Editing Fourier series (section)

== Properties ==
=== Symmetry relations ===
When the real and imaginary parts of a complex function are decomposed into their [[Even and odd functions#Even–odd decomposition|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:{{sfn|Proakis|Manolakis|1996|p=291}}{{sfn|Oppenheim|Schafer|2010|p=55}}

:<math>
\begin{array}{rlcccccccc}   
\mathsf{Time\ domain} & s & = & s_{\mathrm{RE}} & + & s_{\mathrm{RO}} & + & i\ s_{\mathrm{IE}} & + & i\ s_{\mathrm{IO}} \\
&\Bigg\Updownarrow\mathcal{F} & &\Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F} & &\ \ \Bigg\Updownarrow\mathcal{F}\\
\mathsf{Frequency\ domain} & S & = & S_\mathrm{RE} & + & i\ S_\mathrm{IO}\, & + & i\ S_\mathrm{IE} & + & S_\mathrm{RO}
\end{array}
</math> 

From this, various relationships are apparent, for example''':'''

* The transform of a real-valued function <math>(s_\mathrm{RE}+s_\mathrm{RO})</math> is the [[Even and odd functions#Complex-valued functions|''conjugate symmetric'']] function <math>S_\mathrm{RE}+i\ S_\mathrm{IO}.</math>  Conversely, a ''conjugate symmetric'' transform implies a real-valued time-domain.
* The transform of an imaginary-valued function <math>(i\ s_\mathrm{IE}+i\ s_\mathrm{IO})</math> is the [[Even and odd functions#Complex-valued functions|''conjugate antisymmetric'']] function <math>S_\mathrm{RO}+i\ S_\mathrm{IE},</math> and the converse is true.
* The transform of a [[Even and odd functions#Complex-valued functions|''conjugate symmetric'']] function <math>(s_\mathrm{RE}+i\ s_\mathrm{IO})</math> is the real-valued function <math>S_\mathrm{RE}+S_\mathrm{RO},</math> and the converse is true.
* The transform of a [[Even and odd functions#Complex-valued functions|''conjugate antisymmetric'']] function <math>(s_\mathrm{RO}+i\ s_\mathrm{IE})</math> is the imaginary-valued function <math>i\ S_\mathrm{IE}+i\ S_\mathrm{IO},</math> and the converse is true.

=== Riemann–Lebesgue lemma ===
{{main|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>

=== 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="block">\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 ===
{{main|Convolution theorem#Periodic convolution (Fourier series coefficients)}}

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 ===
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, then <math>s</math> belongs to the [[Function_space#Functional_analysis|function space]] <math>C^k(\mathbb{R})</math>.
* If <math>s \in C^k(\mathbb{R})</math>, then the Fourier coefficients of the <math>k^{\text{th}}</math> derivative of <math>s</math> can be expressed in terms of the Fourier coefficients <math>\widehat{s}[n]</math> of <math>s</math>, via the formula <math display="block">\widehat{s^{(k)}}[n] = (in)^k \widehat{s}[n].</math> In particular, since for any 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, i.e., the Fourier coefficients converge to zero faster than the <math>k^{\text{th}}</math> power of <math>|n|</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]]. Typical examples include those [[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]]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 [[atomic orbital]]s of [[chemistry]] are partially described by [[spherical harmonic]]s, which can be used to produce Fourier series on the [[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 [[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 group|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>.