Beginnings

Joseph Fourier wrote^[13]

${\displaystyle \varphi (y)=a_{0}\cos {\frac {\pi y}{2}}+a_{1}\cos 3{\frac {\pi y}{2}}+a_{2}\cos 5{\frac {\pi y}{2}}+\cdots .}$

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

${\displaystyle a_{k}=\int _{-1}^{1}\varphi (y)\cos(2k+1){\frac {\pi y}{2}}\,dy.}$

— Joseph Fourier, Mémoire sur la propagation de la chaleur dans les corps solides (1807).

This immediately gives any coefficient a_k of the trigonometric series for φ(y) for any function which has such an expansion. It works because if φ has such an expansion, then (under suitable convergence assumptions) the integral $${\displaystyle {\begin{aligned}&\int _{-1}^{1}\varphi (y)\cos(2k+1){\frac {\pi y}{2}}\,dy\\&=\int _{-1}^{1}\left(a\cos {\frac {\pi y}{2}}\cos(2k+1){\frac {\pi y}{2}}+a'\cos 3{\frac {\pi y}{2}}\cos(2k+1){\frac {\pi y}{2}}+\cdots \right)\,dy\end{aligned}}}$$ can be carried out term-by-term. But all terms involving ${\displaystyle \cos(2j+1){\frac {\pi y}{2}}\cos(2k+1){\frac {\pi y}{2}}}$ for j ≠ k vanish when integrated from −1 to 1, leaving only the ${\displaystyle k^{\text{th}}}$ term, which is 1.

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".^[14]

Fourier's motivation

The Fourier series expansion of the sawtooth function (below) looks more complicated than the simple formula ${\displaystyle s(x)={\tfrac {x}{\pi }}}$, 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. For example, consider a metal plate in the shape of a square whose sides measure ${\displaystyle \pi }$ meters, with coordinates ${\displaystyle (x,y)\in [0,\pi ]\times [0,\pi ]}$. 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 ${\displaystyle y=\pi }$, is maintained at the temperature gradient ${\displaystyle T(x,\pi )=x}$ degrees Celsius, for ${\displaystyle x}$ in ${\displaystyle (0,\pi )}$, then one can show that the stationary heat distribution (or the heat distribution after a long time has elapsed) is given by

${\displaystyle T(x,y)=2\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}\sin(nx){\sinh(ny) \over \sinh(n\pi )}.}$

Here, sinh is the hyperbolic sine function. This solution of the heat equation is obtained by multiplying each term of the equation from Analysis § Example by ${\displaystyle \sinh(ny)/\sinh(n\pi )}$. While our example function ${\displaystyle s(x)}$ seems to have a needlessly complicated Fourier series, the heat distribution ${\displaystyle T(x,y)}$ is nontrivial. The function ${\displaystyle T}$ cannot be written as a closed-form expression. This method of solving the heat problem was made possible by Fourier's work.

Other applications

Another application 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.

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 ${\displaystyle s(x)}$:^[21]

Sine-cosine form

${\displaystyle s_{N}(x)=a_{0}+\sum _{n=1}^{N}\left(a_{n}\cos \left(2\pi {\tfrac {n}{P}}x\right)+b_{n}\sin \left(2\pi {\tfrac {n}{P}}x\right)\right)}$

Eq.1

Exponential form

${\displaystyle s_{N}(x)=\sum _{n=-N}^{N}c_{n}\ e^{i2\pi {\tfrac {n}{P}}x}}$

Eq.2

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 ${\displaystyle {\tfrac {P}{n}}}$ in the same units as ${\displaystyle x}$.
• a frequency equal to ${\displaystyle {\tfrac {n}{P}}}$ 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 ${\displaystyle N\to \infty }$, 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 ${\displaystyle P}$.^[B][22]

Fourier coefficients

${\displaystyle {\begin{aligned}&a_{0}={\frac {1}{P}}\int _{P}s(x)\,dx&\\&a_{n}={\frac {2}{P}}\int _{P}s(x)\cos \left(2\pi {\tfrac {n}{P}}x\right)\,dx,\ &{\textrm {for}}~n\geq 1\\&b_{n}={\frac {2}{P}}\int _{P}s(x)\sin \left(2\pi {\tfrac {n}{P}}x\right)\,dx,\ &{\text{for}}~n\geq 1\\\end{aligned}}}$

Eq.3

The ${\displaystyle {\tfrac {2}{P}}}$ scale factor follows from substituting Eq.1 into Eq.3 and utilizing the orthogonality of the trigonometric system.^[23] The equivalence of Eq.1 and Eq.2 follows from Euler's formula $${\displaystyle \cos x={\frac {e^{ix}+e^{-ix}}{2}},\quad \sin x={\frac {e^{ix}-e^{-ix}}{2i}},}$$ resulting in:

with ${\displaystyle c_{0}}$ being the mean value of ${\displaystyle s}$ on the interval ${\displaystyle P}$.^[24] Conversely:

Amplitude-phase form

If the function ${\displaystyle s(x)}$ is real-valued then the Fourier series can also be represented as^[25]

Amplitude-phase form

${\displaystyle s_{N}(x)=A_{0}+\sum _{n=1}^{N}A_{n}\cos \left(2\pi {\tfrac {n}{P}}x-\varphi _{n}\right)}$

Eq.4

where ${\displaystyle A_{n}}$ is the amplitude and ${\displaystyle \varphi _{n}}$ is the phase shift of the ${\displaystyle n^{th}}$ harmonic.

The equivalence of Eq.4 and Eq.1 follows from the trigonometric identity: $${\displaystyle \cos \left(2\pi {\tfrac {n}{P}}x-\varphi _{n}\right)=\cos(\varphi _{n})\cos \left(2\pi {\tfrac {n}{P}}x\right)+\sin(\varphi _{n})\sin \left(2\pi {\tfrac {n}{P}}x\right),}$$ which implies^[26] $${\displaystyle a_{n}=A_{n}\cos(\varphi _{n})\quad {\text{and}}\quad b_{n}=A_{n}\sin(\varphi _{n})}$$

are the rectangular coordinates of a vector with polar coordinates ${\displaystyle A_{n}}$ and ${\displaystyle \varphi _{n}}$ given by $${\displaystyle A_{n}={\sqrt {a_{n}^{2}+b_{n}^{2}}}\quad {\text{and}}\quad \varphi _{n}=\operatorname {Arg} (c_{n})=\operatorname {atan2} (b_{n},a_{n})}$$ where ${\displaystyle \operatorname {Arg} (c_{n})}$ is the argument of ${\displaystyle c_{n}}$.

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

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

Fortunately, it is not necessary to evaluate this entire function, because its derivative is zero at the maximum: $${\displaystyle X'(\varphi )=\sin(\varphi )\cdot X(0)-\cos(\varphi )\cdot X(\pi /2)=0,\quad {\textrm {at}}\ \varphi =\varphi _{n}.}$$ Hence $${\displaystyle \varphi _{n}\equiv \arctan(b_{n}/a_{n})=\arctan(X(\pi /2)/X(0)).}$$

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 ${\displaystyle {\widehat {s}}(n)}$ or ${\displaystyle S[n],}$ and functional notation often replaces subscripting:

${\displaystyle {\begin{aligned}s(x)&=\sum _{n=-\infty }^{\infty }{\widehat {s}}(n)\cdot e^{i2\pi {\tfrac {n}{P}}x}&&\scriptstyle {\text{common mathematics notation}}\\&=\sum _{n=-\infty }^{\infty }S[n]\cdot e^{i2\pi {\tfrac {n}{P}}x}&&\scriptstyle {\text{common engineering notation}}\end{aligned}}}$

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:

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

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 ${\displaystyle {\tfrac {1}{P}}}$, which is called the fundamental frequency. ${\displaystyle s(x)}$ can be recovered from this representation by an inverse Fourier transform:

${\displaystyle {\begin{aligned}{\mathcal {F}}^{-1}\{S(f)\}&=\int _{-\infty }^{\infty }\left(\sum _{n=-\infty }^{\infty }S[n]\cdot \delta \left(f-{\frac {n}{P}}\right)\right)e^{i2\pi fx}\,df,\\[6pt]&=\sum _{n=-\infty }^{\infty }S[n]\cdot \int _{-\infty }^{\infty }\delta \left(f-{\frac {n}{P}}\right)e^{i2\pi fx}\,df,\\[6pt]&=\sum _{n=-\infty }^{\infty }S[n]\cdot e^{i2\pi {\tfrac {n}{P}}x}\ \ \triangleq \ s(x).\end{aligned}}}$

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.^[C]

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:^[29][30]

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

From this, various relationships are apparent, for example:

• The transform of a real-valued function ${\displaystyle (s_{\mathrm {RE} }+s_{\mathrm {RO} })}$ is the conjugate symmetric function ${\displaystyle S_{\mathrm {RE} }+i\ S_{\mathrm {IO} }.}$ Conversely, a conjugate symmetric transform implies a real-valued time-domain.
• The transform of an imaginary-valued function ${\displaystyle (i\ s_{\mathrm {IE} }+i\ s_{\mathrm {IO} })}$ is the conjugate antisymmetric function ${\displaystyle S_{\mathrm {RO} }+i\ S_{\mathrm {IE} },}$ and the converse is true.
• The transform of a conjugate symmetric function ${\displaystyle (s_{\mathrm {RE} }+i\ s_{\mathrm {IO} })}$ is the real-valued function ${\displaystyle S_{\mathrm {RE} }+S_{\mathrm {RO} },}$ and the converse is true.
• The transform of a conjugate antisymmetric function ${\displaystyle (s_{\mathrm {RO} }+i\ s_{\mathrm {IE} })}$ is the imaginary-valued function ${\displaystyle i\ S_{\mathrm {IE} }+i\ S_{\mathrm {IO} },}$ and the converse is true.

Riemann–Lebesgue lemma

If ${\displaystyle S}$ is integrable, ${\textstyle \lim _{|n|\to \infty }S[n]=0}$, ${\textstyle \lim _{n\to +\infty }a_{n}=0}$ and ${\textstyle \lim _{n\to +\infty }b_{n}=0.}$

Parseval's theorem

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

Plancherel's theorem

If ${\displaystyle c_{0},\,c_{\pm 1},\,c_{\pm 2},\ldots }$ are coefficients and ${\textstyle \sum _{n=-\infty }^{\infty }|c_{n}|^{2}<\infty }$ 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: $${\displaystyle H[n]=\{S*R\}[n].}$$
• 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 ^[31]

Derivative property

If ${\displaystyle s}$ is a 2π-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 ${\displaystyle {\widehat {s}}[n]}$ of ${\displaystyle s}$, via the formula $${\displaystyle {\widehat {s^{(k)}}}[n]=(in)^{k}{\widehat {s}}[n].}$$ In particular, since for any fixed ${\displaystyle k\geq 1}$ we have ${\displaystyle {\widehat {s^{(k)}}}[n]\to 0}$ as ${\displaystyle n\to \infty }$, it follows that ${\displaystyle |n|^{k}{\widehat {s}}[n]}$ 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 ${\displaystyle \mathbb {R} }$.

Fourier-Stieltjes series

Let ${\displaystyle F(x)}$ be a function of bounded variation defined on the closed interval ${\displaystyle [0,P]\subseteq \mathbb {R} }$. The Fourier series whose coefficients are given by^[32] $${\displaystyle c_{n}={\frac {1}{P}}\int _{0}^{P}\ e^{-i2\pi {\tfrac {n}{P}}x}\,dF(x),\quad \forall n\in \mathbb {Z} ,}$$ is called the Fourier-Stieltjes series. The space of functions of bounded variation ${\displaystyle BV}$ is a subspace of ${\displaystyle L^{1}}$. As any ${\displaystyle F\in BV}$ defines a Radon measure (i.e. a locally finite Borel measure on ${\displaystyle \mathbb {R} }$), this definition can be extended as follows.

Consider the space ${\displaystyle M}$ of all finite Borel measures on the real line; as such ${\displaystyle L^{1}\subset M}$.^[33] If there is a measure ${\displaystyle \mu \in M}$ such that the Fourier-Stieltjes coefficients are given by $${\displaystyle c_{n}={\hat {\mu }}(n)={\frac {1}{P}}\int _{0}^{P}\ e^{-i2\pi {\tfrac {n}{P}}x}\,d\mu (x),\quad \forall n\in \mathbb {Z} ,}$$ then the series is called a Fourier-Stieltjes series. Likewise, the function ${\displaystyle {\hat {\mu }}(n)}$, where ${\displaystyle \mu \in M}$, is called a Fourier-Stieltjes transform.^[34]

The question whether or not ${\displaystyle \mu }$ exists for a given sequence of ${\displaystyle c_{n}}$ forms the basis of the trigonometric moment problem.^[35]

Furthermore, ${\displaystyle M}$ is a strict subspace of the space of (tempered) distributions ${\displaystyle {\mathcal {D}}}$, i.e., ${\displaystyle M\subset {\mathcal {D}}}$. If the Fourier coefficients are determined by a distribution ${\displaystyle F\in {\mathcal {D}}}$ then the series is described as a Fourier-Schwartz series. Contrary to the Fourier-Stieltjes series, deciding whether a given series is a Fourier series or a Fourier-Schwartz series is relatively trivial due to the characteristics of its dual space; the Schwartz space ${\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}$.^[36]

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 ${\displaystyle [-\pi ,\pi ]\times [-\pi ,\pi ]}$: $${\displaystyle {\begin{aligned}f(x,y)&=\sum _{j,k\in \mathbb {Z} }c_{j,k}e^{ijx}e^{iky},\\[5pt]c_{j,k}&={\frac {1}{4\pi ^{2}}}\int _{-\pi }^{\pi }\int _{-\pi }^{\pi }f(x,y)e^{-ijx}e^{-iky}\,dx\,dy.\end{aligned}}}$$

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.^[37]

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,$${\displaystyle g(x_{1},x_{2},x_{3})=g(x_{1}+a_{1},x_{2},x_{3})=g(x_{1},x_{2}+a_{2},x_{3})=g(x_{1},x_{2},x_{3}+a_{3})\quad \forall \;x_{1},x_{2},x_{3}.}$$ 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$${\displaystyle {\begin{aligned}c(m_{1},m_{2},m_{3})={\frac {1}{a_{3}}}\int _{0}^{a_{3}}dx_{3}{\frac {1}{a_{2}}}\int _{0}^{a_{2}}dx_{2}{\frac {1}{a_{1}}}\int _{0}^{a_{1}}dx_{1}\,g(x_{1},x_{2},x_{3})\,e^{-i2\pi \left({\tfrac {m_{1}}{a_{1}}}x_{1}+{\tfrac {m_{2}}{a_{2}}}x_{2}+{\tfrac {m_{3}}{a_{3}}}x_{3}\right)}\end{aligned}},}$$ 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 $${\displaystyle g(x_{1},x_{2},x_{3})=\sum _{m_{1},m_{2},m_{3}\in \mathbb {Z} }\,c(m_{1},m_{2},m_{3})\,e^{i2\pi \left({\tfrac {m_{1}}{a_{1}}}x_{1}+{\tfrac {m_{2}}{a_{2}}}x_{2}+{\tfrac {m_{3}}{a_{3}}}x_{3}\right)}.}$$ 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 ${\displaystyle \mathbf {a} _{1,2,3}}$, $${\displaystyle \mathbf {a} _{i}\cdot \mathbf {b_{j}} =2\pi \delta _{ij},}$$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 the all 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$${\displaystyle c(\mathbf {Q} )={\frac {1}{a_{3}}}\int _{0}^{a_{3}}dx_{3}\,{\frac {1}{a_{2}}}\int _{0}^{a_{2}}dx_{2}\,{\frac {1}{a_{1}}}\int _{0}^{a_{1}}dx_{1}\,f\left(x_{1}{\frac {\mathbf {a} _{1}}{a_{1}}}+x_{2}{\frac {\mathbf {a} _{2}}{a_{2}}}+x_{3}{\frac {\mathbf {a} _{3}}{a_{3}}}\right)e^{-i\mathbf {Q} \cdot \mathbf {r} }.}$$ 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, $${\displaystyle {\begin{bmatrix}x\\y\\z\end{bmatrix}}=\mathbf {J} {\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}}={\begin{bmatrix}{\frac {\mathbf {a} _{1}}{a_{1}}},{\frac {\mathbf {a} _{2}}{a_{2}}},{\frac {\mathbf {a} _{3}}{a_{3}}}\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}}\,,}$$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$${\displaystyle \mathbf {J} ={\begin{bmatrix}{\dfrac {\partial x}{\partial x_{1}}}&{\dfrac {\partial x}{\partial x_{2}}}&{\dfrac {\partial x}{\partial x_{3}}}\\[12pt]{\dfrac {\partial y}{\partial x_{1}}}&{\dfrac {\partial y}{\partial x_{2}}}&{\dfrac {\partial y}{\partial x_{3}}}\\[12pt]{\dfrac {\partial z}{\partial x_{1}}}&{\dfrac {\partial z}{\partial x_{2}}}&{\dfrac {\partial z}{\partial x_{3}}}\end{bmatrix}}\,.}$$