. The function can be analyzed over this interval to produce the Fourier series in the bottom graph. The Fourier series is always a periodic function, even if original function ''s''(''x'') isn't.]]

{{Equation box 1 |title=Fourier series, amplitude-phase form
|indent=:|border|border colour=#0073CF|background colour=#F5FFFA|cellpadding=6
|equation={{NumBlk|
|<math>s_{_N }(x)=D_0 + \sum_{n=1}^N D_n \cos\left( 2\pi \tfrac{ n }{ P } x - \varphi_n\right)</math>
|{{EquationRef|Eq.1}}}}}}<br/>

{{Equation box 1 |title=Fourier series, sine-cosine form
|indent=:|border|border colour=#0073CF|background colour=#F5FFFA|cellpadding=6
|equation={{NumBlk|
|<math>s_{_N}(x) = A_0 + \sum_{n=1}^N \left( 
A_n \cos \left(2 \pi \tfrac{n}{P} x \right) + 
B_n \sin \left(2 \pi \tfrac{n}{P} x  \right) \right)</math>
|{{EquationRef|Eq.2}}}}}}<br/>

{{Equation box 1 |title=Fourier series, exponential form
|indent=:|border|border colour=#0073CF|background colour=#F5FFFA|cellpadding=6
|equation={{NumBlk|
|<math>s_{_N}(x) = \sum_{n=-N}^N C_n \ e^{i 2\pi \tfrac{n}{P}x}</math>
|{{EquationRef|Eq.3}}}}}}

The harmonics are indexed by an integer, <math>n,</math> which is also the number of cycles the corresponding sinusoids make in interval <math> P</math>. Therefore, the sinusoids have''':'''

* a [[Wavelength|wavelength]] equal to <math>\tfrac{P}{n}</math> in the same units as <math>x</math>.
* a [[Physics:Frequency|frequency]] equal to <math>\tfrac{n}{P}</math> in the reciprocal units of <math>x</math>.

Clearly these series can represent functions that are just a sum of one or more of the harmonic frequencies.  The remarkable thing is that it can also represent the intermediate frequencies and/or non-sinusoidal functions because of the infinite number of terms.  The amplitude-phase form is particularly useful for its insight into the rationale for the series coefficients. (see {{slink|#Derivation|nopage=y}})  The exponential form is most easily generalized for complex-valued functions.  (see {{slink||Complex-valued functions|nopage=y}})

The equivalence of these forms requires certain relationships among the coefficients.  For instance, the trigonometric identity''':'''
{{Equation box 1|title=Equivalence of polar and rectangular forms
|indent=:|border|border colour=#0073CF|background colour=#F5FFFA|cellpadding=6
|equation={{NumBlk||<math> \cos\left(2\pi \tfrac{n}{P}x-\varphi_n\right) \ \equiv
\  \cos(\varphi_n)\cdot \cos\left(2\pi \tfrac{n}{P} x\right) 
+  \sin(\varphi_n)\cdot \sin\left(2\pi \tfrac{n}{P} x\right)</math>
|{{EquationRef|Eq.4}}}}}}
means that''':'''

{{Equation box 1|title=
|indent=:|border|border colour=#0073CF|background colour=#F5FFFA|cellpadding=6
|equation={{NumBlk||
<math>\begin{align}
&A_n = D_n \cos(\varphi_n)\quad \text{and}\quad B_n = D_n \sin(\varphi_n)\\
\\
&D_n = \sqrt{A_n^2 + B_n^2}\quad \text{and}\quad \varphi_n = \arctan(B_n, A_n).
\end{align}</math>
|{{EquationRef|Eq.4.1}}}}}}

Therefore <math>A_n</math> and <math>B_n</math> are the rectangular coordinates of a vector with [[Polar coordinate system|polar coordinates]] <math>D_n</math> and <math>\varphi_n.</math>

The coefficients can be given/assumed, such as a music synthesizer or time samples of a waveform.  In the latter case, the exponential form of Fourier series synthesizes a [[Discrete-time Fourier transform|discrete-time Fourier transform]] where variable <math>x</math> represents frequency instead of time.

But typically the coefficients are determined by frequency/harmonic '''''analysis''''' of a given real-valued function <math>s(x),</math> and <math>x</math> represents time''':'''
{{Equation box 1|title=Fourier series analysis
|indent=:|border|border colour=#0073CF|background colour=#F5FFFA|cellpadding=6
|equation={{NumBlk|
|<math>
\begin{aligned}
A_0 &= \frac{1}{P}\int_P s(x)\,dx \\
A_n &= \frac{2}{P}\int_P s(x) \cos \left( 2\pi \tfrac{n}{P} x \right) \,dx\qquad\text{for } n\geq 1\qquad\\
B_n &= \frac{2}{P}\int_P s(x) \sin \left( 2\pi \tfrac{n}{P} x \right) dx,\qquad\text{for } n\geq 1
\end{aligned}</math>
|{{EquationRef|Eq.5}}}}}}

The objective is for <math> s_{\scriptstyle{\infty}}</math> to converge to <math>s(x)</math> at most or all values of <math>x</math> in an interval of length <math>P.</math>    For the well-behaved functions typical of physical processes, equality is customarily assumed, and the [[Dirichlet conditions]] provide sufficient conditions.

The notation<math>\int_P</math> represents integration over the chosen interval.  Typical choices are <math>[-P/2, P/2]</math>  and <math>[0, P]</math>'''.'''  Some authors define <math>P \triangleq 2 \pi</math> because it simplifies the arguments of the sinusoid functions, at the expense of generality.  And some authors assume that <math>s(x)</math> is also <math>P</math>-periodic, in which case <math> s_{\scriptstyle{\infty}}</math> approximates the entire function.  The <math>\tfrac{2}{P}</math> scaling factor is explained by taking a simple case''':''' <math>s(x) = \cos \left( 2\pi \tfrac{k}{P} x \right).</math>  Only the <math>n=k</math> term of {{EquationNote|Eq.2}} is needed for convergence, with <math>A_k =1</math> and <math>B_k = 0.</math>&nbsp; Accordingly {{EquationNote|Eq.5}} provides''':'''

:<math>A_k = \frac{2}{P} \underbrace{\int_P \cos^2 \left( 2\pi \tfrac{k}{P} x \right) \,dx}_{P/2} = 1,</math>  &nbsp;  &nbsp;  &nbsp; as required.

=== Exponential form coefficients ===
Another applicable identity is [[Euler's formula]]''':'''

:<math>
\begin{align}
\cos\left(2\pi \tfrac{n}{P} x - \varphi_n \right) &{}\equiv \tfrac{1}{2} e^{ i \left(2\pi \tfrac{n}{P}x - \varphi_n \right)} + \tfrac{1}{2} e^{-i \left(2\pi \tfrac{n}{P}x - \varphi_n \right)} \\[6pt]
& = \left(\tfrac{1}{2} e^{-i \varphi_n}\right) \cdot e^{i 2\pi \tfrac{+n}{P}x} 
+\left(\tfrac{1}{2} e^{-i \varphi_n}\right)^* \cdot e^{i 2\pi \tfrac{-n}{P}x}
\end{align}
</math>

(Note''':''' the ∗ denotes [[Complex conjugate|complex conjugation]].)

Substituting this into {{EquationNote|Eq.1}} and comparison with {{EquationNote|Eq.3}} ultimately reveals''':'''

{{Equation box 1|title=Exponential form coefficients
|indent=:|border|border colour=#0073CF|background colour=#F5FFFA|cellpadding=6
|equation={{NumBlk||<math>C_n \triangleq \left\{
\begin{array}{lll}
A_0,  \quad  && n = 0\\
\tfrac{ D_n}{2} e^{-i \varphi_n} &= \tfrac{1}{2}(A_n -i B_n), \quad & n > 0\\
C_{|n|}^*, \quad && n < 0
\end{array} \right\}</math>
|{{EquationRef|Eq.6}}}}}}

Conversely''':'''
{{Equation box 1|title=Inverse relationships
|indent=:|border|border colour=#0073CF|background colour=#F5FFFA|cellpadding=6
|equation=<math>\begin{aligned} A_0 &= C_0 &\\
                                    A_n &= C_n+C_{-n} \qquad &\textrm{for}~ n > 0 \\
                                    B_n &= i(C_n-C_{-n}) \qquad &\textrm{for}~ n > 0 \end{aligned}</math>
}}

Substituting {{EquationNote|Eq.5}} into {{EquationNote|Eq.6}} also reveals''':'''<ref>{{cite book |last1=Pinkus |first1=Allan |title=Fourier Series and Integral Transforms |last2=Zafrany |first2=Samy |date=1997 |publisher=Cambridge University Press |isbn=0-521-59771-4 |edition=1st |location=Cambridge, UK |pages=42–44}}</ref>
{{Equation box 1|title=Fourier series analysis
|indent=:|border|border colour=#0073CF|background colour=#F5FFFA|cellpadding=6
|equation={{NumBlk||<math>C_n = \frac{1}{P}\int_P s(x)e^{-i 2\pi \tfrac{n}{P} x}\, dx; \quad \forall\ n \in \mathbb Z\,</math> ([[Number#Integers|all integers]])
|{{EquationRef|Eq.7}}}}}}

=== Complex-valued functions ===
{{EquationNote|Eq.7}} and {{EquationNote|Eq.3}} also apply when <math>s(x)</math> is a complex-valued function.{{efn-ua |But <math>C_n \ne C_{ -n }^*</math>, in general.}}
This follows by expressing <math>\operatorname{Re}(s_N (x))</math> and <math>\operatorname{Im}(s_N(x))</math> as separate real-valued Fourier series, and <math>s_N(x) = \operatorname{Re}(s_N(x)) + i\ \operatorname{Im}(s_N(x)).</math>

=== Derivation ===
The coefficients <math>D_n</math> and <math>\varphi_n</math> can be understood and derived in terms of the [[Cross-correlation|cross-correlation]] between <math>s(x)</math> and a sinusoid at frequency <math>\tfrac{n}{P}</math>.  For a general frequency <math>f,</math> and an analysis interval <math>[x_0,x_0+P],</math> the cross-correlation function''':'''
[[File:Correlation_function.svg|right|thumb|300px|Fig 2. The blue curve is the cross-correlation of a square wave and a cosine function, as the phase lag of the cosine varies over one cycle. The amplitude and phase lag at the maximum value are the polar coordinates of one harmonic in the Fourier series expansion of the square wave. The corresponding rectangular coordinates can be determined by evaluating the cross-correlation at just two phase lags separated by 90º.]]

{{Equation box 1
|title=Derivation of Eq.1
|indent=:|border|border colour=#0073CF|background colour=#F5FFFA|cellpadding=6
|equation={{NumBlk|
|<math>\Chi_f(\tau) = \tfrac{2}{P}\int_{x_0}^{x_0+P} s(x) \cdot \cos\left( 2\pi f (x-\tau) \right)\, dx; 
\quad \tau \in \left[ 0, \tfrac{2\pi}{f} \right]</math>
|{{EquationRef|Eq.8}}}}}}

is essentially a [[Matched filter|matched filter]], with ''template'' <math>\cos(2\pi f x)</math>.
<!--{{efn-ua
|The scale factor <math>\tfrac{2}{P},</math> which could be inserted later, results in a series that converges to <math>s(x)</math> instead of <math>\tfrac{P}{2}s(x).</math>}}-->
The [[Maxima and minima|maximum]] of <math>\Chi_f(\tau)</math> is a measure of the amplitude <math>(D)</math> of frequency <math>f</math> in the function <math>s(x)</math>, and the value of <math>\tau</math> at the maximum determines the phase <math>(\varphi)</math> of that frequency.  Figure 2 is an example, where <math>s(x)</math> is a square wave (not shown), and frequency <math>f</math> is the <math>4^{\text{th}}</math> harmonic.  It is also an example of deriving the maximum from just two samples, instead of searching the entire function.  Combining  {{EquationNote|Eq.8}} with {{EquationNote|Eq.4}} gives''':'''
:<math>\begin{align}
\Chi_n(\varphi) &= \tfrac{2}{P} \int_P s(x) \cdot \cos\left(2\pi \tfrac{n}{P} x-\varphi \right)\, dx
; \quad \varphi \in [0, 2\pi]\\
&=\cos(\varphi)\cdot \underbrace{\tfrac{2}{P}\int_P s(x) \cdot \cos\left(2\pi \tfrac{n}{P} x\right)\, dx}_{A}
+\sin(\varphi)\cdot \underbrace{\tfrac{2}{P}\int_P s(x) \cdot \sin\left(2\pi \tfrac{n}{P} x\right)\, dx}_{B}\\
&=\cos(\varphi)\cdot A + \sin(\varphi)\cdot B
\end{align}</math>

The derivative of <math>\Chi_n(\varphi)</math> is zero at the phase of maximum correlation.

:<math>\Chi'_n(\varphi)=\sin(\varphi)\cdot A - \cos(\varphi)\cdot B = 0
\quad \longrightarrow\quad \tan(\varphi) = \frac{B}{A} \quad \longrightarrow\quad \varphi = \arctan(B, A)</math>

Therefore, computing <math>A_n</math> and <math>B_n</math> according to {{EquationNote|Eq.5}} creates the component's phase <math>\varphi_n</math> of maximum correlation.  And the component's amplitude is''':'''
:<math>
\begin{align}
D_n \triangleq \Chi_n(\varphi_n)\ &= \cos(\varphi_n)\cdot A_n + \sin(\varphi_n)\cdot B_n \\
&=\frac{A_n}{\sqrt{A_n^2+B_n^2}}\cdot A_n + \frac{B _n}{\sqrt{A_n^2+B_n^2}}\cdot B_n =\frac{A_n^2+B_n^2}{\sqrt{A_n^2+B_n^2}}
&= \sqrt{A_n^2+B_n^2}.
\end{align}
</math>

===Other common notations===
The notation <math>C_n</math> is inadequate for discussing the Fourier coefficients of several different functions. Therefore, it is customarily replaced by a modified form of the function {{nowrap|(<math>s,</math> in this case),}} such as <math>\widehat{s}(n)</math> or <math>S[n]</math>, and functional notation often replaces subscripting''':'''

:<math>\begin{align}
s(x) &= \sum_{n=-\infty}^\infty \widehat{s}(n)\cdot e^{i 2\pi \tfrac{n}{P} x} &&  \scriptstyle \text{common mathematics notation} \\
&= \sum_{n=-\infty}^\infty S[n]\cdot e^{i 2\pi \tfrac{n}{P} x} &&  \scriptstyle \text{common engineering notation}
\end{align}</math>

In engineering, particularly when the variable <math>x</math> represents time, the coefficient sequence is called a [[Frequency domain|frequency domain]] representation. Square brackets are often used to emphasize that the domain of this function is a discrete set of frequencies.

Another commonly used frequency domain representation uses the Fourier series coefficients to [[Engineering:Modulation|modulate]] a [[Dirac comb]]:
:<math>S(f) \ \triangleq \ \sum_{n=-\infty}^\infty S[n]\cdot \delta \left(f-\frac{n}{P}\right),</math>

where <math>f</math> represents a continuous frequency domain. When variable <math>x</math> has units of seconds, <math>f</math> has units of [[Hertz|hertz]]. The "teeth" of the comb are spaced at multiples (i.e. harmonics) of <math>\tfrac{1}{P}</math>, which is called the [[Physics:Fundamental frequency|fundamental frequency]]. <math>s_{\infty}(x)</math> can be recovered from this representation by an [[Fourier inversion theorem|inverse Fourier transform]]:

:<math>\begin{align}
\mathcal{F}^{-1}\{S(f)\} &= \int_{-\infty}^\infty \left( \sum_{n=-\infty}^\infty S[n]\cdot \delta \left(f-\frac{n}{P}\right)\right) e^{i 2 \pi f x}\,df, \\[6pt]
&= \sum_{n=-\infty}^\infty S[n]\cdot \int_{-\infty}^\infty \delta\left(f-\frac{n}{P}\right) e^{i 2 \pi f x}\,df, \\[6pt]
&= \sum_{n=-\infty}^\infty S[n]\cdot e^{i 2\pi \tfrac{n}{P} x} \ \ \triangleq \ s_\infty(x).
\end{align}</math>

The constructed function <math>S(f)</math> is therefore commonly referred to as a '''Fourier transform''', even though the Fourier integral of a periodic function is not convergent at the harmonic frequencies.{{efn-ua|
Since the integral defining the Fourier transform of a periodic function is not convergent, it is necessary to view the periodic function and its transform as [[Distribution (mathematics)|distributions]]. In this sense <math>\mathcal{F} \{ e^{i 2\pi \tfrac{n}{P} x} \}</math> is a [[Dirac delta function]], which is an example of a distribution.
}}

=== Analysis example ===
[[File:sawtooth pi.svg|thumb|right|400px|Plot of the [[Sawtooth wave|sawtooth wave]], a periodic continuation of the linear function <math>s(x)=x/\pi</math> on the interval <math>(-\pi,\pi]</math>]]

[[File:Periodic identity function.gif|thumb|right|400px|Animated plot of the first five successive partial Fourier series]]

Consider a sawtooth function''':'''
:<math>s(x) = \frac{x}{\pi}, \quad \mathrm{for } -\pi < x < \pi,</math>
:<math>s(x + 2\pi k) = s(x), \quad \mathrm{for } -\pi < x < \pi \text{ and } k \in \mathbb{Z}.</math>
In this case, the Fourier coefficients are given by
:<math>\begin{align}
A_n & = \frac{1}{\pi}\int_{-\pi}^{\pi}s(x) \cos(nx)\,dx = 0, \quad n \ge 0. \\[4pt]
B_n & = \frac{1}{\pi}\int_{-\pi}^{\pi}s(x) \sin(nx)\, dx\\[4pt]
&= -\frac{2}{\pi n}\cos(n\pi) + \frac{2}{\pi^2 n^2}\sin(n\pi)\\[4pt]
&= \frac{2\,(-1)^{n+1}}{\pi n}, \quad n \ge 1.\end{align}</math>
It can be shown that the Fourier series converges to <math>s(x)</math> at every point <math>x</math> where <math>s</math> is differentiable, and therefore''':'''
{{NumBlk|:
|<math>\begin{align}
s(x) &= \frac{a_0}{2} + \sum_{n=1}^\infty \left[A_n\cos\left(nx\right)+B_n\sin\left(nx\right)\right] \\[4pt]
&=\frac{2}{\pi}\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \sin(nx), \quad \mathrm{for}\ (x-\pi)\ \text{is not a multiple of}\ 2\pi.
\end{align}</math>
|{{EquationRef|Eq.9}}
}}

When <math>x=\pi</math>, the Fourier series converges to 0, which is the half-sum of the left- and right-limit of ''s'' at <math>x=\pi</math>. This is a particular instance of the [[Convergence of Fourier series#Convergence at a given point|Dirichlet theorem]] for Fourier series.

This example leads to a solution of the [[Basel problem]].

=== Convergence ===
{{main|Convergence of Fourier series}}
A proof that a Fourier series is a valid representation of any periodic function (that satisfies the [[Dirichlet conditions]])  is overviewed in {{slink||Fourier theorem proving convergence of Fourier series}}.

In [[Engineering:Engineering|engineering]] applications, the Fourier series is generally assumed to converge except at jump discontinuities since the functions encountered in engineering are better-behaved than functions encountered in other disciplines. In particular, if <math>s</math> is continuous and the derivative of <math>s(x)</math> (which may not exist everywhere) is square integrable, then the Fourier series of <math>s</math> converges absolutely and uniformly to <math>s(x)</math>.<ref>{{cite book|last=Tolstov|first=Georgi P.|url=https://books.google.com/books?id=XqqNDQeLfAkC&q=fourier-series+converges+continuous-function&pg=PA82|title=Fourier Series|publisher=Courier-Dover|year=1976|isbn=0-486-63317-9}}</ref> If a function is [[Square-integrable function|square-integrable]] on the interval <math>[x_0,x_0+P]</math>, then the Fourier series [[Carleson's theorem|converges]] to the function at [[Almost everywhere|almost everywhere]]. It is possible to define Fourier coefficients for more general functions or distributions, in which case point wise convergence often fails, and convergence in norm or [[Weak convergence (Hilbert space)|weak convergence]] is usually studied.

<gallery widths="224" heights="224">
Fourier_series_square_wave_circles_animation.gif|link=//upload.wikimedia.org/wikipedia/commons/b/bd/Fourier_series_square_wave_circles_animation.svg|Four partial sums (Fourier series) of lengths 1, 2, 3, and 4 terms, showing how the approximation to a square wave improves as the number of terms increases [{{filepath:Fourier_series_square_wave_circle_animation.svg}} (animation)]
Fourier_series_sawtooth_wave_circles_animation.gif|link=//upload.wikimedia.org/wikipedia/commons/1/1e/Fourier_series_sawtooth_wave_circles_animation.svg|Four partial sums (Fourier series) of lengths 1, 2, 3, and 4 terms, showing how the approximation to a sawtooth wave improves as the number of terms increases [{{filepath:Fourier_series_sawtooth_wave_circles_animation.svg}} (animation)]
Example_of_Fourier_Convergence.gif |Example of convergence to a somewhat arbitrary function. Note the development of the "ringing" ([[Gibbs phenomenon]]) at the transitions to/from the vertical sections.
</gallery>

==History{{anchor|Historical development}}==

The Fourier series is named in honor of Jean-Baptiste Joseph Fourier (1768–1830), who made important contributions to the study of [[Trigonometric series|trigonometric series]], after preliminary investigations by [[Biography:Leonhard Euler|Leonhard Euler]], [[Biography:Jean le Rond d'Alembert|Jean le Rond d'Alembert]], and [[Biography:Daniel Bernoulli|Daniel Bernoulli]].{{efn-ua|
These three did some [[Physics:Wave equation#Notes|important early work on the wave equation]], especially D'Alembert. Euler's work in this area was mostly [[Physics:Euler–Bernoulli beam theory|comtemporaneous/ in collaboration with Bernoulli]], although the latter made some independent contributions to the theory of waves and vibrations. (See {{harvnb|Fetter|Walecka|2003|pp=209–210}}).
}} Fourier introduced the series for the purpose of solving the [[Heat equation|heat equation]] in a metal plate, publishing his initial results in his 1807 ''Mémoire sur la propagation de la chaleur dans les corps solides'' (''Treatise on the propagation of heat in solid bodies''), and publishing his ''Théorie analytique de la chaleur'' (''Analytical theory of heat'') in 1822. The ''Mémoire'' introduced Fourier analysis, specifically Fourier series. Through Fourier's research the fact was established that an arbitrary (at first, continuous<ref name="Stillwell2013">{{cite book|last=Stillwell|first=John|title=Routledge History of Philosophy|publisher=Routledge|year=2013|isbn=978-1-134-92880-4|editor-last=Ten|editor-first=C. L.|volume=VII: The Nineteenth Century|page=204|chapter=Logic and the philosophy of mathematics in the nineteenth century|chapter-url=https://books.google.com/books?id=91AqBgAAQBAJ&pg=PA204}}</ref> and later generalized to any [[Piecewise|piecewise]]-smooth<ref name="iit.edu" />) function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier in 1807, before the [[Organization:Académie française|French Academy]].<ref name="Cajori1893">{{cite book|last=Cajori|first=Florian|url=https://archive.org/details/ahistorymathema00cajogoog|title=A History of Mathematics|publisher=Macmillan|year=1893|page=[https://archive.org/details/ahistorymathema00cajogoog/page/n254 283]|author-link=Florian Cajori}}</ref> Early ideas of decomposing a periodic function into the sum of simple oscillating functions date back to the 3rd century BC, when ancient astronomers proposed an empiric model of planetary motions, based on [[Astronomy:Deferent and epicycle|deferents and epicycles]].

The [[Heat equation|heat equation]] is a [[Partial differential equation|partial differential equation]]. Prior to Fourier's work, no solution to the heat equation was known in the general case, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a [[Sine|sine]] or [[Cosine|cosine]] wave. These simple solutions are now sometimes called eigensolutions. Fourier's idea was to model a complicated heat source as a superposition (or [[Linear combination|linear combination]]) of simple sine and cosine waves, and to write the [[Superposition principle|solution as a superposition]] of the corresponding [[Eigenfunction|eigensolutions]]. This superposition or linear combination is called the Fourier series.

From a modern point of view, Fourier's results are somewhat informal, due to the lack of a precise notion of [[Function (mathematics)|function]] and [[Integral|integral]] in the early nineteenth century. Later, [[Biography:Peter Gustav Lejeune Dirichlet|Peter Gustav Lejeune Dirichlet]]<ref>{{cite journal|last=Lejeune-Dirichlet|first=Peter Gustav|year=1829|title=Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données|trans-title=On the convergence of trigonometric series which serve to represent an arbitrary function between two given limits|url=https://archive.org/details/arxiv-0806.1294|journal=Journal für die reine und angewandte Mathematik|language=fr|volume=4|pages=157–169|arxiv=0806.1294}}</ref> and [[Biography:Bernhard Riemann|Bernhard Riemann]]<ref>{{cite web|title=Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe|trans-title=About the representability of a function by a trigonometric series|url=http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Trig/|url-status=live|archive-url=https://web.archive.org/web/20080520085248/http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Trig/|archive-date=20 May 2008|access-date=19 May 2008|work=Habilitationsschrift, Göttingen; 1854. Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, vol. 13, 1867. Published posthumously for Riemann by [[Biography:Richard Dedekind|Richard Dedekind]]|language=de}}</ref><ref>{{citation|last1=Mascre|first1=D.|title=Landmark Writings in Western Mathematics 1640–1940|date=1867|url=https://books.google.com/books?id=UdGBy8iLpocC|page=49|publication-date=2005|editor-last=Grattan-Guinness|editor-first=Ivor|chapter=Posthumous Thesis on the Representation of Functions by Trigonometric Series|publisher=Elsevier|isbn=9780080457444|last2=Riemann|first2=Bernhard}}</ref><ref>{{cite book|last=Remmert|first=Reinhold|url=https://books.google.com/books?id=uP8SF4jf7GEC|title=Theory of Complex Functions: Readings in Mathematics|date=1991|publisher=Springer|isbn=9780387971957|page=29}}</ref> expressed Fourier's results with greater precision and formality.

Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems, and especially those involving linear differential equations with constant coefficients, for which the eigensolutions are [[Physics:Sine wave|sinusoid]]s. The Fourier series has many such applications in [[Electrical engineering|electrical engineering]], [[Oscillation|vibration]] analysis, [[Physics:Acoustics|acoustics]], [[Physics:Optics|optics]], [[Signal processing|signal processing]], [[Image processing|image processing]], [[Physics:Quantum mechanics|quantum mechanics]], [[Econometrics|econometrics]],<ref>{{cite book|last1=Nerlove|first1=Marc|url=https://archive.org/details/analysisofeconom0000nerl|title=Analysis of Economic Time Series. Economic Theory, Econometrics, and Mathematical Economics|last2=Grether|first2=David M.|last3=Carvalho|first3=Jose L.|publisher=Elsevier|year=1995|isbn=0-12-515751-7|url-access=registration}}</ref> [[Physics:Membrane theory of shells|shell theory]],<ref>Wilhelm Flügge, ''Stresses in Shells'' (1973) 2nd edition. {{isbn|978-3-642-88291-3}}. Originally published in German as ''Statik und Dynamik der Schalen'' (1937).</ref> etc.

=== Beginnings ===
Joseph Fourier wrote:{{dubious|reason=Did Fourier really write this in English?|date=February 2020}}

{{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)<ref>{{cite book|url=http://gallica.bnf.fr/ark:/12148/bpt6k33707.image.r=Oeuvres+de+Fourier.f223.pagination.langFR |title=Oeuvres de Fourier|date= 1888| pages= 218–219|last= Fourier|first= Jean-Baptiste-Joseph|editor=Gaston Darboux|publisher=Gauthier-Villars et Fils|location=Paris |language=fr |via=Gallica|trans-title=The Works of Fourier }}</ref>{{efn-ua|
These words are not strictly Fourier's. Whilst the cited article does list the author as Fourier, a footnote indicates that the article was actually written by Poisson (that it was not written by Fourier is also clear from the consistent use of the third person to refer to him) and that it is, "for reasons of historical interest", presented as though it were Fourier's original memoire.}}}}

This immediately gives any coefficient ''a<sub>k</sub>'' of the trigonometrical 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}
a_k&=\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.

In these few lines, which are close to the modern formalism used in Fourier series, Fourier revolutionized both mathematics and physics. Although similar trigonometric series were previously used by Euler, [[Biography:Jean le Rond d'Alembert|d'Alembert]], [[Biography:Daniel Bernoulli|Daniel Bernoulli]] and [[Biography: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|function space]]s, and [[Physics:Harmonic analysis|harmonic analysis]].

When Fourier submitted a later competition essay in 1811, the committee (which included Lagrange, Laplace, [[Biography:Étienne-Louis Malus|Malus]] and [[Biography: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 rigour''.{{citation needed|date=November 2012}}

===Fourier's motivation===
[[File:Fourier heat in a plate.png|thumb|right|Heat distribution in a metal plate, using Fourier's method]]
The Fourier series expansion of the sawtooth function (above) looks more complicated than the simple formula <math>s(x)=\tfrac{x}{\pi}</math>, so it is not immediately apparent why one would need the Fourier series. While there are many applications, Fourier's motivation was in solving the [[Heat equation|heat equation]]. For example, consider a metal plate in the shape of a square whose sides measure <math>\pi</math> meters, with coordinates <math>(x,y) \in [0,\pi] \times [0,\pi]</math>. If there is no heat source within the plate, and if three of the four sides are held at 0 degrees Celsius, while the fourth side, given by <math>y=\pi</math>, is maintained at the temperature gradient <math>T(x,\pi)=x</math> degrees Celsius, for <math>x</math> in <math>(0,\pi)</math>, then one can show that the stationary heat distribution (or the heat distribution after a long period of time has elapsed) is given by
: <math>T(x,y) = 2\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \sin(nx) {\sinh(ny) \over \sinh(n\pi)}.</math>
Here, sinh is the hyperbolic sine function. This solution of the heat equation is obtained by multiplying each term of {{EquationNote|Eq.9}} by <math>\sinh(ny)/\sinh(n\pi)</math>. While our example function <math>s(x)</math> seems to have a needlessly complicated Fourier series, the heat distribution <math>T(x,y)</math> is nontrivial. The function <math>T</math> cannot be written as a [[Closed-form expression|closed-form expression]]. This method of solving the heat problem was made possible by Fourier's work.

===Complex Fourier series animation===
[[File:Complex Fourier series tracing and stereo playing the letter 'e'.webm|thumb|Complex Fourier series tracing the letter 'e'. (The [[Julia (programming language)|Julia]] source code that generates the frames of this animation is here<ref>{{cite web |last1=Sepesi |first1=G |title=Zeno's Enduring Example |date=13 February 2022 |url=https://towardsdatascience.com/zenos-illustrative-example-bb371b99f25a?source=friends_link&sk=8a6f90c96adf5b4a28d223a4a0db7350 |publisher=Towards Data Science |pages=Appendix B}}</ref> in Appendix B.)]]

An example of the ability of the complex Fourier series to trace any two dimensional closed figure is shown in the adjacent animation of the complex Fourier series tracing the letter 'e' (for exponential). Note that the animation uses the variable 't' to parameterize the letter 'e' in the complex plane, which is equivalent to using the parameter 'x' in this article's subsection on complex valued functions.

In the animation's back plane, the rotating vectors are aggregated in an order that alternates between a vector rotating in the positive (counter clockwise) direction and a vector rotating at the same frequency but in the negative (clockwise) direction, resulting in a single tracing arm with lots of zigzags. This perspective shows how the addition of each pair of rotating vectors (one rotating in the positive direction and one rotating in the negative direction) nudges the previous trace (shown as a light gray dotted line) closer to the shape of the letter 'e'.

In the animation's front plane, the rotating vectors are aggregated into two sets, the set of all the positive rotating vectors and the set of all the negative rotating vectors (the non-rotating component is evenly split between the two), resulting in two tracing arms rotating in opposite directions. The animation's small circle denotes the midpoint between the two arms and also the midpoint between the origin and the current tracing point denoted by '+'. This perspective shows how the complex Fourier series is an extension (the addition of an arm) of the complex geometric series which has just one arm. It also shows how the two arms coordinate with each other. For example, as the tracing point is rotating in the positive direction, the negative direction arm stays parked. Similarly, when the tracing point is rotating in the negative direction, the positive direction arm stays parked.

In between the animation's back and front planes are rotating trapezoids whose areas represent the values of the complex Fourier series terms. This perspective shows the amplitude, frequency, and phase of the individual terms of the complex Fourier series in relation to the series sum spatially converging to the letter 'e' in the back and front planes. The audio track's left and right channels correspond respectively to the real and imaginary components of the current tracing point '+' but increased in frequency by a factor of 3536 so that the animation's fundamental frequency (n=1) is a 220&nbsp;Hz tone (A220).

===Other applications===
Another application is to solve the [[Basel problem]] by using [[Parseval's theorem]]. The example generalizes and one may compute [[Riemann zeta function|ζ]](2''n''), for any positive integer ''n''.

== Table of common Fourier series ==
Some common pairs of periodic functions and their Fourier series coefficients are shown in the table below.

*<math>s(x)</math> designates a periodic function with period <math>P</math>.
*<math>A_0, A_n, B_n</math> designate the Fourier series coefficients (sine-cosine form) of the periodic function <math>s(x)</math>.

{| class="wikitable"
|-
! Time domain <br /> <math>s(x)</math>
! Plot
! Frequency domain (sine-cosine form) <br /> <math>\begin{align}& A_0 \\ & A_n \quad \text{for } n \ge 1 \\ & B_n \quad \text{for } n \ge 1\end{align}</math>
! Remarks
! Reference
|-
|<math>s(x)=A \left| \sin\left(\frac{2\pi}{P}x\right)\right| \quad \text{for } 0 \le x < P</math>
|[[File:PlotRectifiedSineSignal.svg|250px|center]]
|<math>\begin{align}
A_0 = & \frac{2A}{\pi}\\
A_n = & \begin{cases}
  \frac{-4A}{\pi}\frac{1}{n^2-1} & \quad n \text{ even} \\
  0 & \quad n \text{ odd}
  \end{cases}\\
B_n = & 0\\
\end{align}</math>
|Full-wave rectified sine
|<ref name="Papula">{{cite book|last=Papula|first=Lothar|title=Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler|publisher=Vieweg+Teubner Verlag|year=2009|isbn=978-3834807571|trans-title=Mathematical Functions for Engineers and Physicists|language=de}}</ref>{{rp|p. 193}}
|-
|<math>s(x)=\begin{cases}
A \sin\left(\frac{2\pi}{P}x\right) & \quad \text{for } 0 \le x < P/2 \\
0 & \quad \text{for } P/2 \le x < P\\
\end{cases}
</math>
|[[File:PlotHalfRectifiedSineSignal.svg|250px|center]]
|<math>\begin{align}
A_0 = & \frac{A}{\pi}\\
A_n = & \begin{cases}
  \frac{-2A}{\pi}\frac{1}{n^2-1} & \quad n \text{ even} \\
  0 & \quad n \text{ odd}
  \end{cases}\\
B_n = & \begin{cases}
  \frac{A}{2} & \quad n=1 \\
  0 & \quad n > 1
  \end{cases}\\
\end{align}</math>
|Half-wave rectified sine
|<ref name=Papula/>{{rp|p. 193}}
|-
|<math>s(x)=\begin{cases}
A & \quad \text{for } 0 \le x < D \cdot P \\
0 & \quad \text{for } D \cdot P \le x < P\\
\end{cases}
</math>
|[[File:PlotRectangleSignal.svg|250px|center]]
|<math>\begin{align}
A_0 = & AD\\
A_n = & \frac{A}{n \pi} \sin \left( 2 \pi n D \right)\\
B_n = & \frac{2A}{n \pi} \left( \sin \left( \pi n D \right) \right) ^2\\
\end{align}</math>
| <math>0 \le D \le 1</math>
|
|-
|<math>s(x)=\frac{Ax}{P} \quad \text{for } 0 \le x < P</math>
|[[File:PlotSawtooth1Signal.svg|250px|center]]
|<math>\begin{align}
A_0 = & \frac{A}{2}\\
A_n = & 0\\
B_n = & \frac{-A}{n \pi}\\
\end{align}</math>
| 
|<ref name=Papula/>{{rp|p. 192}}
|-
|<math>s(x)=A-\frac{Ax}{P} \quad \text{for } 0 \le x < P</math>
|[[File:PlotSawtooth2Signal.svg|250px|center]]
|<math>\begin{align}
A_0 = & \frac{A}{2}\\
A_n = & 0\\
B_n = & \frac{A}{n \pi}\\
\end{align}</math>
| 
|<ref name=Papula/>{{rp|p. 192}}
|-
|<math>s(x)=\frac{4A}{P^2}\left( x-\frac{P}{2} \right)^2 \quad \text{for } 0 \le x < P</math>
|[[File:PlotParabolaSignal.svg|250px|center]]
|<math>\begin{align}
A_0 = & \frac{A}{3}\\
A_n = & \frac{4A}{\pi^2 n^2}\\
B_n = & 0\\
\end{align}</math>
| 
|<ref name=Papula/>{{rp|p. 193}}
|}

== Table of basic properties ==
This table shows some mathematical operations in the time domain and the corresponding effect in the Fourier series coefficients. Notation:
* [[Complex conjugate|Complex conjugation]] is denoted by an asterisk.
*<math>s(x),r(x)</math> designate <math>P</math>-periodic functions '''or''' functions defined only for <math>x \in [0,P]. </math>
* <math>S[n], R[n]</math> designate the Fourier series coefficients (exponential form) of <math>s</math> and <math>r.</math>

{| class="wikitable"
|-
! Property
! Time domain
! Frequency domain (exponential form)
! Remarks
! Reference
|-
| Linearity
| <math>a\cdot s(x) + b\cdot r(x)</math>
| <math>a\cdot S[n] + b\cdot R[n]</math>
| <math>a,b \in \mathbb{C}</math>
|
|-
| Time reversal / Frequency reversal
| <math>s(-x)</math>
| <math>S[-n]</math>
|
| <ref name="Shmaliy" />{{rp|p. 610}}
|-
| Time conjugation
| <math>s^*(x)</math>
| <math>S^*[-n]</math>
|
| <ref name="Shmaliy" />{{rp|p. 610}}
|-
| Time reversal & conjugation
| <math>s^*(-x)</math>
| <math>S^*[n]</math>
|
|
|-
| Real part in time
| <math>\operatorname{Re}{(s(x))}</math>
| <math>\frac{1}{2}(S[n] + S^*[-n])</math>
|
|
|-
| Imaginary part in time
| <math>\operatorname{Im}{(s(x))}</math>
| <math>\frac{1}{2i}(S[n] - S^*[-n])</math>
|
|
|-
| Real part in frequency
| <math>\frac{1}{2}(s(x)+s^*(-x))</math>
| <math>\operatorname{Re}{(S[n])}</math>
|
|
|-
| Imaginary part in frequency
| <math>\frac{1}{2i}(s(x)-s^*(-x))</math>
| <math>\operatorname{Im}{(S[n])}</math>
|
|
|-
| Shift in time / Modulation in frequency
| <math>s(x-x_0)</math>
| <math>S[n] \cdot e^{-i 2\pi\tfrac{x_0}{P}n}</math>
| <math>x_0 \in \mathbb{R}</math>
| <ref name="Shmaliy">{{cite book | author=Shmaliy, Y.S.| title=Continuous-Time Signals| publisher=Springer | year=2007 | isbn=978-1402062711}}</ref>{{rp|p.610}}
|-
| Shift in frequency / Modulation in time
| <math>s(x) \cdot e^{i 2\pi \frac{n_0}{P}x}</math>
| <math>S[n-n_0] \!</math>
| <math>n_0 \in \mathbb{Z}</math>
| <ref name="Shmaliy" />{{rp|p. 610}}
|}

== Symmetry properties ==
When the real and imaginary parts of a complex function are decomposed into their even and odd parts, there are four components, denoted below by the subscripts RE, RO, IE, and IO. And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform:<ref name="ProakisManolakis1996">{{cite book|last1=Proakis|first1=John G. |last2=Manolakis|first2=Dimitris G.|title=Digital Signal Processing: Principles, Algorithms, and Applications|url=https://archive.org/details/digitalsignalpro00proa|url-access=registration|year=1996|publisher=Prentice Hall|isbn=978-0-13-373762-2|edition=3rd|page=[https://archive.org/details/digitalsignalpro00proa/page/291 291]}}</ref>

:<math>
\begin{array}{rccccccccc}
\text{Time domain} & s & = & s_{_{\text{RE}}} & + & s_{_{\text{RO}}} & + & i s_{_{\text{IE}}} & + & \underbrace{i\ s_{_{\text{IO}}}} \\
&\Bigg\Updownarrow\mathcal{F} & &\Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F}\\
\text{Frequency domain} & S & = & S_\text{RE} & + & \overbrace{\,i\ S_\text{IO}\,} & + & i S_\text{IE} & + & S_\text{RO}
\end{array}
</math>

From this, various relationships are apparent, for example:
*The transform of a real-valued function ({{math|''s''{{sub|{{sub|RE}}}} + ''s''{{sub|{{sub|RO}}}}}}) is the [[Even and odd functions#Complex-valued functions|even symmetric]] function {{math|''S''{{sub|{{sub|RE}}}} + ''i'' ''S''{{sub|{{sub|IO}}}}}}.  Conversely, an even-symmetric transform implies a real-valued time-domain.
*The transform of an imaginary-valued function ({{math|''i'' ''s''{{sub|{{sub|IE}}}} + ''i'' ''s''{{sub|{{sub|IO}}}}}}) is the [[Even and odd functions#Complex-valued functions|odd symmetric]] function {{math|''S''{{sub|{{sub|RO}}}} + ''i'' ''S''{{sub|{{sub|IE}}}}}}, and the converse is true.
*The transform of an even-symmetric function ({{math|''s''{{sub|{{sub|RE}}}} + ''i'' ''s''{{sub|{{sub|IO}}}}}}) is the real-valued function {{math|''S''{{sub|{{sub|RE}}}} + ''S''{{sub|{{sub|RO}}}}}}, and the converse is true.
*The transform of an odd-symmetric function ({{math|''s''{{sub|{{sub|RO}}}} + ''i'' ''s''{{sub|{{sub|IE}}}}}}) is the imaginary-valued function {{math|''i'' ''S''{{sub|{{sub|IE}}}} + ''i'' ''S''{{sub|{{sub|IO}}}}}}, and the converse is true.

== Other properties ==

=== Riemann–Lebesgue lemma ===
If <math>S</math> is integrable, <math display="inline">\lim_{|n| \to \infty} S[n]=0</math>, <math display="inline">\lim_{n \to +\infty} a_n=0</math> and <math display="inline"> \lim_{n \to +\infty} b_n=0.</math> This result is known as the [[Riemann–Lebesgue lemma]].

=== Parseval's theorem ===
{{main|Parseval's theorem}}
If <math>s</math> belongs to <math>L^2(P)</math> (periodic over an interval of length <math>P</math>) then''':''' <math display="inline">\frac{1}{P}\int_{P} |s(x)|^2 \, dx = \sum_{n=-\infty}^\infty \Bigl|S[n]\Bigr|^2.</math>

=== Plancherel's theorem ===
{{main|Plancherel theorem}}
If <math>c_0,\, c_{\pm 1},\, c_{\pm 2}, \ldots</math> are coefficients and <math display="inline">\sum_{n=-\infty}^\infty |c_n|^2 < \infty</math> then there is a unique function <math>s\in L^2(P)</math> such that <math>S[n] = c_n</math> for every <math>n</math>.

=== Convolution theorems ===

Given <math>P</math>-periodic functions, <math>s_{_P}</math> and <math>r_{_P}</math> with Fourier series coefficients <math>S[n]</math> and <math>R[n],</math> <math>n \in \mathbb{Z},</math>

*The pointwise product''':''' <math display="block">h_{_P}(x) \triangleq s_{_P}(x)\cdot r_{_P}(x)</math> is also <math>P</math>-periodic, and its Fourier series coefficients are given by the discrete convolution of the <math>S</math> and <math>R</math> sequences''':''' <math display="block">H[n] = \{S*R\}[n].</math>
*The periodic convolution''':''' <math display="block">h_{_P}(x) \triangleq \int_{P} s_{_P}(\tau)\cdot r_{_P}(x-\tau)\, d\tau</math> is also <math>P</math>-periodic, with Fourier series coefficients''':''' <math display="block">H[n] = P \cdot S[n]\cdot R[n].</math>
*A doubly infinite sequence <math>\left \{c_n \right \}_{n \in Z}</math> in <math>c_0(\mathbb{Z})</math> is the sequence of Fourier coefficients of a function in <math>L^1([0,2\pi])</math> if and only if it is a convolution of two sequences in <math>\ell^2(\mathbb{Z})</math>. See <ref>{{cite web|url=https://mathoverflow.net/q/46626 |title= Characterizations of a linear subspace associated with Fourier series |publisher=MathOverflow |date=2010-11-19 |access-date=2014-08-08}}</ref>

=== Derivative property ===
We say that <math>s</math> belongs to 
<math>C^k(\mathbb{T})</math> if <math>s</math> is a 2{{pi}}-periodic function on <math>\mathbb{R}</math> which is <math>k</math> times differentiable, and its <math>k^{\text{th}}</math> derivative is continuous.
* If <math>s \in C^1(\mathbb{T})</math>, then the Fourier coefficients <math>\widehat{s'}[n]</math> of the derivative <math>s'</math> can be expressed in terms of the Fourier coefficients <math>\widehat{s}[n]</math> of the function <math>s</math>, via the formula <math>\widehat{s'}[n] = in \widehat{s}[n]</math>.
* If <math>s \in C^k(\mathbb{T})</math>, then <math>\widehat{s^{(k)}}[n] = (in)^k \widehat{s}[n]</math>. In particular, since for a fixed <math>k\geq 1</math> we have <math>\widehat{s^{(k)}}[n]\to 0</math> as <math>n\to\infty</math>, it follows that <math>|n|^k\widehat{s}[n]</math> tends to zero, which means that the Fourier coefficients converge to zero faster than the ''k''th power of ''n'' for any <math>k\geq 1</math>.

=== Compact groups ===
{{main|Compact group|Lie group|Peter–Weyl theorem}}

One of the interesting properties of the Fourier transform which we have mentioned, is that it carries convolutions to pointwise products. If that is the property which we seek to preserve, one can produce Fourier series on any [[Compact group|compact group]]. Typical examples include those [[Classical group|classical group]]s that are compact. This generalizes the Fourier transform to all spaces of the form ''L''<sup>2</sup>(''G''), where ''G'' is a compact group, in such a way that the Fourier transform carries [[Convolution|convolution]]s to pointwise products. The Fourier series exists and converges in similar ways to the {{closed-closed|−''π'',''π''}} case.

An alternative extension to compact groups is the [[Peter–Weyl theorem]], which proves results about representations of compact groups analogous to those about finite groups.

[[File:F orbital.png|thumb|right|The [[Physics:Atomic orbital|atomic orbital]]s of chemistry are partially described by spherical harmonics, which can be used to produce Fourier series on the [[Sphere|sphere]].]]

=== Riemannian manifolds ===
{{main|Laplace operator|Riemannian manifold}}

If the domain is not a group, then there is no intrinsically defined convolution. However, if <math>X</math> is a [[Compact space|compact]] [[Riemannian manifold]], it has a [[Laplace–Beltrami operator]]. The Laplace–Beltrami operator is the differential operator that corresponds to [[Laplace operator]] for the Riemannian manifold <math>X</math>. Then, by analogy, one can consider heat equations on <math>X</math>. Since Fourier arrived at his basis by attempting to solve the heat equation, the natural generalization is to use the eigensolutions of the Laplace–Beltrami operator as a basis. This generalizes Fourier series to spaces of the type <math>L^2(X)</math>, where <math>X</math> is a Riemannian manifold. The Fourier series converges in ways similar to the <math>[-\pi,\pi]</math> case. A typical example is to take <math>X</math> to be the sphere with the usual metric, in which case the Fourier basis consists of [[Physics:Spherical harmonics|spherical harmonics]].

=== Locally compact Abelian groups ===
{{main|Pontryagin duality}}

The generalization to compact groups discussed above does not generalize to noncompact, [[Non-abelian group|nonabelian group]]s. However, there is a straightforward generalization to Locally Compact Abelian (LCA) groups.

This generalizes the Fourier transform to <math>L^1(G)</math> or <math>L^2(G)</math>, where <math>G</math> is an LCA group. If <math>G</math> is compact, one also obtains a Fourier series, which converges similarly to the <math>[-\pi,\pi]</math> case, but if <math>G</math> is noncompact, one obtains instead a Fourier integral. This generalization yields the usual [[Fourier transform]] when the underlying locally compact Abelian group is <math>\mathbb{R}</math>.

== Extensions ==
=== Fourier series on a square ===
We can also define the Fourier series for functions of two variables <math>x</math> and <math>y</math> in the square <math>[-\pi,\pi]\times[-\pi,\pi]</math>:
<math display="block">\begin{align}
f(x,y) & = \sum_{j,k \in \Z} c_{j,k}e^{ijx}e^{iky},\\[5pt]
c_{j,k} & = \frac{1}{4 \pi^2} \int_{-\pi}^\pi \int_{-\pi}^\pi f(x,y) e^{-ijx}e^{-iky}\, dx \, dy.
\end{align}</math>

Aside from being useful for solving partial differential equations such as the heat equation, one notable application of Fourier series on the square is in [[Image compression|image compression]]. In particular, the [[JPEG]] image compression standard uses the two-dimensional [[Discrete cosine transform|discrete cosine transform]], a discrete form of the [[Sine and cosine transforms|Fourier cosine transform]], which uses only cosine as the basis function.

For two-dimensional arrays with a staggered appearance, half of the Fourier series coefficients disappear, due to additional symmetry.<ref>[https://www.youtube.com/watch?v=V7l9Im9zneg Vanishing of Half the Fourier Coefficients in Staggered Arrays]</ref>

=== Fourier series of Bravais-lattice-periodic-function ===
<!--Linked from Reciprocal Lattice to anchor Multidimensional-->
A three-dimensional [[Bravais lattice]] is defined as the set of vectors of the form:
<math display="block">\mathbf{R} = n_1\mathbf{a}_1 + n_2\mathbf{a}_2 + n_3\mathbf{a}_3</math>
where <math>n_i</math> are integers and <math>\mathbf{a}_i</math> are three linearly independent vectors. Assuming we have some function, <math>f(\mathbf{r})</math>, such that it obeys the condition of periodicity for any Bravais lattice vector <math>\mathbf{R}</math>, <math>f(\mathbf{r}) = f(\mathbf{R}+\mathbf{r})</math>, we could make a Fourier series of it. This kind of function can be, for example, the effective potential that one electron "feels" inside a periodic crystal. It is useful to make the Fourier series of the potential when applying Bloch's theorem. First, we may write any arbitrary position vector <math>\mathbf{r}</math> in the coordinate-system of the lattice:
<math display="block">\mathbf{r} = x_1\frac{\mathbf{a}_{1}}{a_1}+ x_2\frac{\mathbf{a}_{2}}{a_2}+ x_3\frac{\mathbf{a}_{3}}{a_3},</math>
where <math>a_i \triangleq |\mathbf{a}_i|,</math> meaning that <math>a_i</math> is defined to be the magnitude of <math>\mathbf{a}_i</math>, so <math>\hat{\mathbf{a}_{i}} = \frac {\mathbf{a}_{i}}{a_i}</math> is the unit vector directed along <math>\mathbf{a}_i</math>.

Thus we can define a new function,
<math display="block">g(x_1,x_2,x_3) \triangleq f(\mathbf{r}) = f \left (x_1\frac{\mathbf{a}_{1}}{a_1}+x_2\frac{\mathbf{a}_{2}}{a_2}+x_3\frac{\mathbf{a}_{3}}{a_3} \right ).</math>

This new function, <math>g(x_1,x_2,x_3)</math>, is now a function of three-variables, each of which has periodicity <math>a_1</math>, <math>a_2</math>, and <math>a_3</math> respectively:
<math display="block">g(x_1,x_2,x_3) = g(x_1+a_1,x_2,x_3) = g(x_1,x_2+a_2,x_3) = g(x_1,x_2,x_3+a_3).</math>

This enables us to build up a set of Fourier coefficients, each being indexed by three independent integers <math>m_1,m_2,m_3</math>. In what follows, we use function notation to denote these coefficients, where previously we used subscripts. If we write a series for <math>g</math> on the interval <math>\left [ 0, a_1\right ]</math> for <math>x_1</math>, we can define the following:
<math display="block">h^\mathrm{one}(m_1, x_2, x_3) \triangleq \frac{1}{a_1}\int_0^{a_1} g(x_1, x_2, x_3)\cdot e^{-i 2\pi \tfrac{m_1}{a_1} x_1}\, dx_1</math>

And then we can write:
<math display="block">g(x_1, x_2, x_3)=\sum_{m_1=-\infty}^\infty h^\mathrm{one}(m_1, x_2, x_3) \cdot e^{i 2\pi \tfrac{m_1}{a_1} x_1}</math>

Further defining:
<math display="block">\begin{align}
h^\mathrm{two}(m_1, m_2, x_3) & \triangleq \frac{1}{a_2}\int_0^{a_2} h^\mathrm{one}(m_1, x_2, x_3)\cdot e^{-i 2\pi \tfrac{m_2}{a_2} x_2}\, dx_2 \\[12pt]
& = \frac{1}{a_2}\int_0^{a_2} dx_2 \frac{1}{a_1}\int_0^{a_1} dx_1 g(x_1, x_2, x_3)\cdot e^{-i 2\pi \left(\tfrac{m_1}{a_1} x_1+\tfrac{m_2}{a_2} x_2\right)}
\end{align}</math>

We can write <math>g</math> once again as:
<math display="block">g(x_1, x_2, x_3)=\sum_{m_1=-\infty}^\infty \sum_{m_2=-\infty}^\infty h^\mathrm{two}(m_1, m_2, x_3) \cdot e^{i 2\pi \tfrac{m_1}{a_1} x_1} \cdot e^{i 2\pi \tfrac{m_2}{a_2} x_2}</math>

Finally applying the same for the third coordinate, we define:
<math display="block">\begin{align}
h^\mathrm{three}(m_1, m_2, m_3) & \triangleq \frac{1}{a_3}\int_0^{a_3} h^\mathrm{two}(m_1, m_2, x_3)\cdot e^{-i 2\pi \tfrac{m_3}{a_3} x_3}\, dx_3 \\[12pt]
& = \frac{1}{a_3}\int_0^{a_3} dx_3 \frac{1}{a_2}\int_0^{a_2} dx_2 \frac{1}{a_1}\int_0^{a_1} dx_1 g(x_1, x_2, x_3)\cdot e^{-i 2\pi \left(\tfrac{m_1}{a_1} x_1+\tfrac{m_2}{a_2} x_2 + \tfrac{m_3}{a_3} x_3\right)}
\end{align}</math>

We write <math>g</math> as:
<math display="block">g(x_1, x_2, x_3)=\sum_{m_1=-\infty}^\infty \sum_{m_2=-\infty}^\infty \sum_{m_3=-\infty}^\infty h^\mathrm{three}(m_1, m_2, m_3) \cdot e^{i 2\pi \tfrac{m_1}{a_1} x_1} \cdot e^{i 2\pi \tfrac{m_2}{a_2} x_2}\cdot e^{i 2\pi \tfrac{m_3}{a_3} x_3}</math>

Re-arranging:
<math display="block">g(x_1, x_2, x_3)=\sum_{m_1, m_2, m_3 \in \Z } h^\mathrm{three}(m_1, m_2, m_3) \cdot e^{i 2\pi \left( \tfrac{m_1}{a_1} x_1+ \tfrac{m_2}{a_2} x_2 + \tfrac{m_3}{a_3} x_3\right)}. </math>

Now, every ''reciprocal'' lattice vector can be written (but does not mean that it is the only way of writing) as <math>\mathbf{G} = m_1\mathbf{g}_1 + m_2\mathbf{g}_2 + m_3\mathbf{g}_3</math>, where <math>m_i</math> are integers and <math>\mathbf{g}_i</math> are reciprocal lattice vectors to satisfy <math>\mathbf{g_i} \cdot \mathbf{a_j}=2\pi\delta_{ij}</math> (<math>\delta_{ij} = 1</math> for <math>i = j</math>, and <math>\delta_{ij} = 0</math> for <math>i \neq j</math>). Then for any arbitrary reciprocal lattice vector <math>\mathbf{G}</math> and arbitrary position vector <math>\mathbf{r}</math> in the original Bravais lattice space, their scalar product is:
<math display="block">\mathbf{G} \cdot \mathbf{r} = \left ( m_1\mathbf{g}_1 + m_2\mathbf{g}_2 + m_3\mathbf{g}_3 \right ) \cdot \left (x_1\frac{\mathbf{a}_1}{a_1}+ x_2\frac{\mathbf{a}_2}{a_2} +x_3\frac{\mathbf{a}_3}{a_3} \right ) = 2\pi \left( x_1\frac{m_1}{a_1}+x_2\frac{m_2}{a_2}+x_3\frac{m_3}{a_3} \right ).</math>

So it is clear that in our expansion of <math>g(x_1,x_2,x_3) = f(\mathbf{r})</math>, the sum is actually over reciprocal lattice vectors:
<math display="block">f(\mathbf{r})=\sum_{\mathbf{G}} h(\mathbf{G}) \cdot e^{i \mathbf{G} \cdot \mathbf{r}}, </math>

where <math display="block">h(\mathbf{G}) = \frac{1}{a_3} \int_0^{a_3} dx_3 \, \frac{1}{a_2}\int_0^{a_2} dx_2 \, \frac{1}{a_1}\int_0^{a_1} dx_1 \, f\left(x_1\frac{\mathbf{a}_1}{a_1} + x_2\frac{\mathbf{a}_2}{a_2} + x_3\frac{\mathbf{a}_3}{a_3} \right)\cdot e^{-i \mathbf{G} \cdot \mathbf{r}}. </math>

Assuming
<math display="block">\mathbf{r} = (x,y,z) = x_1\frac{\mathbf{a}_1}{a_1}+x_2\frac{\mathbf{a}_2}{a_2}+x_3\frac{\mathbf{a}_3}{a_3},</math>
we can solve this system of three linear equations for <math>x</math>, <math>y</math>, and <math>z</math> in terms of <math>x_1</math>, <math>x_2</math> and <math>x_3</math> in order to calculate the volume element in the original rectangular coordinate system. Once we have <math>x</math>, <math>y</math>, and <math>z</math> in terms of <math>x_1</math>, <math>x_2</math> and <math>x_3</math>, we can calculate the [[Jacobian matrix and determinant|Jacobian determinant]]:
<math display="block">\begin{vmatrix}
\dfrac{\partial x_1}{\partial x} & \dfrac{\partial x_1}{\partial y} & \dfrac{\partial x_1}{\partial z} \\[12pt]
\dfrac{\partial x_2}{\partial x} & \dfrac{\partial x_2}{\partial y} & \dfrac{\partial x_2}{\partial z} \\[12pt]
\dfrac{\partial x_3}{\partial x} & \dfrac{\partial x_3}{\partial y} & \dfrac{\partial x_3}{\partial z}
\end{vmatrix}</math>
which after some calculation and applying some non-trivial cross-product identities can be shown to be equal to:
<math display="block">\frac{a_1 a_2 a_3}{\mathbf{a}_1\cdot(\mathbf{a}_2 \times \mathbf{a}_3)}</math>

(it may be advantageous for the sake of simplifying calculations, to work in such a rectangular coordinate system, in which it just so happens that <math>\mathbf{a}_1</math> is parallel to the ''x'' axis, <math>\mathbf{a}_2</math> lies in the ''xy''-plane, and <math>\mathbf{a}_3</math> has components of all three axes). The denominator is exactly the volume of the primitive unit cell which is enclosed by the three primitive-vectors <math>\mathbf{a}_1</math>, <math>\mathbf{a}_2</math> and <math>\mathbf{a}_3</math>. In particular, we now know that
<math display="block">dx_1 \, dx_2 \, dx_3 = \frac{a_1 a_2 a_3}{\mathbf{a}_1\cdot(\mathbf{a}_2 \times \mathbf{a}_3)} \cdot dx \, dy \, dz. </math>

We can write now <math>h(\mathbf{G})</math> as an integral with the traditional coordinate system over the volume of the primitive cell, instead of with the <math>x_1</math>, <math>x_2</math> and <math>x_3</math> variables:
<math display="block">h(\mathbf{G}) = \frac{1}{\mathbf{a}_1\cdot(\mathbf{a}_2 \times \mathbf{a}_3)}\int_{C} d\mathbf{r} f(\mathbf{r})\cdot e^{-i \mathbf{G} \cdot \mathbf{r}} </math>
writing <math>d\mathbf{r}</math> for the volume element <math>dx \, dy \, dz</math>; and where <math>C</math> is the primitive unit cell, thus, <math>\mathbf{a}_1\cdot(\mathbf{a}_2 \times \mathbf{a}_3)</math> is the volume of the primitive unit cell.

=== Hilbert space interpretation ===
{{main|Hilbert space}}

In the language of [[Hilbert space]]s, the set of functions <math>\left\{e_n=e^{inx}: n \in \Z\right\}</math> is an [[Orthonormal basis|orthonormal basis]] for the space <math>L^2([-\pi,\pi])</math> of square-integrable functions on <math>[-\pi,\pi]</math>. This space is actually a Hilbert space with an inner product given for any two elements <math>f</math> and <math>g</math> by:
:<math>\langle f,\, g \rangle \;\triangleq \; \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x)g^*(x)\,dx,</math> where <math>g^{*}(x)</math> is the complex conjugate of <math>g(x).</math>
The basic Fourier series result for Hilbert spaces can be written as
:<math>f=\sum_{n=-\infty}^\infty \langle f,e_n \rangle \, e_n.</math>
[[File:Fourier series integral identities.gif|thumb|400px|right|Sines and cosines form an orthogonal set, as illustrated above. The integral of sine, cosine and their product is zero (green and red areas are equal, and cancel out) when <math>m</math>, <math>n</math> or the functions are different, and π only if <math>m</math> and <math>n</math> are equal, and the function used is the same. They would form an orthonormal set, if the integral equaled 1 (that is, each function would need to be scaled by <math>1/\sqrt{\pi}</math>).]]
This corresponds exactly to the complex exponential formulation given above. The version with sines and cosines is also justified with the Hilbert space interpretation. Indeed, the sines and cosines form an orthogonal set:
<math display="block">\int_{-\pi}^{\pi} \cos(mx)\, \cos(nx)\, dx = \frac{1}{2}\int_{-\pi}^{\pi} \cos((n-m)x)+\cos((n+m)x)\, dx = \pi \delta_{mn}, \quad m, n \ge 1, </math>
<math display="block">\int_{-\pi}^{\pi} \sin(mx)\, \sin(nx)\, dx = \frac{1}{2}\int_{-\pi}^{\pi} \cos((n-m)x)-\cos((n+m)x)\, dx = \pi \delta_{mn}, \quad m, n \ge 1</math>
(where ''δ''<sub>''mn''</sub> is the [[Kronecker delta]]), and
<math display="block">\int_{-\pi}^{\pi} \cos(mx)\, \sin(nx)\, dx = \frac{1}{2}\int_{-\pi}^{\pi} \sin((n+m)x)+\sin((n-m)x)\, dx = 0;</math>
furthermore, the sines and cosines are orthogonal to the constant function <math>1</math>. An ''orthonormal basis'' for <math>L^2([-\pi,\pi])</math> consisting of real functions is formed by the functions <math>1</math> and <math>\sqrt{2} \cos (nx)</math>, <math>\sqrt{2} \sin (nx)</math> with ''n''= 1,2,.... The density of their span is a consequence of the [[Stone–Weierstrass theorem]], but follows also from the properties of classical kernels like the [[Fejér kernel]].

== Fourier theorem proving convergence of Fourier series ==
{{main|Convergence of Fourier series}}

These theorems, and informal variations of them that don't specify the convergence conditions, are sometimes referred to generically as ''Fourier's theorem'' or ''the Fourier theorem''.<ref>{{cite book |last=Siebert |first=William McC. |url=https://books.google.com/books?id=zBTUiIrb2WIC&q=%22fourier%27s+theorem%22&pg=PA402 |title=Circuits, signals, and systems |publisher=MIT Press |year=1985 |isbn=978-0-262-19229-3 |page=402}}</ref><ref>{{cite book |last1=Marton |first1=L. |url=https://books.google.com/books?id=27c1WOjCBX4C&q=%22fourier+theorem%22&pg=PA369 |title=Advances in Electronics and Electron Physics |last2=Marton |first2=Claire |publisher=Academic Press |year=1990 |isbn=978-0-12-014650-5 |page=369}}</ref><ref>{{cite book |last=Kuzmany |first=Hans |url=https://books.google.com/books?id=-laOoZitZS8C&q=%22fourier+theorem%22&pg=PA14 |title=Solid-state spectroscopy |publisher=Springer |year=1998 |isbn=978-3-540-63913-8 |page=14}}</ref><ref>{{cite book |last1=Pribram |first1=Karl H. |url=https://books.google.com/books?id=nsD4L2zsK4kC&q=%22fourier+theorem%22&pg=PA26 |title=Brain and perception |last2=Yasue |first2=Kunio |last3=Jibu |first3=Mari |publisher=Lawrence Erlbaum Associates |year=1991 |isbn=978-0-89859-995-4 |page=26}}</ref>

:<math>s_{_N}(x) = \sum_{n=-N}^N S[n]\ e^{i 2\pi\tfrac{n}{P} x},</math>
is a [[Trigonometric polynomial|trigonometric polynomial]] of degree <math>N</math> that can be generally expressed as''':'''
:<math>p_{_N}(x)=\sum_{n=-N}^N p[n]\ e^{i 2\pi\tfrac{n}{P}x}.</math>

===Least squares property===
[[Parseval's theorem]] implies that:

{{math theorem | math_statement=The trigonometric polynomial <math>s_{_N}</math> is the unique best trigonometric polynomial of degree <math>N</math> approximating <math>s(x)</math>, in the sense that, for any trigonometric polynomial <math>p_{_N} \neq s_{_N}</math> of degree <math>N</math>, we have:
<math display="block">\|s_{_N} - s\|_2 < \|p_{_N} - s\|_2,</math>
where the Hilbert space norm is defined as:
<math display="block">\| g \|_2 = \sqrt{{1 \over P} \int_P |g(x)|^2 \, dx}.</math>
}}

===Convergence theorems ===
Because of the least squares property, and because of the completeness of the Fourier basis, we obtain an elementary convergence result.

{{math theorem | math_statement= If <math>s</math> belongs to <math>L^2 (P)</math> (an interval of length <math>P</math>), then <math>s_{\infty}</math> converges to <math>s</math> in <math>L^2 (P)</math>, that is,&thinsp; <math>\|s_{_N} - s\|_2</math> converges to 0 as <math>N \to \infty</math>.}}

We have already mentioned that if <math>s</math> is continuously differentiable, then <math>(i\cdot n) S[n]</math> is the <math>n^{\text{th}}</math> Fourier coefficient of the derivative <math>s'</math>. It follows, essentially from the [[Cauchy–Schwarz inequality]], that <math>s_{\infty}</math> is absolutely summable. The sum of this series is a continuous function, equal to <math>s</math>, since the Fourier series converges in the mean to <math>s</math>:

{{math theorem| math_statement= If <math>s \in C^1(\mathbb{T})</math>, then <math>s_{\infty}</math> converges to <math>s</math> [[Uniform convergence|uniformly]] (and hence also [[Pointwise convergence|pointwise]].)}}

This result can be proven easily if <math>s</math> is further assumed to be <math>C^2</math>, since in that case <math>n^2S[n]</math> tends to zero as <math>n \rightarrow \infty</math>. More generally, the Fourier series is absolutely summable, thus converges uniformly to <math>s</math>, provided that <math>s</math> satisfies a [[Hölder condition]] of order <math>\alpha > 1/2</math>. In the absolutely summable case, the inequality:
:<math>\sup_x |s(x) - s_{_N}(x)| \le \sum_{|n| > N} |S[n]|</math>

proves uniform convergence.

Many other results concerning the [[Convergence of Fourier series|convergence of Fourier series]] are known, ranging from the moderately simple result that the series converges at <math>x</math> if <math>s</math> is differentiable at <math>x</math>, to [[Biography:Lennart Carleson|Lennart Carleson]]'s much more sophisticated result that the Fourier series of an <math>L^2</math> function actually converges [[Almost everywhere|almost everywhere]].

=== Divergence ===
Since Fourier series have such good convergence properties, many are often surprised by some of the negative results. For example, the Fourier series of a continuous ''T''-periodic function need not converge pointwise.{{Citation needed|date=September 2020}} The [[Uniform boundedness principle|uniform boundedness principle]] yields a simple non-constructive proof of this fact.

In 1922, [[Biography:Andrey Kolmogorov|Andrey Kolmogorov]] published an article titled ''Une série de Fourier-Lebesgue divergente presque partout'' in which he gave an example of a Lebesgue-integrable function whose Fourier series diverges almost everywhere. He later constructed an example of an integrable function whose Fourier series diverges everywhere.<ref>{{cite book |last1=Katznelson |first1=Yitzhak |title=An introduction to Harmonic Analysis |publisher=Dover Publications, Inc. |edition=2nd corrected |date=1976 |location=New York, NY |isbn=0-486-63331-4}}</ref>

==See also==
{{cols|colwidth=26em}}
* [[ATS theorem]]
* [[Carleson's theorem]]
* [[Dirichlet kernel]]
* [[Discrete Fourier transform]]
* [[Fast Fourier transform]]
* [[Fejér's theorem]]
* [[Fourier analysis]]
* [[Fourier sine and cosine series]]
* [[Fourier transform]]
* [[Gibbs phenomenon]]
* [[Half range Fourier series]]
* [[Laurent series]] – the substitution ''q''&nbsp;=&nbsp;''e''<sup>''ix''</sup> transforms a Fourier series into a Laurent series, or conversely. This is used in the ''q''-series expansion of the [[J-invariant|''j''-invariant]].
* [[Least-squares spectral analysis]]
* [[Multidimensional transform]]
* [[Spectral theory]]
* [[Sturm–Liouville theory]]
* [[Residue theorem]] integrals of ''f''(''z''), singularities, poles
{{colend}}

== Notes ==
{{notelist-ua}}

==References==
{{Reflist|refs=

<ref name=iit.edu>{{cite web |title=Fourier Series and Boundary Value Problems|last=Fasshauer |first=Greg |work=Math 461 Course Notes, Ch 3 |publisher=Department of Applied Mathematics, Illinois Institute of Technology |date=2015 |access-date=6 November 2020 |url= http://www.math.iit.edu/~fass/Notes461_Ch3Print.pdf }}</ref>
}}

===Further reading===
{{refbegin|2|indent=yes}}
*{{cite book |author1 =William E. Boyce |author2 =Richard C. DiPrima |title=Elementary Differential Equations and Boundary Value Problems |edition=8th |publisher=John Wiley & Sons, Inc. |location=New Jersey |year=2005 |isbn=0-471-43338-1}}
*{{cite book | author = Joseph Fourier, translated by Alexander Freeman | title = The Analytical Theory of Heat | publisher = Dover Publications | year = 2003 | isbn = 0-486-49531-0 }} 2003 unabridged republication of the 1878 English translation by Alexander Freeman of Fourier's work ''Théorie Analytique de la Chaleur'', originally published in 1822.
*{{cite journal |author=Enrique A. Gonzalez-Velasco |title=Connections in Mathematical Analysis: The Case of Fourier Series |journal=American Mathematical Monthly |volume=99 |year=1992 |pages=427–441 |issue=5 |doi=10.2307/2325087|jstor=2325087 }}
*{{cite book|last1=Fetter|first1=Alexander L. |last2=Walecka|first2=John Dirk |title=Theoretical Mechanics of Particles and Continua|url=https://books.google.com/books?id=olMpStYOlnoC&pg=PA209|date= 2003|publisher=Courier |isbn=978-0-486-43261-8}}
*[[Biography:Felix Klein|Felix Klein]], ''Development of mathematics in the 19th century''. Mathsci Press Brookline, Mass, 1979. Translated by M. Ackerman from ''Vorlesungen über die Entwicklung der Mathematik im 19 Jahrhundert'', Springer, Berlin, 1928.
*{{cite book |author=Walter Rudin |title=Principles of mathematical analysis |url=https://archive.org/details/principlesofmath00rudi |url-access=registration |edition=3rd |publisher=McGraw-Hill, Inc. |location=New York |year=1976 |isbn=0-07-054235-X}}
*{{cite book | author=A. Zygmund |  title= Trigonometric Series | title-link = Trigonometric Series | edition=third | publisher = Cambridge University Press | location=Cambridge | year=2002 | isbn=0-521-89053-5}} The first edition was published in 1935.
{{refend}}

==External links==
*{{springer|title=Fourier series|id=p/f041090}}
*{{MathWorld | urlname= FourierSeries | title= Fourier Series}}
*{{webarchive |url=https://web.archive.org/web/20011205152434/http://www.shsu.edu/~icc_cmf/bio/fourier.html |date=December 5, 2001 |title=Joseph Fourier – A site on Fourier's life which was used for the historical section of this article }}

{{series (mathematics)}}

{{DEFAULTSORT:Fourier Series}}
[[Category:Fourier series| ]]

{{Sourceattribution|Fourier series}}