In mathematics, a Fourier series is a periodic function composed of harmonically related sinusoids, combined by a weighted summation. With appropriate weights, one cycle (or period) of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). As such, the summation is a synthesis of another function. The discretetime Fourier transform is an example of Fourier series. The process of deriving weights that describe a given function is a form of Fourier analysis. For functions on unbounded intervals, the analysis and synthesis analogies are Fourier transform and inverse transform.
The Fourier series is named in honor of JeanBaptiste Joseph Fourier (1768–1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli. Fourier introduced the series for the purpose of solving the 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 ^{[1]} and later generalized to any piecewisesmooth) function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier in 1807, before the French Academy.^{[2]} 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 deferents and epicycles.
The heat equation is a 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 or 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) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding 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 and integral in the early nineteenth century. Later, Peter Gustav Lejeune Dirichlet^{[3]} and Bernhard Riemann^{[4]} ^{[5]} 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 sinusoids. The Fourier series has many such applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics,^{[6]} shell theory,^{[7]} etc.
Consider a realvalued function,
s(x),
P,
is essentially a matched filter, with template\Chi_{f(\tau)}\triangleq\int_{P}s(x) ⋅ \cos\left(2\pif(x\tau)\right)dx ; \tau\in\left[ 0, 2\pi/f \right]
\cos(2\pifx).
f
s.
(\tauf).
N
s(x)
x
n,
n
P.
x,
P/n.
n/P,
n
\cos\left(2\pix\tfrac{n}{P}\right).
P=2\pi
Rather than computationally intensive crosscorrelation, Fourier analysis customarily exploits a trigonometric identity:
A_{n ⋅ }\cos\left(\tfrac{2\pi}{P}nx\varphi_{n\right)} \equiv \underbrace{A_{n}\cos(\varphi_{n)}}
a_{n} 
⋅ \cos\left(\tfrac{2\pi}{P}nx\right)+\underbrace{A_{n}\sin(\varphi_{n)}}
b_{n} 
⋅ \sin\left(\tfrac{2\pi}{P}nx\right),
a_{n}
b_{n}
A_{n}
\varphi_{n,}
Then:
A_{n}=
2}  
\sqrt{a  
n 
\varphi_{n}=\operatorname{arctan2}(b_{n,a}_{n)}
And note that
a_{0}
b_{0}
a_{0}=
2  
P 
\int_{P}s(x)dx
b_{0}=0.
Another applicable identity is Euler's formula. Here, complex conjugation is denoted by an asterisk:
\begin{array}{lll} \cos\left(\tfrac{2\pi}{P}nx\varphi_{n}\right)&{}\equiv\tfrac{1}{2}e^{}{P}nx\varphi_{n}\right)}&{}+\tfrac{1}{2}e^{i}{P}nx\varphi_{n}\right)}\\ &=\left(\tfrac{1}{2}
i\varphi_{n}  
e 
\right) ⋅ e^{i}{P}(+n)x}&{}+\left(\tfrac{1}{2}
i\varphi_{n}  
e 
\right)^{*} ⋅ e^{i}{P}(n)x}. \end{array}
c_{n}\triangleq\left\{ \begin{array}{lll} A_{0/2}&=a_{0/2,} &n=0\\ \tfrac{A_{n}{2}}
i\varphi_{n}  
e 
&=\tfrac{1}{2}(a_{n}ib_{n),} &n>
*,  
0\\ c  
n 
&&n<0 \end{array}\right\} =
1  
P 
\int_{P}s(x) ⋅ e^{i}{P}nx} dx,
the final result is:
This is the customary form for generalizing to complexvalued
s(x)
If
s(x)
x,
c  
_{Rn} 
=
1  
P 
\int_{P}\operatorname{Re}\{s(x)\} ⋅ e^{i}{P}nx} dx
c  
_{In} 
=
1  
P 
\int_{P}\operatorname{Im}\{s(x)\} ⋅ e^{i}{P}nx} dx
s  
_{N} 
(x)=
N  
\sum  
n=N 
c  
_{Rn} 
⋅ e^{i}{P}nx}+i ⋅
N  
\sum  
n=N 
c  
_{In} 
⋅ e^{i}{P}nx}
N  
=\sum  
n=N 
\left(c  
_{Rn} 
+i ⋅
c  
_{In} 
\right) ⋅ e^{}{P}nx}.
Defining
c_{n}\triangleq
c  
_{Rn} 
+i ⋅
c  
_{In} 
This is identical to except
c_{n}
c_{n}
c_{n}
\begin{align} c_{n}&=
1  
P 
\int_{P}\operatorname{Re}\{s(x)\} ⋅ e^{i}{P}nx} dx+i ⋅
1  
P 
\int_{P}\operatorname{Im}\{s(x)\} ⋅ e^{i}{P}nx} dx\\[4pt] &=
1  
P 
\int_{P}\left(\operatorname{Re}\{s(x)\}+i ⋅ \operatorname{Im}\{s(x)\}\right) ⋅ e^{i}{P}nx} dx =
1  
P 
\int_{P}s(x) ⋅ e^{i}{P}nx} dx. \end{align}
The notation
c_{n}
s
\hat{s}[n]
S[n]
\begin{align} s_{infty(x)}&=
infty  
\sum  
n=infty 
\hat{s}[n] ⋅ e^{i2\pi}\\[6pt] &=
infty  
\sum  
n=infty 
S[n] ⋅ e^{i2\pi}&&\scriptstylecommon engineering notation \end{align}
In engineering, particularly when the variable
x
Another commonly used frequency domain representation uses the Fourier series coefficients to modulate a Dirac comb:
S(f) \triangleq
infty  
\sum  
n=infty 
S[n] ⋅ \delta\left(f
n  
P 
\right),
where
f
x
f
1/P
s_{infty}(x)
\begin{align} l{F}^{1}\{S(f)\}&=
infty  
\int  
infty 
\left(
infty  
\sum  
n=infty 
S[n] ⋅ \delta\left(f
n  
P 
\right)\right)e^{i}df,\\[6pt] &=
infty  
\sum  
n=infty 
S[n] ⋅
infty  
\int  \delta\left(f  
infty 
n  
P 
\right)e^{i}df,\\[6pt] &=
infty  
\sum  
n=infty 
S[n] ⋅ e^{i2\pi} \triangleq s_{infty(x). \end{align}}
The constructed function
S(f)
See main article: Convergence of Fourier series. In engineering applications, the Fourier series is generally presumed to converge almost everywhere (the exceptions being at discrete discontinuities) since the functions encountered in engineering are betterbehaved than the functions that mathematicians can provide as counterexamples to this presumption. In particular, if
s
s(x)
s
s(x)
[x_{0,x}_{0+P]}
An interactive animation can be seen here.
]]We now use the formula above to give a Fourier series expansion of a very simple function. Consider a sawtooth wave
s(x)=
x  
\pi 
, for\pi<x<\pi,
s(x+2\pik)=s(x), for\pi<x<\piandk\inZ.
\begin{align} a_{n}&=
1  
\pi 
\pi  
\int  
\pi 
s(x)\cos(nx)dx=0, n\ge0.\\[4pt] b_{n}&=
1  
\pi 
\pi  
\int  
\pi 
s(x)\sin(nx)dx\\[4pt] &=
2  
\pin 
\cos(n\pi)+
2  
\pi^{2}n^{2} 
\sin(n\pi)\\[4pt] &=
2(1)^{n+1}  
\pin 
, n\ge1.\end{align}
s(x)
x
s
x=\pi
x=\pi
This example leads us to a solution to the Basel problem.
The Fourier series expansion of our function in Example 1 looks more complicated than the simple formula
s(x)=x/\pi
\pi
(x,y)\in[0,\pi] x [0,\pi]
y=\pi
T(x,\pi)=x
x
(0,\pi)
T(x,y)=
infty  
2\sum  
n=1 
(1)^{n+1}  
n 
\sin(nx){\sinh(ny)\over\sinh(n\pi)}.
\sinh(ny)/\sinh(n\pi)
s(x)
T(x,y)
T
An example of the ability of the complex Fourier series to draw any two dimensional closed figure is shown in the adjacent animation of the complex Fourier series converging to a drawing in the complex plane of the letter 'e' (for exponential). The animation alternates between fast rotations to take less time and slow rotations to show more detail. The terms of the complex Fourier series are shown in two rotating arms: one arm is an aggregate of all the complex Fourier series terms that rotate in the positive direction (counter clockwise, according to the right hand rule), the other arm is an aggregate of all the complex Fourier series terms that rotate in the negative direction. The constant term that does not rotate at all is evenly split between the two arms. The animation's small circle represents the midpoint between the extent of the two arms, which is also the midpoint between the origin and the complex Fourier series approximation which is the '+' symbol in the animation. (The GNU Octave source code for generating this animation is here.^{[10]} Note that the animation uses the variable 't' to parameterize the drawing in the complex plane, equivalent to the use of the parameter 'x' in this article's subsection on complex valued functions.)
Another application of this Fourier series is to solve the Basel problem by using Parseval's theorem. The example generalizes and one may compute ζ(2n), for any positive integer n.
Joseph Fourier wrote:
This immediately gives any coefficient a_{k} 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
\begin{align} a_{k&=\int}
 
1 
dy\\ &=
 
\int  \cos(2k+1)  
1 
\piy  
2 
+a'\cos3
\piy  \cos(2k+1)  
2 
\piy  
2 
+ … \right)dy \end{align}
can be carried out termbyterm. But all terms involving
\cos(2j+1)  \piy  \cos(2k+1) 
2 
\piy  
2 
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, d'Alembert, Daniel Bernoulli and 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 convergence, function spaces, and harmonic analysis.
When Fourier submitted a later competition essay in 1811, the committee (which included Lagrange, Laplace, Malus and 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.
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 at the time Fourier completed his original work. Fourier originally defined the Fourier series for realvalued functions of real arguments, and using the sine and cosine functions as the basis set for the decomposition.
Many other Fourierrelated transforms have since been defined, extending the initial idea to other applications. This general area of inquiry is now sometimes called harmonic analysis. A Fourier series, however, can be used only for periodic functions, or for functions on a bounded (compact) interval.
We can also define the Fourier series for functions of two variables
x
y
[\pi,\pi] x [\pi,\pi]
\begin{align} f(x,y)&=\sum_{j,k}c_{j,k}e^{ijx}e^{iky},\\[5pt] c_{j,k}&=
1  
4\pi^{2} 
\pi  
\int  
\pi 
\pi  
\int  
\pi 
f(x,y)e^{ijx}e^{iky}dxdy. \end{align}
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. In particular, the jpeg image compression standard uses the twodimensional discrete cosine transform, which is a Fourierrelated transform using only the cosine basis functions.
Fourier series of BravaislatticeperiodicfunctionR=n_{1a}_{1}+n_{2a}_{2}+n_{3a}_{3}
n_{i}
a_{i}
f(r)
R:f(r)=f(R+r)
r
r=
x  

+
x  

+
x  

,
where
a_{i}\triangleqa_{i.}
Thus we can define a new function,
g(x_{1,x}_{2,x}_{3)}\triangleqf(r)=f\left
(x  

+x  

+x  

\right).
This new function,
g(x_{1,x}_{2,x}_{3)}
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).}
This enables us to build up a set of Fourier coefficients, each being indexed by three independent integers
m_{1,m}_{2,m}_{3}
one(m  
h  
1, 
x_{2,}x_{3)}\triangleq
1  
a_{1} 
a_{1}  
\int  
0 
g(x_{1,}x_{2,}x_{3) ⋅ }
 
e 
dx_{1}
And then we can write:
g(x_{1,}x_{2,}x_{3)=\sum}
infty  
m_{1=infty} 
one(m  
h  
1, 
x_{2,}x_{3)} ⋅
 
e 
Further defining:
two(m  
\begin{align} h  
1, 
m_{2,}x_{3)}&\triangleq
1  
a_{2} 
a_{2}  
\int  
0 
one(m  
h  
1, 
x_{2,}x_{3) ⋅ }
 
e 
dx_{2}\\[12pt] &=
1  
a_{2} 
a_{2}  
\int  
0 
dx_{2}
1  
a_{1} 
a_{1}  
\int  
0 
dx_{1}g(x_{1,}x_{2,}x_{3) ⋅ }
 
e 
\end{align}
We can write
g
g(x_{1,}x_{2,}x_{3)=\sum}
infty  
m_{1=infty} 
infty  
\sum  
m_{2=infty} 
two(m  
h  
1, 
m_{2,}x_{3)} ⋅
 
e 
⋅
 
e 
Finally applying the same for the third coordinate, we define:
three(m  
\begin{align} h  
1, 
m_{2,}m_{3)}&\triangleq
1  
a_{3} 
a_{3}  
\int  
0 
two(m  
h  
1, 
m_{2,}x_{3) ⋅ }
 
e 
dx_{3}\\[12pt] &=
1  
a_{3} 
a_{3}  
\int  
0 
dx_{3}
1  
a_{2} 
a_{2}  
\int  
0 
dx_{2}
1  
a_{1} 
a_{1}  
\int  
0 
dx_{1}g(x_{1,}x_{2,}x_{3) ⋅ }
 
e 
\end{align}
We write
g
g(x_{1,}x_{2,}x_{3)=\sum}
infty  
m_{1=infty} 
infty  
\sum  
m_{2=infty} 
infty  
\sum  
m_{3=infty} 
three(m  
h  
1, 
m_{2,}m_{3)} ⋅
 
e 
⋅
 
e 
⋅
 
e 
Rearranging:
g(x_{1,}x_{2,}x_{3)=\sum}
m_{1,}m_{2,}m_{3}\in\Z 
three(m  
h  
1, 
m_{2,}m_{3)} ⋅
 
e 
.
Now, every reciprocal lattice vector can be written as
G=\ell_{1g}_{1}+\ell_{2g}_{2}+\ell_{3g}_{3}
l_{i}
g_{i}
g_{i} 
⋅
a_{j}=2\pi\delta  
ij 
G
r
G ⋅ r=\left(\ell_{1g}_{1}+\ell_{2g}_{2}+\ell_{3g}_{3}\right) ⋅ \left
(x  

+
x  

+x  

\right)=2\pi\left(
x  

+x  

+x  

\right).
And so it is clear that in our expansion, the sum is actually over reciprocal lattice vectors:
f(r)=\sum_{G}h(G) ⋅ e^{i},
where
h(G)=
1  
a_{3} 
a_{3}  
\int  
0 
dx_{3}
1  
a_{2} 
a_{2}  
\int  
0 
dx_{2}
1  
a_{1} 
a_{1}  
\int  
0 
dx_{1}
f\left(x  

+
x  

+
x  

\right) ⋅ e^{i}.
Assuming
r=(x,y,z)=
x  

+x  

+x  

,
we can solve this system of three linear equations for
x
y
z
x_{1}
x_{2}
x_{3}
x
y
z
x_{1}
x_{2}
x_{3}
\begin{vmatrix} \dfrac{\partialx_{1}{\partial}x}&\dfrac{\partialx_{1}{\partial}y}&\dfrac{\partialx_{1}{\partial}z}\\[12pt] \dfrac{\partialx_{2}{\partial}x}&\dfrac{\partialx_{2}{\partial}y}&\dfrac{\partialx_{2}{\partial}z}\\[12pt] \dfrac{\partialx_{3}{\partial}x}&\dfrac{\partialx_{3}{\partial}y}&\dfrac{\partialx_{3}{\partial}z} \end{vmatrix}
which after some calculation and applying some nontrivial crossproduct identities can be shown to be equal to:
a_{1}a_{2}a_{3}  
a_{1 ⋅ (a}_{2} x a_{3)} 
(it may be advantageous for the sake of simplifying calculations, to work in such a cartesian coordinate system, in which it just so happens that
a_{1}
a_{2}
a_{3}
a_{1}
a_{2}
a_{3}
dx_{1}dx_{2}dx_{3}=
a_{1}a_{2}a_{3}  
a_{1 ⋅ (a}_{2} x a_{3)} 
⋅ dxdydz.
We can write now
h(G)
x_{1}
x_{2}
x_{3}
h(G)=
1  
a_{1 ⋅ (a}_{2} x a_{3)} 
\int_{C}drf(r) ⋅ e^{i}
writing
dr
dxdydz
C
a_{1 ⋅ (a}_{2} x a_{3)}
See main article: Hilbert space. In the language of Hilbert spaces, the set of functions
inx  
\left\{e  
n=e 
:n\in\Z\right\}
L^{2([\pi,\pi])}
[\pi,\pi]
f
g
\langlef,g\rangle \triangleq
1  
2\pi 
\pi  
\int  
\pi 
f(x)g^{*(x)dx,}
g^{*}(x)
g(x).
infty  
f=\sum  
n=infty 
\langlef,e_{n}\ranglee_{n.}
\pi  
\int  
\pi 
\cos(mx)\cos(nx)dx=
1  
2 
\pi  
\int  
\pi 
\cos((nm)x)+\cos((n+m)x)dx=\pi\delta_{mn}, m,n\ge1,
\pi  
\int  
\pi 
\sin(mx)\sin(nx)dx=
1  
2 
\pi  
\int  
\pi 
\cos((nm)x)\cos((n+m)x)dx=\pi\delta_{mn}, m,n\ge1
\pi  
\int  
\pi 
\cos(mx)\sin(nx)dx=
1  
2 
\pi  
\int  
\pi 
\sin((n+m)x)+\sin((nm)x)dx=0;
1
L^{2([\pi,\pi])}
1
\sqrt{2}\cos(nx)
\sqrt{2}\sin(nx)
This table shows some mathematical operations in the time domain and the corresponding effect in the Fourier series coefficients. Notation:
f(x),g(x)
P
x\in[0,P]
F[n],G[n]
f
g
Property  Time domain  Frequency domain (exponential form)  Remarks  Reference  

Linearity  a ⋅ f(x)+b ⋅ g(x)  a ⋅ F[n]+b ⋅ G[n]  a,b\inC  
Time reversal / Frequency reversal  f(x)  F[n]  
Time conjugation  f^{*(x)}  F^{*[n]}  
Time reversal & conjugation  f^{*(x)}  F^{*[n]}  
Real part in time  \operatorname{Re}{(f(x))} 
(F[n]+F^{*[n])}  
Imaginary part in time  \operatorname{Im}{(f(x))} 
(F[n]F^{*[n])}  
Real part in frequency 
(f(x)+f^{*(x))}  \operatorname{Re}{(F[n])}  
Imaginary part in frequency 
(f(x)f^{*(x))}  \operatorname{Im}{(F[n])}  
Shift in time / Modulation in frequency  f(xx_{0)}  F[n] ⋅
 x_{0}\inR  ^{[11]}  
Shift in frequency / Modulation in time  f(x) ⋅
 F[nn_{0]}  n_{0}\inZ 
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 onetoone mapping between the four components of a complex time function and the four components of its complex frequency transform:^{[12]}
\begin{array}{rccccccccc} Timedomain&f&=&
f  
_{RE} 
&+&
f  
_{RO} 
&+&i
f  
_{IE} 
&+
&\underbrace{i f  
_{IO} 
From this, various relationships are apparent, for example:
If
f
\to \infty 
If
f
L^{2(P)}
P
If
c_{0,}c_{\pm},c_{\pm},\ldots
f\inL^{2(P)}
F[n]=c_{n}
n
Given
P
f  
_{P} 
g  
_{P} 
F[n]
G[n],
n\inZ,
P
F
G
P
\left\{c_{n}\right\}_{n}
c_{0(Z)}
L^{1([0,2\pi])}
\ell^{2(Z)}
We say that
f
C^{k(T)}
f
R
k
f\inC^{1(T)}
\widehat{f'}[n]
f'
\widehat{f}[n]
f
\widehat{f'}[n]=in\widehat{f}[n]
f\inC^{k(T)}
\widehat{f^{(k)}
k\geq1
\widehat{f^{(k)}
n\toinfty
n^{k\widehat{f}[n]}
k\geq1
See main article: Compact group, Lie group and 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. Typical examples include those classical groups that are compact. This generalizes the Fourier transform to all spaces of the form L^{2}(G), where G is a compact group, in such a way that the Fourier transform carries convolutions to pointwise products. The Fourier series exists and converges in similar ways to the 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.
See main article: Laplace operator and Riemannian manifold.
If the domain is not a group, then there is no intrinsically defined convolution. However, if
X
X
X
L^{2(X)}
X
[\pi,\pi]
X
See main article: Pontryagin duality.
The generalization to compact groups discussed above does not generalize to noncompact, nonabelian groups. However, there is a straightforward generalization to Locally Compact Abelian (LCA) groups.
This generalizes the Fourier transform to
L^{1(G)}
L^{2(G)}
G
G
[\pi,\pi]
G
R
Some common pairs of periodic functions and their Fourier Series coefficients are shown in the table below. The following notation applies:
f(x)
0<x\leP
a_{0,}a_{n,}b_{n}
f
Time domain f(x)  Plot  Frequency domain (sinecosine form) \begin{align}&a_{0}\ &a_{n} forn\ge1\ &b_{n} forn\ge1\end{align}  Remarks  Reference  

f(x)=A\left  \sin\left(\fracx\right)\right  \quad \text 0 \le x < P  \begin{align} a_{0}=&
\\ a_{n}=&\begin{cases}
& neven\\ 0& nodd \end{cases}\\ b_{n}=&0\\ \end{align}  Fullwave rectified sine  ^{[14]}  
f(x)=\begin{cases} A\sin\left(
x\right)& for0\lex<P/2\\ 0& forP/2\lex<P\\ \end{cases}  \begin{align} a_{0}=&
\\ a_{n}=&\begin{cases}
& neven\\ 0& nodd \end{cases}\\ b_{n}=&\begin{cases}
& n=1\\ 0& n>1 \end{cases}\\ \end{align}  Halfwave rectified sine  
f(x)=\begin{cases} A& for0\lex<D ⋅ P\\ 0& forD ⋅ P\lex<P\\ \end{cases}  \begin{align} a_{0}=&2AD\\ a_{n}=&
\sin\left(2\pinD\right)\\ b_{n}=&
\left(\sin\left(\pinD\right)\right)^{2\\ \end{align}}  0\leD\le1  
for0\lex<P  \begin{align} a_{0}=&A\\ a_{n}=&0\\ b_{n}=&
\\ \end{align}  
for0\lex<P  \begin{align} a_{0}=&A\\ a_{n}=&0\\ b_{n}=&
\\ \end{align}  
\left(x
\right)^{2} for0\lex<P  \begin{align} a_{0}=&
\\ a_{n}=&
\\ b_{n}=&0\\ \end{align} 
See main article: Convergence of Fourier series.
Recalling,
s  
_{N} 
(x)=
N  
\sum  
n=N 
S[n] e^{i\tfrac{2\pi}{P}nx},
it is a trigonometric polynomial of degree
N
p  
_{N} 
N  
(x)=\sum  
n=N 
p[n] e^{i\tfrac{2\pi}{P}nx}.
Parseval's theorem implies that:
Theorem. The trigonometric polynomialis the unique best trigonometric polynomial of degree
s _{N} approximatingN
, in the sense that, for any trigonometric polynomials(x)
of degree
p _{N} ≠
s _{N} , we have:N
where the Hilbert space norm is defined as:
\s _{N} s\_{2}<
\p _{N} s\_{2,}
\g\_{2}=\sqrt{{1\overP}\int_{P}g(x)^{2}dx}.
See also: Gibbs phenomenon. Because of the least squares property, and because of the completeness of the Fourier basis, we obtain an elementary convergence result.
Theorem. If
s
L^{2}(P)
P
s_{infty}
s
L^{2}(P)
\s  
_{N} 
s\_{2}
N → infty
We have already mentioned that if
s
(i ⋅ n)S[n]
s'
s_{infty}
s
s
Theorem. If
s\inC^{1(T)}
s_{infty}
s
This result can be proven easily if
s
C^{2}
n^{2S[n]}
n → infty
s
s
\alpha>1/2
\sup_{x}s(x)
s  
_{N} 
(x)\le\sum_{n}S[n]
Many other results concerning the convergence of Fourier series are known, ranging from the moderately simple result that the series converges at
x
s
x
L^{2}
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".^{[15]} ^{[16]} ^{[17]} ^{[18]}
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 Tperiodic function need not converge pointwise. The uniform boundedness principle yields a simple nonconstructive proof of this fact.
In 1922, Andrey Kolmogorov published an article titled Une série de FourierLebesgue divergente presque partout in which he gave an example of a Lebesgueintegrable function whose Fourier series diverges almost everywhere. He later constructed an example of an integrable function whose Fourier series diverges everywhere .