Fourier series

Definition

The Fourier series of a complex-valued P-periodic function ${\displaystyle s(x)}$, integrable over the interval ${\displaystyle [0,P]}$ on the real line, is defined as a trigonometric series of the form $${\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_n{e}^{i2\pi \frac{n}{P}x},}$$ such that the Fourier coefficients ${\displaystyle c_n}$ are complex numbers defined by the integral $${\displaystyle c_n=\frac{1}{P}{\int }_0^{P}s(x)e^{-i2\pi \frac{n}{P}x}dx.}$$ The series does not necessarily converge (in the pointwise sense) and, even if it does, it is not necessarily equal to ${\displaystyle s(x)}$. Only when certain conditions are satisfied (e.g. if ${\displaystyle s(x)}$ is continuously differentiable) does the Fourier series converge to ${\displaystyle s(x)}$, i.e., $${\displaystyle s(x)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_n{e}^{i2\pi \frac{n}{P}x}.}$$ For functions satisfying the Dirichlet sufficiency conditions, pointwise convergence holds. However, these are not necessary conditions and there are many theorems about different types of convergence of Fourier series (e.g. uniform convergence or mean convergence). The definition naturally extends to the Fourier series of a (periodic) distribution ${\displaystyle s}$ (also called Fourier-Schwartz series). Then the Fourier series converges to ${\displaystyle s(x)}$ in the distribution sense.

The process of determining the Fourier coefficients of a given function or signal is called analysis, while forming the associated trigonometric series (or its various approximations) is called synthesis.

Synthesis

A Fourier series can be written in several equivalent forms, shown here as the ${\displaystyle N^{\text{th}}}$ partial sums ${\displaystyle s_N(x)}$ of the Fourier series of ${\displaystyle s(x)}$:

Sine-cosine form

Exponential form

The harmonics are indexed by an integer, ${\displaystyle n,}$ which is also the number of cycles the corresponding sinusoids make in interval ${\displaystyle P}$. Therefore, the sinusoids have:

• a wavelength equal to ${\displaystyle \frac{P}{n}}$ in the same units as ${\displaystyle x}$.
• a frequency equal to ${\displaystyle \frac{n}{P}}$ in the reciprocal units of ${\displaystyle x}$.

These series can represent functions that are just a sum of one or more frequencies in the harmonic spectrum. In the limit ${\displaystyle N\to \mathrm{\infty }}$, a trigonometric series can also represent the intermediate frequencies or non-sinusoidal functions because of the infinite number of terms.

Analysis

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 where variable ${\displaystyle x}$ represents frequency instead of time. In general, the coefficients are determined by analysis of a given function ${\displaystyle s(x)}$ whose domain of definition is an interval of length ${\displaystyle P}$.

Fourier coefficients

The ${\displaystyle \frac{2}{P}}$ scale factor follows from substituting Eq.1 into Eq.3 and utilizing the orthogonality of the trigonometric system. The equivalence of Eq.1 and Eq.2 follows from Euler's formula $${\displaystyle \cos x=\frac{e^{ix}+e^{-ix}}{2},\sin x=\frac{e^{ix}-e^{-ix}}{2i},}$$ resulting in:

Exponential form coefficients

${\displaystyle c_n=\{\begin{array}{ll}\frac{1}{2}(a_n+i{b}_n)& \text{if }n>0,\\ a_n& \text{if }n=0,\\ \frac{1}{2}(a_{-n}-i{b}_{-n})& \text{if }n<0,\end{array}}$

with ${\displaystyle c_0}$ being the mean value of ${\displaystyle s}$ on the interval ${\displaystyle P}$. Conversely:

Inverse relationships

${\displaystyle \begin{array}{ccc}{a}_0& =c_0& \\ a_n& =c_n+c_{-n}& \text{for}n>0\\ b_n& =(c_n-c_{-n})/i& \text{for}n>0\end{array}}$

Example

Consider a sawtooth function: $${\displaystyle s(x)=s(x+2\pi k)=\frac{x}{\pi },\mathrm{f}\mathrm{o}\mathrm{r}-\pi <x<\pi ,\text{ and }k\in \mathbb{Z}.}$$ In this case, the Fourier coefficients are given by $${\displaystyle \begin{array}{cc}{a}_0& =0.\\ a_n& =\frac{1}{\pi }{\int }_{-\pi }^{\pi }s(x)\cos (nx)dx=0,n\ge 1.\\ b_n& =\frac{1}{\pi }{\int }_{-\pi }^{\pi }s(x)\sin (nx)dx\\ & =-\frac{2}{\pi n}\cos (n\pi )+\frac{2}{{\pi }^2{n}^2}\sin (n\pi )\\ & =\frac{2(-1{)}^{n+1}}{\pi n},n\ge 1.\end{array}}$$ It can be shown that the Fourier series converges to ${\displaystyle s(x)}$ at every point ${\displaystyle x}$ where ${\displaystyle s}$ is differentiable, and therefore: $${\displaystyle \begin{array}{cc}s(x)& =a_0+\sum _{n=1}^{\mathrm{\infty }}\left[a_n\cos \left(nx\right)+b_n\sin \left(nx\right)\right]\\ & =\frac{2}{\pi }\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n+1}}{n}\sin (nx),\mathrm{f}\mathrm{o}\mathrm{r}(x-\pi )\text{is not a multiple of}2\pi .\end{array}}$$ When ${\displaystyle x=\pi }$, the Fourier series converges to 0, which is the half-sum of the left- and right-limit of ${\displaystyle s}$ at ${\displaystyle x=\pi }$. This is a particular instance of the Dirichlet theorem for Fourier series.

This example leads to a solution of the Basel problem.

Amplitude-phase form

If the function ${\displaystyle s(x)}$ is real-valued then the Fourier series can also be represented as

Amplitude-phase form

where ${\displaystyle A_n}$ is the amplitude and ${\displaystyle {\phi }_n}$ is the phase shift of the ${\displaystyle n^{th}}$ harmonic.

The equivalence of Eq.4 and Eq.1 follows from the trigonometric identity: $${\displaystyle \cos \left(2\pi \frac{n}{P}x-{\phi }_n\right)=\cos ({\phi }_n)\cos \left(2\pi \frac{n}{P}x\right)+\sin ({\phi }_n)\sin \left(2\pi \frac{n}{P}x\right),}$$ which implies $${\displaystyle a_n=A_n\cos ({\phi }_n)\text{and}{b}_n=A_n\sin ({\phi }_n)}$$

are the rectangular coordinates of a vector written in polar coordinates as $${\displaystyle A_n\mathrm{\angle }{\phi }_n=a_n+i{b}_n}$$ where $${\displaystyle A_n=\sqrt{a_n^2+b_n^2}\text{and}{\phi }_n=\mathrm{atan2}(b_n,a_n)=-\mathrm{Arg}(c_n)}$$

An example of determining the parameter ${\displaystyle {\phi }_n}$ for one value of ${\displaystyle n}$ is shown in Figure 2. It is the value of ${\displaystyle \phi }$ at the maximum correlation between ${\displaystyle s(x)}$ and a cosine template, ${\displaystyle \cos (2\pi \frac{n}{P}x-\phi )}$. The blue graph is the cross-correlation function, also known as a matched filter:

${\displaystyle \begin{array}{cc}\mathrm{X}(\phi )& ={\int }_{P}s(x)\cdot \cos \left(2\pi \frac{n}{P}x-\phi \right)dx\phi \in \left[0,2\pi \right]\\ & =\cos (\phi )\underset{X(0)}{\underbrace{{\int }_{P}s(x)\cdot \cos \left(2\pi \frac{n}{P}x\right)dx}}+\sin (\phi )\underset{X(\pi /2)}{\underbrace{{\int }_{P}s(x)\cdot \sin \left(2\pi \frac{n}{P}x\right)dx}}\end{array}}$

Fortunately, it is not necessary to evaluate this entire function, because its derivative is zero at the maximum: $${\displaystyle X^{\prime }(\phi )=\sin (\phi )\cdot X(0)-\cos (\phi )\cdot X(\pi /2)=0,\text{at}\phi ={\phi }_n.}$$ Hence $${\displaystyle {\phi }_n\equiv \arctan (b_n/a_n)=\arctan (X(\pi /2)/X(0)).}$$

Common notations

The notation ${\displaystyle c_n}$ is inadequate for discussing the Fourier coefficients of several different functions. Therefore, it is customarily replaced by a modified form of the function (${\displaystyle s,}$ in this case), such as ${\displaystyle \hat{s}(n)}$ or ${\displaystyle S[n],}$ and functional notation often replaces subscripting:

${\displaystyle \begin{array}{cccc}s(x)& =\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\hat{s}(n)\cdot e^{i2\pi \frac{n}{P}x}& & \text{common mathematics notation}\\ & =\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}S[n]\cdot e^{i2\pi \frac{n}{P}x}& & \text{common engineering notation}\end{array}}$

In engineering, particularly when the variable ${\displaystyle x}$ represents time, the coefficient sequence is called a 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 modulate a Dirac comb:

${\displaystyle S(f)\triangleq \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}S[n]\cdot \delta \left(f-\frac{n}{P}\right),}$

where ${\displaystyle f}$ represents a continuous frequency domain. When variable ${\displaystyle x}$ has units of seconds, ${\displaystyle f}$ has units of hertz. The "teeth" of the comb are spaced at multiples (i.e. harmonics) of ${\displaystyle \frac{1}{P}}$, which is called the fundamental frequency. ${\displaystyle s(x)}$ can be recovered from this representation by an inverse Fourier transform:

${\displaystyle \begin{array}{cc}{\mathcal{F}}^{-1}\{S(f)\}& ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left(\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}S[n]\cdot \delta \left(f-\frac{n}{P}\right)\right)e^{i2\pi fx}df,\\ & =\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}S[n]\cdot {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta \left(f-\frac{n}{P}\right)e^{i2\pi fx}df,\\ & =\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}S[n]\cdot e^{i2\pi \frac{n}{P}x}\triangleq s(x).\end{array}}$

The constructed function ${\displaystyle S(f)}$ 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.

Table of common Fourier series

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

• ${\displaystyle s(x)}$ designates a periodic function with period ${\displaystyle P.}$
• ${\displaystyle a_0,a_n,b_n}$ designate the Fourier series coefficients (sine-cosine form) of the periodic function ${\displaystyle s(x).}$

Table of basic transformation rules

This table shows some mathematical operations in the time domain and the corresponding effect in the Fourier series coefficients. Notation:

• Complex conjugation is denoted by an asterisk.
• ${\displaystyle s(x),r(x)}$ designate ${\displaystyle P}$-periodic functions or functions defined only for ${\displaystyle x\in [0,P].}$
• ${\displaystyle S[n],R[n]}$ designate the Fourier series coefficients (exponential form) of ${\displaystyle s}$ and ${\displaystyle r.}$

Properties

Symmetry relations

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:

${\displaystyle \begin{array}{cccccccccc}\mathsf{T}\mathsf{i}\mathsf{m}\mathsf{e}\mathsf{d}\mathsf{o}\mathsf{m}\mathsf{a}\mathsf{i}\mathsf{n}& s& =& s_{\mathrm{R}\mathrm{E}}& +& s_{\mathrm{R}\mathrm{O}}& +& i{s}_{\mathrm{I}\mathrm{E}}& +& i{s}_{\mathrm{I}\mathrm{O}}\\ & \Updownarrow \mathcal{F}& & \Updownarrow \mathcal{F}& & \Updownarrow \mathcal{F}& & \Updownarrow \mathcal{F}& & \Updownarrow \mathcal{F}\\ \mathsf{F}\mathsf{r}\mathsf{e}\mathsf{q}\mathsf{u}\mathsf{e}\mathsf{n}\mathsf{c}\mathsf{y}\mathsf{d}\mathsf{o}\mathsf{m}\mathsf{a}\mathsf{i}\mathsf{n}& S& =& S_{\mathrm{R}\mathrm{E}}& +& i{S}_{\mathrm{I}\mathrm{O}}& +& i{S}_{\mathrm{I}\mathrm{E}}& +& S_{\mathrm{R}\mathrm{O}}\end{array}}$

From this, various relationships are apparent, for example:

• The transform of a real-valued function ${\displaystyle (s_{\mathrm{R}\mathrm{E}}+s_{\mathrm{R}\mathrm{O}})}$ is the conjugate symmetric function ${\displaystyle S_{\mathrm{R}\mathrm{E}}+i{S}_{\mathrm{I}\mathrm{O}}.}$ Conversely, a conjugate symmetric transform implies a real-valued time-domain.
• The transform of an imaginary-valued function ${\displaystyle (i{s}_{\mathrm{I}\mathrm{E}}+i{s}_{\mathrm{I}\mathrm{O}})}$ is the conjugate antisymmetric function ${\displaystyle S_{\mathrm{R}\mathrm{O}}+i{S}_{\mathrm{I}\mathrm{E}},}$ and the converse is true.
• The transform of a conjugate symmetric function ${\displaystyle (s_{\mathrm{R}\mathrm{E}}+i{s}_{\mathrm{I}\mathrm{O}})}$ is the real-valued function ${\displaystyle S_{\mathrm{R}\mathrm{E}}+S_{\mathrm{R}\mathrm{O}},}$ and the converse is true.
• The transform of a conjugate antisymmetric function ${\displaystyle (s_{\mathrm{R}\mathrm{O}}+i{s}_{\mathrm{I}\mathrm{E}})}$ is the imaginary-valued function ${\displaystyle i{S}_{\mathrm{I}\mathrm{E}}+i{S}_{\mathrm{I}\mathrm{O}},}$ and the converse is true.

Riemann–Lebesgue lemma

If ${\displaystyle S}$ is integrable, $\underset{|n|\to \mathrm{\infty }}{lim}S[n]=0$, $\underset{n\to +\mathrm{\infty }}{lim}{a}_n=0$ and $\underset{n\to +\mathrm{\infty }}{lim}{b}_n=0.$

Parseval's theorem

If ${\displaystyle s}$ belongs to ${\displaystyle L^2(P)}$ (periodic over an interval of length ${\displaystyle P}$) then: $${\displaystyle \frac{1}{P}{\int }_P|s(x){|}^{2}dx=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}|S[n]{|}^2.}$$

Plancherel's theorem

If ${\displaystyle c_0,c_{\pm 1},c_{\pm 2},\dots }$ are coefficients and $\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}|c_n{|}^2<\mathrm{\infty }$ then there is a unique function ${\displaystyle s\in L^2(P)}$ such that ${\displaystyle S[n]=c_n}$ for every ${\displaystyle n}$.

Convolution theorems

Given ${\displaystyle P}$-periodic functions, ${\displaystyle s_P}$ and ${\displaystyle r_P}$ with Fourier series coefficients ${\displaystyle S[n]}$ and ${\displaystyle R[n],}$ ${\displaystyle n\in \mathbb{Z},}$

• The pointwise product: $${\displaystyle h_P(x)\triangleq s_P(x)\cdot r_P(x)}$$ is also ${\displaystyle P}$-periodic, and its Fourier series coefficients are given by the discrete convolution of the ${\displaystyle S}$ and ${\displaystyle R}$ sequences: $${\displaystyle H[n]=\{S\ast R\}[n].}$$
• The periodic convolution: $${\displaystyle h_P(x)\triangleq {\int }_P{s}_P(\tau )\cdot r_P(x-\tau )d\tau }$$ is also ${\displaystyle P}$-periodic, with Fourier series coefficients: $${\displaystyle H[n]=P\cdot S[n]\cdot R[n].}$$
• A doubly infinite sequence ${\displaystyle {\left\{c_n\right\}}_{n\in Z}}$ in ${\displaystyle c_0(\mathbb{Z})}$ is the sequence of Fourier coefficients of a function in ${\displaystyle L^1([0,2\pi ])}$ if and only if it is a convolution of two sequences in ${\displaystyle {\ell }^2(\mathbb{Z})}$. See

Derivative property

If ${\displaystyle s}$ is a ${\displaystyle P}$-periodic function on ${\displaystyle \mathbb{R}}$ which is ${\displaystyle k}$ times differentiable, and its ${\displaystyle k^{\text{th}}}$ derivative is continuous, then ${\displaystyle s}$ belongs to the function space ${\displaystyle C^k(\mathbb{R})}$.

• If ${\displaystyle s\in C^k(\mathbb{R})}$, then the Fourier coefficients of the ${\displaystyle k^{\text{th}}}$ derivative of ${\displaystyle s}$ can be expressed in terms of the Fourier coefficients ${\displaystyle \hat{s}[n]}$ of ${\displaystyle s}$, via the formula $${\displaystyle \widehat{s^{(k)}}[n]={\left(i\frac{2\pi n}{P}\right)}^k\hat{s}[n].}$$ In particular, since for any fixed ${\displaystyle k\ge 1}$ we have ${\displaystyle \widehat{s^{(k)}}[n]\to 0}$ as ${\displaystyle n\to \mathrm{\infty }}$, it follows that ${\displaystyle |n{|}^k\hat{s}[n]}$ tends to zero, i.e., the Fourier coefficients converge to zero faster than the ${\displaystyle k^{\text{th}}}$ power of ${\displaystyle |n|}$.

Compact groups

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 L²(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 [−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.

Riemannian manifolds

If the domain is not a group, then there is no intrinsically defined convolution. However, if ${\displaystyle X}$ is a 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 ${\displaystyle X}$. Then, by analogy, one can consider heat equations on ${\displaystyle X}$. 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 ${\displaystyle L^2(X)}$, where ${\displaystyle X}$ is a Riemannian manifold. The Fourier series converges in ways similar to the ${\displaystyle [-\pi ,\pi ]}$ case. A typical example is to take ${\displaystyle X}$ to be the sphere with the usual metric, in which case the Fourier basis consists of spherical harmonics.

Locally compact Abelian groups

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

Extensions

Fourier-Stieltjes series

Formally, the Fourier-Stieltjes series can be defined as the Fourier series whose coefficients are given by $${\displaystyle c_n=\hat{\mu }(n)=\frac{1}{P}{\int }_0^P{e}^{-i2\pi \frac{n}{P}x}d\mu (x),\mathrm{\forall }n\in \mathbb{Z},}$$ for any ${\displaystyle \mu \in M}$, where ${\displaystyle M}$ is the space finite Borel measures on the interval ${\displaystyle [0,P]}$. As such, when ${\displaystyle \mu \in M}$, the function ${\displaystyle \hat{\mu }(n)}$ is also referred to as a Fourier-Stieltjes transform.

This follows from an earlier and more concrete representation of a Radon measure (i.e. a locally finite Borel measure) on ${\displaystyle \mathbb{R}}$, given by F. Riesz. That is, if ${\displaystyle F}$ is a function of bounded variation on the interval ${\displaystyle [0,P]}$ then the Fourier coefficients can be expressed by the Riemann-Stieltjes integral $${\displaystyle c_n=\frac{1}{P}{\int }_0^P{e}^{-i2\pi \frac{n}{P}x}dF(x),\mathrm{\forall }n\in \mathbb{Z},}$$ called the Fourier-Stieltjes coefficients of ${\displaystyle F}$. As the distributional derivative of ${\displaystyle F}$ is a Radon measure, it is subject to the Lebesgue decomposition and can be expressed as ${\displaystyle dF=F^{\prime }dx+d{F}_s}$. If ${\displaystyle d{F}_s=0}$ the expression reduces to the original definition of the Fourier coefficients, hence a Fourier series is a Fourier-Stieltjes series.

The question whether or not ${\displaystyle \mu }$ exists for a given sequence of ${\displaystyle c_n}$ forms the basis of the trigonometric moment problem.

The Fourier series can be generalized still further from measures to distributions. If the Fourier coefficients are determined by a distribution ${\displaystyle F\in {\mathcal{D}}^{\prime }}$ then the series is sometimes described as a Fourier-Schwartz series.

While it is often extremely difficult to decide whether a given series is a Fourier or a Fourier-Stieltjes series, deciding whether or not it is a Fourier-Schwartz series is relatively trivial.

Fourier series on a square

We can also define the Fourier series for functions of two variables ${\displaystyle x}$ and ${\displaystyle y}$ in the square ${\displaystyle [-\pi ,\pi ]\times [-\pi ,\pi ]}$: $${\displaystyle \begin{array}{cc}f(x,y)& =\sum _{j,k\in \mathbb{Z}}{c}_{j,k}{e}^{ijx}{e}^{iky},\\ c_{j,k}& =\frac{1}{4{\pi }^2}{\int }_{-\pi }^{\pi }{\int }_{-\pi }^{\pi }f(x,y)e^{-ijx}{e}^{-iky}dxdy.\end{array}}$$

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 two-dimensional discrete cosine transform, a discrete form of the 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.

Fourier series of a Bravais-lattice-periodic function

A three-dimensional Bravais lattice is defined as the set of vectors of the form $${\displaystyle \mathbf{R}=n_1{\mathbf{a}}_1+n_2{\mathbf{a}}_2+n_3{\mathbf{a}}_3}$$ where ${\displaystyle n_i}$ are integers and ${\displaystyle {\mathbf{a}}_i}$ are three linearly independent but not necessarily orthogonal vectors. Let us consider some function ${\displaystyle f(\mathbf{r})}$ with the same periodicity as the Bravais lattice, i.e. ${\displaystyle f(\mathbf{r})=f(\mathbf{R}+\mathbf{r})}$ for any lattice vector ${\displaystyle \mathbf{R}}$. This situation frequently occurs in solid-state physics where ${\displaystyle f(\mathbf{r})}$ might, for example, represent the effective potential that an electron "feels" inside a periodic crystal. In presence of such a periodic potential, the quantum-mechanical description of the electron results in a periodically modulated plane-wave commonly known as Bloch state.

In order to develop ${\displaystyle f(\mathbf{r})}$ in a Fourier series, it is convenient to introduce an auxiliary function $${\displaystyle g(x_1,x_2,x_3)\triangleq f(\mathbf{r})=f\left(x_1\frac{{\mathbf{a}}_1}{a_1}+x_2\frac{{\mathbf{a}}_2}{a_2}+x_3\frac{{\mathbf{a}}_3}{a_3}\right).}$$ Both ${\displaystyle f(\mathbf{r})}$ and ${\displaystyle g(x_1,x_2,x_3)}$ contain essentially the same information. However, instead of the position vector ${\displaystyle \mathbf{r}}$, the arguments of ${\displaystyle g}$ are coordinates ${\displaystyle x_{1,2,3}}$ along the unit vectors ${\displaystyle {\mathbf{a}}_i/a_i}$ of the Bravais lattice, such that ${\displaystyle g}$ is an ordinary periodic function in these variables,$${\displaystyle g(x_1,x_2,x_3)=g(x_1+a_1,x_2,x_3)=g(x_1,x_2+a_2,x_3)=g(x_1,x_2,x_3+a_3)\mathrm{\forall }{x}_1,x_2,x_3.}$$ This trick allows us to develop ${\displaystyle g}$ as a multi-dimensional Fourier series, in complete analogy with the square-periodic function discussed in the previous section. Its Fourier coefficients are$${\displaystyle \begin{array}{c}c(m_1,m_2,m_3)=\frac{1}{a_3}{\int }_0^{a_3}d{x}_3\frac{1}{a_2}{\int }_0^{a_2}d{x}_2\frac{1}{a_1}{\int }_0^{a_1}d{x}_{1}g(x_1,x_2,x_3)e^{-i2\pi \left(\frac{m_1}{a_1}{x}_1+\frac{m_2}{a_2}{x}_2+\frac{m_3}{a_3}{x}_3\right)}\end{array},}$$ where ${\displaystyle m_1,m_2,m_3}$ are all integers. ${\displaystyle c(m_1,m_2,m_3)}$ plays the same role as the coefficients ${\displaystyle c_{j,k}}$ in the previous section but in order to avoid double subscripts we note them as a function.

Once we have these coefficients, the function ${\displaystyle g}$ can be recovered via the Fourier series $${\displaystyle g(x_1,x_2,x_3)=\sum _{m_1,m_2,m_3\in \mathbb{Z}}c(m_1,m_2,m_3)e^{i2\pi \left(\frac{m_1}{a_1}{x}_1+\frac{m_2}{a_2}{x}_2+\frac{m_3}{a_3}{x}_3\right)}.}$$ We would now like to abandon the auxiliary coordinates ${\displaystyle x_{1,2,3}}$ and to return to the original position vector ${\displaystyle \mathbf{r}}$. This can be achieved by means of the reciprocal lattice whose vectors ${\displaystyle {\mathbf{b}}_{1,2,3}}$ are defined such that they are orthonormal (up to a factor ${\displaystyle 2\pi }$) to the original Bravais vectors ${\displaystyle {\mathbf{a}}_{1,2,3}}$, $${\displaystyle {\mathbf{a}}_i\cdot {\mathbf{b}}_{\mathbf{j}}=2\pi {\delta }_{ij},}$$with ${\displaystyle {\delta }_{ij}}$ the Kronecker delta. With this, the scalar product between a reciprocal lattice vector ${\displaystyle \mathbf{Q}}$ and an arbitrary position vector ${\displaystyle \mathbf{r}}$ written in the Bravais lattice basis becomes $${\displaystyle \mathbf{Q}\cdot \mathbf{r}=\left(m_1{\mathbf{b}}_1+m_2{\mathbf{b}}_2+m_3{\mathbf{b}}_3\right)\cdot \left(x_1\frac{{\mathbf{a}}_1}{a_1}+x_2\frac{{\mathbf{a}}_2}{a_2}+x_3\frac{{\mathbf{a}}_3}{a_3}\right)=2\pi \left(x_1\frac{m_1}{a_1}+x_2\frac{m_2}{a_2}+x_3\frac{m_3}{a_3}\right),}$$which is exactly the expression occurring in the Fourier exponents. The Fourier series for ${\displaystyle f(\mathbf{r})=g(x_1,x_2,x_3)}$ can therefore be rewritten as a sum over all the reciprocal lattice vectors ${\displaystyle \mathbf{Q}=m_1{\mathbf{b}}_1+m_2{\mathbf{b}}_2+m_3{\mathbf{b}}_3}$,$${\displaystyle f(\mathbf{r})=\sum _{\mathbf{Q}}c(\mathbf{Q})e^{i\mathbf{Q}\cdot \mathbf{r}},}$$ and the coefficients are$${\displaystyle c(\mathbf{Q})=\frac{1}{a_3}{\int }_0^{a_3}d{x}_3\frac{1}{a_2}{\int }_0^{a_2}d{x}_2\frac{1}{a_1}{\int }_0^{a_1}d{x}_{1}f\left(x_1\frac{{\mathbf{a}}_1}{a_1}+x_2\frac{{\mathbf{a}}_2}{a_2}+x_3\frac{{\mathbf{a}}_3}{a_3}\right)e^{-i\mathbf{Q}\cdot \mathbf{r}}.}$$ The remaining task will be to convert this integral over lattice coordinates back into a volume integral. The relation between the lattice coordinates ${\displaystyle x_{1,2,3}}$ and the original cartesian coordinates ${\displaystyle \mathbf{r}=(x,y,z)}$ is a linear system of equations, $${\displaystyle \mathbf{r}=x_1\frac{{\mathbf{a}}_1}{a_1}+x_2\frac{{\mathbf{a}}_2}{a_2}+x_3\frac{{\mathbf{a}}_3}{a_3},}$$which, when written in matrix form, $${\displaystyle \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\mathbf{J}\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]=\left[\begin{array}{c}\frac{{\mathbf{a}}_1}{a_1},\frac{{\mathbf{a}}_2}{a_2},\frac{{\mathbf{a}}_3}{a_3}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right],}$$involves a constant matrix ${\displaystyle \mathbf{J}}$ whose columns are the unit vectors ${\displaystyle {\mathbf{a}}_j/a_j}$ of the Bravais lattice. When changing variables from ${\displaystyle \mathbf{r}}$ to ${\displaystyle (x_1,x_2,x_3)}$ in an integral, the same matrix ${\displaystyle \mathbf{J}}$ appears as a Jacobian matrix$${\displaystyle \mathbf{J}=\left[\begin{array}{ccc}{\displaystyle \frac{\mathrm{\partial }x}{\mathrm{\partial }{x}_1}}& {\displaystyle \frac{\mathrm{\partial }x}{\mathrm{\partial }{x}_2}}& {\displaystyle \frac{\mathrm{\partial }x}{\mathrm{\partial }{x}_3}}\\ {\displaystyle \frac{\mathrm{\partial }y}{\mathrm{\partial }{x}_1}}& {\displaystyle \frac{\mathrm{\partial }y}{\mathrm{\partial }{x}_2}}& {\displaystyle \frac{\mathrm{\partial }y}{\mathrm{\partial }{x}_3}}\\ {\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{x}_1}}& {\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{x}_2}}& {\displaystyle \frac{\mathrm{\partial }z}{\mathrm{\partial }{x}_3}}\end{array}\right].}$$

Its determinant ${\displaystyle J}$ is therefore also constant and can be inferred from any integral over any domain; here we choose to calculate the volume of the primitive unit cell ${\displaystyle \mathrm{\Gamma }}$ in both coordinate systems: $${\displaystyle V_{\mathrm{\Gamma }}={\int }_{\mathrm{\Gamma }}{d}^{3}r=J{\int }_0^{a_1}d{x}_1{\int }_0^{a_2}d{x}_2{\int }_0^{a_3}d{x}_3=J{a}_1{a}_2{a}_3}$$ The unit cell being a parallelepiped, we have ${\displaystyle V_{\mathrm{\Gamma }}={\mathbf{a}}_1\cdot ({\mathbf{a}}_2\times {\mathbf{a}}_3)}$ and thus $${\displaystyle d^{3}r=Jd{x}_{1}d{x}_{2}d{x}_3=\frac{{\mathbf{a}}_1\cdot ({\mathbf{a}}_2\times {\mathbf{a}}_3)}{a_1{a}_2{a}_3}d{x}_{1}d{x}_{2}d{x}_3.}$$ This allows us to write ${\displaystyle c(\mathbf{Q})}$ as the desired volume integral over the primitive unit cell ${\displaystyle \mathrm{\Gamma }}$ in ordinary cartesian coordinates: $${\displaystyle c(\mathbf{Q})=\frac{1}{{\mathbf{a}}_1\cdot ({\mathbf{a}}_2\times {\mathbf{a}}_3)}{\int }_{\mathrm{\Gamma }}{d}^{3}rf(\mathbf{r})\cdot e^{-i\mathbf{Q}\cdot \mathbf{r}}.}$$

Hilbert space

As the trigonometric series is a special class of orthogonal system, Fourier series can naturally be defined in the context of Hilbert spaces. For example, the space of square-integrable functions on ${\displaystyle [-\pi ,\pi ]}$ forms the Hilbert space ${\displaystyle L^2([-\pi ,\pi ])}$. Its inner product, defined for any two elements ${\displaystyle f}$ and ${\displaystyle g}$, is given by: $${\displaystyle ⟨f,g⟩=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(x)\overline{g(x)}dx.}$$ This space is equipped with the orthonormal basis ${\displaystyle \left\{e_n=e^{inx}:n\in \mathbb{Z}\right\}}$. Then the (generalized) Fourier series expansion of ${\displaystyle f\in L^2([-\pi ,\pi ])}$, given by $${\displaystyle f(x)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_n{e}^{inx},}$$ can be written as $${\displaystyle f=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}⟨f,e_n⟩e_n.}$$

The sine-cosine form follows in a similar fashion. Indeed, the sines and cosines form an orthogonal set: $${\displaystyle {\int }_{-\pi }^{\pi }\cos (mx)\cos (nx)dx=\frac{1}{2}{\int }_{-\pi }^{\pi }\cos ((n-m)x)+\cos ((n+m)x)dx=\pi {\delta }_{mn},m,n\ge 1,}$$ $${\displaystyle {\int }_{-\pi }^{\pi }\sin (mx)\sin (nx)dx=\frac{1}{2}{\int }_{-\pi }^{\pi }\cos ((n-m)x)-\cos ((n+m)x)dx=\pi {\delta }_{mn},m,n\ge 1}$$ (where δ_mn is the Kronecker delta), and $${\displaystyle {\int }_{-\pi }^{\pi }\cos (mx)\sin (nx)dx=\frac{1}{2}{\int }_{-\pi }^{\pi }\sin ((n+m)x)+\sin ((n-m)x)dx=0;}$$ Hence, the set $${\displaystyle \left\{\frac{1}{\sqrt{2}},\frac{\cos x}{\sqrt{2}},\frac{\sin x}{\sqrt{2}},\dots ,\frac{\cos (nx)}{\sqrt{2}},\frac{\sin (nx)}{\sqrt{2}},\dots \right\},}$$ also forms an orthonormal basis for ${\displaystyle L^2([-\pi ,\pi ])}$. 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

In engineering, the Fourier series is generally assumed to converge except at jump discontinuities since the functions encountered in engineering are usually better-behaved than those in other disciplines. In particular, if ${\displaystyle s}$ is continuous and the derivative of ${\displaystyle s(x)}$ (which may not exist everywhere) is square integrable, then the Fourier series of ${\displaystyle s}$ converges absolutely and uniformly to ${\displaystyle s(x)}$. If a function is square-integrable on the interval ${\displaystyle [x_0,x_0+P]}$, then the Fourier series converges to the function almost everywhere. It is possible to define Fourier coefficients for more general functions or distributions, in which case pointwise convergence often fails, and convergence in norm or weak convergence is usually studied.

The theorems proving that a Fourier series is a valid representation of any periodic function (that satisfies the Dirichlet conditions), and informal variations of them that do not specify the convergence conditions, are sometimes referred to generically as Fourier's theorem or the Fourier theorem.

Least squares property

The earlier Eq.2:

${\displaystyle s_N(x)=\sum _{n=-N}^{N}S[n]e^{i2\pi \frac{n}{P}x},}$

is a trigonometric polynomial of degree ${\displaystyle N}$ that can be generally expressed as:

${\displaystyle p_N(x)=\sum _{n=-N}^{N}p[n]e^{i2\pi \frac{n}{P}x}.}$

Parseval's theorem implies that:

Theorem—The trigonometric polynomial ${\displaystyle s_N}$ is the unique best trigonometric polynomial of degree ${\displaystyle N}$ approximating ${\displaystyle s(x)}$, in the sense that, for any trigonometric polynomial ${\displaystyle p_N\ne s_N}$ of degree ${\displaystyle N}$, we have: $${\displaystyle \Vert s_N-s{\Vert }_2<\Vert p_N-s{\Vert }_2,}$$ where the Hilbert space norm is defined as: $${\displaystyle \Vert g{\Vert }_2=\sqrt{\frac{1}{P}{\int }_P|g(x){|}^{2}dx}.}$$

Convergence theorems

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

Theorem—If ${\displaystyle s}$ belongs to ${\displaystyle L^2(P)}$, then ${\displaystyle s_N}$ converges to ${\displaystyle s}$ in ${\displaystyle L^2(P)}$ as ${\displaystyle N\to \mathrm{\infty }}$, that is: $${\displaystyle \underset{N\to \mathrm{\infty }}{lim}\Vert s_N-s{\Vert }_2=0.}$$

If ${\displaystyle s}$ is continuously differentiable, then ${\displaystyle (in)S[n]}$ is the ${\displaystyle n}$th Fourier coefficient of the first derivative ${\displaystyle s^{\prime }}$. Since ${\displaystyle s^{\prime }}$ is continuous, and therefore bounded, it is square-integrable and its Fourier coefficients are square-summable. Then, by the Cauchy–Schwarz inequality,

${\displaystyle (\sum _{n\ne 0}|S[n]|{)}^2\le \sum _{n\ne 0}\frac{1}{n^2}\cdot \sum _{n\ne 0}|nS[n]{|}^2.}$

This means that ${\displaystyle s}$ is absolutely summable. The sum of this series is a continuous function, equal to ${\displaystyle s}$, since the Fourier series converges in ${\displaystyle L^1}$ to ${\displaystyle s}$:

Theorem—If ${\displaystyle s\in C^1(\mathbb{R})}$, then ${\displaystyle s_N}$ converges to ${\displaystyle s}$ uniformly.

This result can be proven easily if ${\displaystyle s}$ is further assumed to be ${\displaystyle C^2}$, since in that case ${\displaystyle n^{2}S[n]}$ tends to zero as ${\displaystyle n\to \mathrm{\infty }}$. More generally, the Fourier series is absolutely summable, thus converges uniformly to ${\displaystyle s}$, provided that ${\displaystyle s}$ satisfies a Hölder condition of order ${\displaystyle \alpha >\frac{1}{2}}$. In the absolutely summable case, the inequality:

${\displaystyle \underset{x}{sup}|s(x)-s_N(x)|\le \sum _{|n|>N}|S[n]|}$

proves uniform convergence.

Many other results concerning the convergence of Fourier series are known, ranging from the moderately simple result that the series converges at ${\displaystyle x}$ if ${\displaystyle s}$ is differentiable at ${\displaystyle x}$, to more sophisticated results such as Carleson's theorem which states that the Fourier series of an ${\displaystyle L^2}$ function converges 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. The uniform boundedness principle yields a simple non-constructive proof of this fact.

In 1922, 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.

It is possible to give explicit examples of a continuous function whose Fourier series diverges at 0: for instance, the even and 2π-periodic function f defined for all x in [0,π] by

${\displaystyle f(x)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}\sin \left[\left(2^{n^3}+1\right)\frac{x}{2}\right].}$

Because the function is even the Fourier series contains only cosines:

${\displaystyle \sum _{m=0}^{\mathrm{\infty }}{C}_m\cos (mx).}$

The coefficients are:

${\displaystyle C_m=\frac{1}{\pi }\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}\left\{\frac{2}{2^{n^3}+1-2m}+\frac{2}{2^{n^3}+1+2m}\right\}}$

As m increases, the coefficients will be positive and increasing until they reach a value of about ${\displaystyle C_m\approx 2/(n^2\pi )}$ at ${\displaystyle m=2^{n^3}/2}$ for some n and then become negative (starting with a value around ${\displaystyle -2/(n^2\pi )}$) and getting smaller, before starting a new such wave. At ${\displaystyle x=0}$ the Fourier series is simply the running sum of ${\displaystyle C_m,}$ and this builds up to around

${\displaystyle \frac{1}{n^2\pi }\sum _{k=0}^{2^{n^3}/2}\frac{2}{2k+1}\sim \frac{1}{n^2\pi }\ln 2^{n^3}=\frac{n}{\pi }\ln 2}$

in the nth wave before returning to around zero, showing that the series does not converge at zero but reaches higher and higher peaks. Note that though the function is continuous, it is not differentiable.