Fourier series

The Fourier series can be represented in different forms. The amplitude-phase form, sine-cosine form, and exponential form are commonly used and are expressed here for a real-valued function ${\displaystyle s(x)}$. (See § Complex-valued functions and § Other common notations for alternative forms).

The number of terms summed, ${\displaystyle N}$, is a potentially infinite integer. Even so, the series might not converge or exactly equate to ${\displaystyle s(x)}$ at all values of ${\displaystyle x}$ (such as a single-point discontinuity) in the analysis interval. For the well-behaved functions typical of physical processes, equality is customarily assumed, and the Dirichlet conditions provide sufficient conditions.

The integer index, ${\displaystyle n}$, is also the number of cycles the ${\displaystyle n^{\text{th}}}$ harmonic makes in the function's period ${\displaystyle P}$.[upper-alpha 3] Therefore:

• The ${\displaystyle n^{\text{th}}}$ harmonic's wavelength is ${\displaystyle {\tfrac {P}{n}}}$ and in units of ${\displaystyle x}$.
• The ${\displaystyle n^{\text{th}}}$ harmonic's frequency is ${\displaystyle {\tfrac {n}{P}}}$ and in reciprocal units of ${\displaystyle x}$.

Fig 1. The top graph shows a non-periodic function s(x) in blue defined only over the red interval from 0 to P. The function can be analyzed over this interval to produce the Fourier series in the bottom graph. The Fourier series is always a periodic function, even if original function s(x) wasn't.

The Fourier series in amplitude-phase form is:

Fourier series, amplitude-phase form

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

(Eq.1)

• Its ${\displaystyle n^{\text{th}}}$ harmonic is ${\displaystyle A_{n}\cdot \cos \left({\tfrac {2\pi }{P}}nx-\varphi _{n}\right)}$.
• ${\displaystyle A_{n}}$ is the ${\displaystyle n^{\text{th}}}$ harmonic's amplitude and ${\displaystyle \varphi _{n}}$ is its phase shift.
• The fundamental frequency of ${\displaystyle s_{\scriptscriptstyle N}(x)}$ is the term for when ${\displaystyle n}$ equals 1, and can be referred to as the ${\displaystyle 1^{\text{st}}}$ harmonic.
• ${\displaystyle {\tfrac {A_{o}}{2}}}$ is sometimes called the ${\displaystyle 0^{\text{th}}}$ harmonic or DC component. It is the mean value of ${\displaystyle s(x)}$.

Clearly Eq.1 can represent functions that are just a sum of one or more of the harmonic frequencies. The remarkable thing, for those not yet familiar with this concept, is that it can also represent the intermediate frequencies and/or non-sinusoidal functions because of the potentially infinite number of terms (${\displaystyle N}$).

Fig 2. The blue curve is the cross-correlation of a square wave and a cosine function, as the phase lag of the cosine varies over one cycle. The amplitude and phase lag at the maximum value are the polar coordinates of one harmonic in the Fourier series expansion of the square wave. The corresponding Cartesian coordinates can be determined by evaluating the cross-correlation at just two phase lags separated by 90º.

The coefficients ${\displaystyle A_{n}}$ and ${\displaystyle \varphi _{n}}$ can be understood and derived in terms of the cross-correlation between ${\displaystyle s(x)}$ and a sinusoid at frequency ${\displaystyle {\tfrac {n}{P}}}$. For a general frequency ${\displaystyle f,}$ and an analysis interval ${\displaystyle [x_{0},x_{0}+P],}$ the cross-correlation function:

${\displaystyle \mathrm {X} _{f}(\tau )={\tfrac {2}{P}}\int _{x_{0}}^{x_{0}+P}s(x)\cdot \cos \left(2\pi f(x-\tau )\right)\,dx;\quad \tau \in \left[0,{\tfrac {2\pi }{f}}\right]}$

(Eq.2)

is essentially a matched filter, with template ${\displaystyle \cos(2\pi fx)}$.[upper-alpha 4] The maximum of ${\displaystyle \mathrm {X} _{f}(\tau )}$ is a measure of the amplitude ${\displaystyle (A)}$ of frequency ${\displaystyle f}$ in the function ${\displaystyle s(x)}$, and the value of ${\displaystyle \tau }$ at the maximum determines the phase ${\displaystyle (\varphi )}$ of that frequency. Figure 2 is an example, where ${\displaystyle s(x)}$ is a square wave (not shown), and frequency ${\displaystyle f}$ is the ${\displaystyle 4^{\text{th}}}$ harmonic. It is also an example of deriving the maximum from just two samples, instead of searching the entire function. That is made possible by a trigonometric identity:

Equivalence of polar and Cartesian forms

${\displaystyle \cos \left({\tfrac {2\pi }{P}}nx-\varphi _{n}\right)\ \equiv \ \cos(\varphi _{n})\cdot \cos \left({\tfrac {2\pi }{P}}nx\right)+\sin(\varphi _{n})\cdot \sin \left({\tfrac {2\pi }{P}}nx\right)}$

(Eq.3)

Substituting this into Eq.2 gives:

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

which introduces the definitions of ${\displaystyle a_{n}}$ and ${\displaystyle b_{n}}$.[2]  And we note for later reference that ${\displaystyle a_{0}}$ and ${\displaystyle b_{0}}$ can be simplified:

$${\displaystyle a_{0}={\tfrac {2}{P}}\int _{P}s(x)dx~,\quad b_{0}=0~.}$$

The derivative of ${\displaystyle \mathrm {X} _{n}(\varphi )}$ is zero at the phase of maximum correlation.

$${\displaystyle \mathrm {X} '_{n}(\varphi _{n})=\sin(\varphi _{n})\cdot a_{n}-\cos(\varphi _{n})\cdot b_{n}=0\quad \longrightarrow \quad \tan(\varphi _{n})={\frac {b_{n}}{a_{n}}}\quad \longrightarrow \quad \varphi _{n}=\arctan(b_{n},a_{n})}$$

And the correlation peak value is:

${\displaystyle {\begin{aligned}A_{n}\triangleq \mathrm {X} _{n}(\varphi _{n})\ &=\cos(\varphi _{n})\cdot a_{n}+\sin(\varphi _{n})\cdot b_{n}\\&={\frac {a_{n}}{\sqrt {a_{n}^{2}+b_{n}^{2}}}}\cdot a_{n}+{\frac {b_{n}}{\sqrt {a_{n}^{2}+b_{n}^{2}}}}\cdot b_{n}={\frac {a_{n}^{2}+b_{n}^{2}}{\sqrt {a_{n}^{2}+b_{n}^{2}}}}&={\sqrt {a_{n}^{2}+b_{n}^{2}}}.\end{aligned}}}$

Therefore ${\displaystyle a_{n}}$ and ${\displaystyle b_{n}}$ are the Cartesian coordinates of a vector with polar coordinates ${\displaystyle A_{n}}$ and ${\displaystyle \varphi _{n}.}$

Substituting Eq.3 into Eq.1 gives:

${\displaystyle {\displaystyle s_{\scriptscriptstyle N}(x)={\frac {A_{0}}{2}}+\sum _{n=1}^{N}\left[A_{n}\cos(\varphi _{n})\cdot \cos \left({\tfrac {2\pi }{P}}nx\right)+A_{n}\sin(\varphi _{n})\cdot \sin \left({\tfrac {2\pi }{P}}nx\right)\right]}}$

In terms of the readily computed quantities, ${\displaystyle a_{n}}$ and ${\displaystyle b_{n}}$, recall that:

${\displaystyle \cos(\varphi _{n})=a_{n}/A_{n}}$

${\displaystyle \sin(\varphi _{n})=b_{n}/A_{n}}$

${\displaystyle A_{0}={\sqrt {a_{0}^{2}+b_{0}^{2}}}={\sqrt {a_{0}^{2}}}=a_{0}}$

Therefore an alternative form of the Fourier series, using the Cartesian coordinates, is the sine-cosine form:[upper-alpha 5]

Fourier series, sine-cosine form

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

(Eq.4)

Another applicable identity is Euler's formula:

${\displaystyle {\begin{aligned}\cos \left({\tfrac {2\pi }{P}}nx-\varphi _{n}\right)&{}\equiv {\tfrac {1}{2}}e^{i\left(2\pi nx/P-\varphi _{n}\right)}+{\tfrac {1}{2}}e^{-i\left(2\pi nx/P-\varphi _{n}\right)}\\[6pt]&=\left({\tfrac {1}{2}}e^{-i\varphi _{n}}\right)\cdot e^{i2\pi (+n)x/P}+\left({\tfrac {1}{2}}e^{-i\varphi _{n}}\right)^{*}\cdot e^{i2\pi (-n)x/P}\end{aligned}}}$

(Note: the ∗ denotes complex conjugation.)

Therefore, with definitions:

${\displaystyle c_{n}\triangleq \left\{{\begin{array}{lll}A_{0}/2&=a_{0}/2,\quad &n=0\\{\tfrac {A_{n}}{2}}e^{-i\varphi _{n}}&={\tfrac {1}{2}}(a_{n}-ib_{n}),\quad &n>0\\c_{|n|}^{*},\quad &&n<0\end{array}}\right\}={\tfrac {1}{P}}\int _{P}s(x)\cdot e^{-i2\pi nx/P}\,dx,}$

the final result is:

Fourier series, exponential form

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

(Eq.5)

This is the customary form for generalizing to § Complex-valued functions. Negative values of ${\displaystyle n}$ correspond to negative frequency (explained in Fourier transform § Use of complex sinusoids to represent real sinusoids).

Plot of the sawtooth wave, a periodic continuation of the linear function ${\displaystyle s(x)=x/\pi }$ on the interval ${\displaystyle (-\pi ,\pi ]}$

Animated plot of the first five successive partial Fourier series

Consider a sawtooth function:

${\displaystyle s(x)={\frac {x}{\pi }},\quad \mathrm {for} -\pi <x<\pi ,}$

${\displaystyle s(x+2\pi k)=s(x),\quad \mathrm {for} -\pi <x<\pi {\text{ and }}k\in \mathbb {Z} .}$

In this case, the Fourier coefficients are given by

${\displaystyle {\begin{aligned}a_{n}&={\frac {1}{\pi }}\int _{-\pi }^{\pi }s(x)\cos(nx)\,dx=0,\quad n\geq 0.\\[4pt]b_{n}&={\frac {1}{\pi }}\int _{-\pi }^{\pi }s(x)\sin(nx)\,dx\\[4pt]&=-{\frac {2}{\pi n}}\cos(n\pi )+{\frac {2}{\pi ^{2}n^{2}}}\sin(n\pi )\\[4pt]&={\frac {2\,(-1)^{n+1}}{\pi n}},\quad n\geq 1.\end{aligned}}}$

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:

${\displaystyle {\begin{aligned}s(x)&={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }\left[a_{n}\cos \left(nx\right)+b_{n}\sin \left(nx\right)\right]\\[4pt]&={\frac {2}{\pi }}\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}\sin(nx),\quad \mathrm {for} \quad x-\pi \notin 2\pi \mathbb {Z} .\end{aligned}}}$

(Eq.6)

When ${\displaystyle x=\pi }$, the Fourier series converges to 0, which is the half-sum of the left- and right-limit of 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.

A proof that a Fourier series is a valid representation of any periodic function (that satisfies the Dirichlet conditions) is overviewed in § Fourier theorem proving 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 better-behaved than the functions that mathematicians can provide as counter-examples to this presumption. 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 ${\displaystyle s(x)}$.[3] If a function is square-integrable on the interval ${\displaystyle [x_{0},x_{0}+P]}$, then the Fourier series converges to the function at almost every point. It is possible to define Fourier coefficients for more general functions or distributions, in such cases convergence in norm or weak convergence is usually of interest.

• 
  Four partial sums (Fourier series) of lengths 1, 2, 3, and 4 terms, showing how the approximation to a square wave improves as the number of terms increases (animation)
• 
  Four partial sums (Fourier series) of lengths 1, 2, 3, and 4 terms, showing how the approximation to a sawtooth wave improves as the number of terms increases (animation)
• 
  Example of convergence to a somewhat arbitrary function. Note the development of the "ringing" (Gibbs phenomenon) at the transitions to/from the vertical sections.

If ${\displaystyle s(x)}$ is a complex-valued function of a real variable ${\displaystyle x,}$ both components (real and imaginary part) are real-valued functions that can be represented by a Fourier series. The two sets of coefficients and the partial sum are given by:

${\displaystyle c_{_{Rn}}={\frac {1}{P}}\int _{P}\operatorname {Re} \{s(x)\}\cdot e^{-i{\tfrac {2\pi }{P}}nx}\ dx}$ and ${\displaystyle c_{_{In}}={\frac {1}{P}}\int _{P}\operatorname {Im} \{s(x)\}\cdot e^{-i{\tfrac {2\pi }{P}}nx}\ dx}$

${\displaystyle s_{_{N}}(x)=\sum _{n=-N}^{N}c_{_{Rn}}\cdot e^{i{\tfrac {2\pi }{P}}nx}+i\cdot \sum _{n=-N}^{N}c_{_{In}}\cdot e^{i{\tfrac {2\pi }{P}}nx}=\sum _{n=-N}^{N}\left(c_{_{Rn}}+i\cdot c_{_{In}}\right)\cdot e^{i{\tfrac {2\pi }{P}}nx}.}$

Defining ${\displaystyle c_{n}\triangleq c_{_{Rn}}+i\cdot c_{_{In}}}$ yields:[4][5]

${\displaystyle s_{_{N}}(x)=\sum _{n=-N}^{N}c_{n}\cdot e^{i{\tfrac {2\pi }{P}}nx}}$

(Eq.7)

This is identical to Eq.5 except ${\displaystyle c_{n}}$ and ${\displaystyle c_{-n}}$ are no longer complex conjugates. The formula for ${\displaystyle c_{n}}$ is also unchanged:

${\displaystyle {\begin{aligned}c_{n}&={\tfrac {1}{P}}\int _{P}\operatorname {Re} \{s(x)\}\cdot e^{-i{\frac {2\pi }{P}}nx}\ dx+i\cdot {\tfrac {1}{P}}\int _{P}\operatorname {Im} \{s(x)\}\cdot e^{-i{\tfrac {2\pi }{P}}nx}\ dx\\[4pt]&={\tfrac {1}{P}}\int _{P}\left(\operatorname {Re} \{s(x)\}+i\cdot \operatorname {Im} \{s(x)\}\right)\cdot e^{-i{\tfrac {2\pi }{P}}nx}\ dx\ =\ {\tfrac {1}{P}}\int _{P}s(x)\cdot e^{-i{\tfrac {2\pi }{P}}nx}\ dx.\end{aligned}}}$

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:

${\displaystyle {\begin{aligned}s_{\infty }(x)&=\sum _{n=-\infty }^{\infty }{\hat {s}}[n]\cdot e^{i\,2\pi nx/P}\\[6pt]&=\sum _{n=-\infty }^{\infty }S[n]\cdot e^{i\,2\pi nx/P}&&\scriptstyle {\mathsf {common\ engineering\ notation}}\end{aligned}}}$

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. ${\displaystyle s_{\infty }(x)}$ can be recovered from this representation by an inverse Fourier transform:

${\displaystyle {\begin{aligned}{\mathcal {F}}^{-1}\{S(f)\}&=\int _{-\infty }^{\infty }\left(\sum _{n=-\infty }^{\infty }S[n]\cdot \delta \left(f-{\frac {n}{P}}\right)\right)e^{i2\pi fx}\,df,\\[6pt]&=\sum _{n=-\infty }^{\infty }S[n]\cdot \int _{-\infty }^{\infty }\delta \left(f-{\frac {n}{P}}\right)e^{i2\pi fx}\,df,\\[6pt]&=\sum _{n=-\infty }^{\infty }S[n]\cdot e^{i\,2\pi nx/P}\ \ \triangleq \ s_{\infty }(x).\end{aligned}}}$

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.[upper-alpha 6]

Joseph Fourier wrote:

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

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)[15][upper-alpha 8]

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

$${\displaystyle {\begin{aligned}a_{k}&=\int _{-1}^{1}\varphi (y)\cos(2k+1){\frac {\pi y}{2}}\,dy\\&=\int _{-1}^{1}\left(a\cos {\frac {\pi y}{2}}\cos(2k+1){\frac {\pi y}{2}}+a'\cos 3{\frac {\pi y}{2}}\cos(2k+1){\frac {\pi y}{2}}+\cdots \right)\,dy\end{aligned}}}$$

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.

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.

Heat distribution in a metal plate, using Fourier's method

The Fourier series expansion of the sawtooth function (above) looks more complicated than the simple formula ${\displaystyle s(x)={\tfrac {x}{\pi }}}$, 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 period of 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 Eq.6 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.

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

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

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

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

The discrete-time Fourier transform is an example of a Fourier series.

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 integern.

If ${\displaystyle S}$ is integrable, ${\textstyle \lim _{|n|\to \infty }S[n]=0}$, ${\textstyle \lim _{n\to +\infty }a_{n}=0}$ and ${\textstyle \lim _{n\to +\infty }b_{n}=0.}$ This result is known as the Riemann–Lebesgue lemma.

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

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 ${\displaystyle n}$.

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 [20]

We say that ${\displaystyle s}$ belongs to ${\displaystyle C^{k}(\mathbb {T} )}$ 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.

• If ${\displaystyle s\in C^{1}(\mathbb {T} )}$, then the Fourier coefficients ${\displaystyle {\widehat {s'}}[n]}$ of the derivative ${\displaystyle s'}$ can be expressed in terms of the Fourier coefficients ${\displaystyle {\widehat {s}}[n]}$ of the function ${\displaystyle s}$, via the formula ${\displaystyle {\widehat {s'}}[n]=in{\widehat {s}}[n]}$.
• If ${\displaystyle s\in C^{k}(\mathbb {T} )}$, then ${\displaystyle {\widehat {s^{(k)}}}[n]=(in)^{k}{\widehat {s}}[n]}$. In particular, since for a 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, which means that the Fourier coefficients converge to zero faster than the kth power of n for any ${\displaystyle k\geq 1}$.

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.

The atomic orbitals of chemistry are partially described by spherical harmonics, which can be used to produce Fourier series on the sphere.

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.

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 ${\displaystyle \mathbb {R} }$.

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.[21]

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 vectors. Assuming we have some function, ${\displaystyle f(\mathbf {r} )}$, such that it obeys the condition of periodicity for any Bravais lattice vector ${\displaystyle \mathbf {R} }$, ${\displaystyle f(\mathbf {r} )=f(\mathbf {R} +\mathbf {r} )}$, we could make a Fourier series of it. This kind of function can be, for example, the effective potential that one electron "feels" inside a periodic crystal. It is useful to make the Fourier series of the potential when applying Bloch's theorem. First, we may write any arbitrary position vector ${\displaystyle \mathbf {r} }$ in the coordinate-system of the lattice:

$${\displaystyle \mathbf {r} =x_{1}{\frac {\mathbf {a} _{1}}{a_{1}}}+x_{2}{\frac {\mathbf {a} _{2}}{a_{2}}}+x_{3}{\frac {\mathbf {a} _{3}}{a_{3}}},}$$

where ${\displaystyle a_{i}\triangleq |\mathbf {a} _{i}|,}$ meaning that ${\displaystyle a_{i}}$ is defined to be the magnitude of ${\displaystyle \mathbf {a} _{i}}$, so ${\displaystyle {\hat {\mathbf {a} _{i}}}={\frac {\mathbf {a} _{i}}{a_{i}}}}$ is the unit vector directed along ${\displaystyle \mathbf {a} _{i}}$.