A Fourier series () is an expansion of a

periodic function A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the t ...

into a sum of

trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...

. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by

Joseph Fourier Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre, Burgundy and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analys ...

to find solutions to the

heat equation In mathematics and physics (more specifically thermodynamics), the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quanti ...

. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by

integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...

s of the function multiplied by trigonometric functions, described in . The study of the convergence of Fourier series focus on the behaviors of the ''partial sums'', which means studying the behavior of the sum as more and more terms from the series are summed. The figures below illustrate some partial Fourier series results for the components of a square wave. File:SquareWaveFourierArrows,rotated,nocaption 20fps.gif, A square wave (represented as the blue dot) is approximated by its sixth partial sum (represented as the purple dot), formed by summing the first six terms (represented as arrows) of the square wave's Fourier series. Each arrow starts at the vertical sum of all the arrows to its left (i.e. the previous partial sum). File:Fourier Series.svg, The first four partial sums of the Fourier series for a square wave. As more harmonics are added, the partial sums ''converge to'' (become more and more like) the square wave. File:Fourier series and transform.gif, Function

s_6(x)

(in red) is a Fourier series sum of 6 harmonically related sine waves (in blue). Its Fourier transform

S(f)

is a frequency-domain representation that reveals the amplitudes of the summed sine waves. Fourier series are closely related to the

Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...

, a more general tool that can even find the frequency information for functions that are ''not'' periodic. Periodic functions can be identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by

\mathbb

S_1

. The Fourier transform is also part of Fourier analysis, but is defined for functions on

\mathbb^n

. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the decomposition. Many other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis.

History

The Fourier series is named in honor of Jean-Baptiste Joseph Fourier (1768–1830), who made important contributions to the study of trigonometric series, after preliminary investigations by

Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...

, Jean le Rond d'Alembert, and

Daniel Bernoulli Daniel Bernoulli ( ; ; – 27 March 1782) was a Swiss people, Swiss-France, French mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applicati ...

. Fourier introduced the series for the purpose of solving the

in a metal plate, publishing his initial results in his 1807 '' Mémoire sur la propagation de la chaleur dans les corps solides'' (''Treatise on the propagation of heat in solid bodies''), and publishing his ''Théorie analytique de la chaleur'' (''Analytical theory of heat'') in 1822. The ''Mémoire'' introduced Fourier analysis, specifically Fourier series. Through Fourier's research the fact was established that an arbitrary (at first, continuous and later generalized to any piecewise-smooth) function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier in 1807, before the French Academy. Early ideas of decomposing a periodic function into the sum of simple oscillating functions date back to the 3rd century BC, when ancient astronomers proposed an empiric model of planetary motions, based on deferents and epicycles. Independently of Fourier, astronomer Friedrich Wilhelm Bessel introduced Fourier series to solve

Kepler's equation In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force. It was derived by Johannes Kepler in 1609 in Chapter 60 of his ''Astronomia nova'', and in book V of his ''Epitome of ...

. His work was published in 1819, unaware of Fourier's work which remained unpublished until 1822. The

is a

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

. Prior to Fourier's work, no solution to the heat equation was known in the general case, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a sine or

cosine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that ...

wave. These simple solutions are now sometimes called eigensolutions. Fourier's idea was to model a complicated heat source as a superposition (or

linear combination In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...

) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. This superposition or linear combination is called the Fourier series. From a modern point of view, Fourier's results are somewhat informal, due to the lack of a precise notion of function and

in the early nineteenth century. Later,

Peter Gustav Lejeune Dirichlet Johann Peter Gustav Lejeune Dirichlet (; ; 13 February 1805 – 5 May 1859) was a German mathematician. In number theory, he proved special cases of Fermat's last theorem and created analytic number theory. In analysis, he advanced the theory o ...

and

Bernhard Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...

expressed Fourier's results with greater precision and formality. Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems, and especially those involving linear differential equations with constant coefficients, for which the eigensolutions are sinusoids. The Fourier series has many such applications in

electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems that use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...

vibration Vibration () is a mechanical phenomenon whereby oscillations occur about an equilibrium point. Vibration may be deterministic if the oscillations can be characterised precisely (e.g. the periodic motion of a pendulum), or random if the os ...

analysis,

acoustics Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...

optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...

signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...

image processing An image or picture is a visual representation. An image can be two-dimensional, such as a drawing, painting, or photograph, or three-dimensional, such as a carving or sculpture. Images may be displayed through other media, including a pr ...

quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

econometrics Econometrics is an application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics", '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8 ...

, shell theory, etc.

Beginnings

Joseph Fourier wrote 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

\begin
&\int_^1\varphi(y)\cos(2k+1)\frac\,dy \\
&= \int_^1\left(a\cos\frac\cos(2k+1)\frac+a'\cos 3\frac\cos(2k+1)\frac+\cdots\right)\,dy
\end

can be carried out term-by-term. But all terms involving

\cos(2j+1)\frac \cos(2k+1)\frac

for vanish when integrated from −1 to 1, leaving only the

k^

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 Jean-Baptiste le Rond d'Alembert ( ; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanics, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the ''E ...

and

Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...

, 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 Convergence may refer to: Arts and media Literature *''Convergence'' (book series), edited by Ruth Nanda Anshen *Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics: **A four-part crossover storyline that ...

function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...

s, and harmonic analysis. When Fourier submitted a later competition essay in 1811, the committee (which included

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaLaplace,

Malus ''Malus'' ( or ) is a genus of about 32–57 species of small deciduous trees or shrubs in the family Rosaceae, including the domesticated orchard apple, crab apples (sometimes known in North America as crabapples) and wild apples. The genus i ...

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".

Fourier's motivation

The Fourier series expansion of the sawtooth function (below) looks more complicated than the simple formula

s(x)=\tfrac

, 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

. For example, consider a metal plate in the shape of a square whose sides measure

\pi

meters, with coordinates

(x,y) \in,\pi \times,\pi /math>. 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 y=\pi, is maintained at the temperature gradient T(x,\pi)=x degrees Celsius, for x in (0,\pi), then one can show that the stationary heat distribution (or the heat distribution after a long time has elapsed) is given by
: T(x,y) = 2\sum_^\infty \frac \sin(nx) . Here, sinh is the

hyperbolic sine In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a unit circle, circle with a unit radius, the points form the right ha ...

function. This solution of the heat equation is obtained by multiplying each term of the equation from Analysis § Example by

\sinh(ny)/\sinh(n\pi)

. While our example function

s(x)

seems to have a needlessly complicated Fourier series, the heat distribution

T(x,y)

is nontrivial. The function

T

cannot be written as a

closed-form expression In mathematics, an expression or equation is in closed form if it is formed with constants, variables, and a set of functions considered as ''basic'' and connected by arithmetic operations (, and integer powers) and function composition. ...

. 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(2''n''), for any positive integer ''n''.

Definition

The Fourier series of a complex-valued -periodic function

s(x)

, integrable over the interval

,P /math> on the real line, is defined as a trigonometric series of the form \sum_^\infty c_n e^, such that the ''Fourier coefficients'' c_n are complex numbers defined by the integral c_n = \frac\int_0^P s(x)\ e^\,dx. The series does not necessarily converge (in the

pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some Function (mathematics), function f. An important class of pointwise concepts are the ''pointwise operations'', that ...

sense) and, even if it does, it is not necessarily equal to

s(x)

. Only when certain conditions are satisfied (e.g. if

s(x)

is continuously differentiable) does the Fourier series converge to

s(x)

, i.e.,

s(x) = \sum_^\infty c_n e^.

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

s

(also called ''Fourier-Schwartz series''). Then the Fourier series converges to

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

N^\text

partial sums

s_N(x)

of the Fourier series of

s(x)

The harmonics are indexed by an integer,

n,

which is also the number of cycles the corresponding sinusoids make in interval

P

. Therefore, the sinusoids have: * a

wavelength In physics and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats. In other words, it is the distance between consecutive corresponding points of the same ''phase (waves ...

equal to

\tfrac

in the same units as

x

. * a

frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...

equal to

\tfrac

in the reciprocal units of

x

. These series can represent functions that are just a sum of one or more frequencies in the harmonic spectrum. In the limit

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

x

represents frequency instead of time. In general, the coefficients are determined by ''analysis'' of a given function

s(x)

whose domain of definition is an interval of length

P

. The

\tfrac

scale factor follows from substituting into and utilizing the orthogonality of the trigonometric system. The equivalence of and follows from Euler's formula

\cos x = \frac, \quad \sin x = \frac,

resulting in: with

c_

being the

mean value A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...

s

on the interval

P

. Conversely:

Example

Consider a sawtooth function:

s(x) = s(x + 2\pi k)  = \frac, \quad \mathrm -\pi < x < \pi,\text k \in \mathbb.

In this case, the Fourier coefficients are given by

\begin
a_0 &= 0.\\
a_n & = \frac\int_^s(x) \cos(nx)\,dx = 0, \quad n \ge 1. \\
b_n & = \frac\int_^s(x) \sin(nx)\, dx\\
&= -\frac\cos(n\pi) + \frac\sin(n\pi)\\
&= \frac, \quad n \ge 1.\end

It can be shown that the Fourier series converges to

s(x)

at every point

x

where

s

is differentiable, and therefore:

&=\frac\sum_^\infty \frac \sin(nx), \quad \mathrm\ (x-\pi)\ \text\ 2\pi. \end

When

x=\pi

, the Fourier series converges to 0, which is the half-sum of the left- and right-limit of

s

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

s(x)

is real-valued then the Fourier series can also be represented as where

A_

is the amplitude and

\varphi_

is the phase shift of the

n^

harmonic. The equivalence of and follows from the trigonometric identity:

\cos\left(2\pi \tfracx-\varphi_n\right) = \cos(\varphi_n)\cos\left(2\pi \tfrac x\right) 
+  \sin(\varphi_n)\sin\left(2\pi \tfrac x\right),

which implies

are the rectangular coordinates of a vector with

polar coordinates In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are *the point's distance from a reference ...

A_n

and

\varphi_n

given by

A_n = \sqrt\quad \text\quad \varphi_n = \operatorname(c_n) = \operatorname(b_n, a_n)

where

\operatorname(c_n)

is the

argument An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persu ...

c_

. An example of determining the parameter

\varphi_n

for one value of

n

is shown in Figure 2. It is the value of

\varphi

at the maximum correlation between

s(x)

and a cosine ''template,''

\cos(2\pi \tfrac x -  \varphi)

. The blue graph is the cross-correlation function, also known as a matched filter: :

\ &=\cos(\varphi) \underbrace_ + \sin(\varphi) \underbrace_ \end

Fortunately, it is not necessary to evaluate this entire function, because its derivative is zero at the maximum:

X'(\varphi) = \sin(\varphi)\cdot X(0) - \cos(\varphi)\cdot X(\pi/2) = 0,  \quad \textrm\ \varphi = \varphi_n.

Hence

\varphi_n  \equiv  \arctan(b_n/a_n) = \arctan(X(\pi/2)/X(0)).

Common notations

The notation

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 (

s,

in this case), such as

\widehat(n)

S

and functional notation often replaces subscripting: :

cdot e^ && \scriptstyle \text \end

In engineering, particularly when the variable

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 In mathematics, a Dirac comb (also known as sha function, impulse train or sampling function) is a periodic function, periodic Function (mathematics), function with the formula \operatorname_(t) \ := \sum_^ \delta(t - k T) for some given perio ...

: :

cdot \delta \left(f-\frac\right),

where

f

represents a continuous frequency domain. When variable

x

has units of seconds,

f

has units of

hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), often described as being equivalent to one event (or Cycle per second, cycle) per second. The hertz is an SI derived unit whose formal expression in ter ...

. The "teeth" of the comb are spaced at multiples (i.e. harmonics) of

\tfrac

, which is called the

fundamental frequency The fundamental frequency, often referred to simply as the ''fundamental'' (abbreviated as 0 or 1 ), is defined as the lowest frequency of a Periodic signal, periodic waveform. In music, the fundamental is the musical pitch (music), pitch of a n ...

s(x)

can be recovered from this representation by an inverse Fourier transform: :

cdot e^ \ \ \triangleq \ s(x). \end

The constructed function

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. *

s(x)

designates a periodic function with period

P.

a_0, a_n, b_n

designate the Fourier series coefficients (sine-cosine form) of the periodic function

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 In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...

is denoted by an asterisk. *

s(x),r(x)

designate

P

-periodic functions or functions defined only for

x \in,P

S R /math> designate the Fourier series coefficients (exponential form) of s and 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: :

\begin   
\mathsf & s & = & s_ & + & s_ & + & i\ s_ & + & i\ s_ \\
&\Bigg\Updownarrow\mathcal & &\Bigg\Updownarrow\mathcal & &\ \ \Bigg\Updownarrow\mathcal & &\ \ \Bigg\Updownarrow\mathcal & &\ \ \Bigg\Updownarrow\mathcal\\
\mathsf & S & = & S_\mathrm & + & i\ S_\mathrm\, & + & i\ S_\mathrm & + & S_\mathrm
\end

From this, various relationships are apparent, for example: * The transform of a real-valued function

(s_\mathrm+s_\mathrm)

is the ''conjugate symmetric'' function

S_\mathrm+i\ S_\mathrm.

Conversely, a ''conjugate symmetric'' transform implies a real-valued time-domain. * The transform of an imaginary-valued function

(i\ s_\mathrm+i\ s_\mathrm)

is the ''conjugate antisymmetric'' function

S_\mathrm+i\ S_\mathrm,

and the converse is true. * The transform of a ''conjugate symmetric'' function

(s_\mathrm+i\ s_\mathrm)

is the real-valued function

S_\mathrm+S_\mathrm,

and the converse is true. * The transform of a ''conjugate antisymmetric'' function

(s_\mathrm+i\ s_\mathrm)

is the imaginary-valued function

i\ S_\mathrm+i\ S_\mathrm,

and the converse is true.

Riemann–Lebesgue lemma

S

is integrable,

\lim_ S 0

\lim_ a_n=0

and

\lim_ b_n=0.

Parseval's theorem

s

belongs to

L^2(P)

(periodic over an interval of length

P

) then:

\frac\int_ , s(x), ^2 \, dx = \sum_^\infty \Bigl, S Bigr, ^2.

Plancherel's theorem

c_0,\, c_,\, c_, \ldots

are coefficients and

\sum_^\infty , c_n, ^2 < \infty

then there is a unique function

s\in L^2(P)

such that

S = c_n

for every

n

Convolution theorems

Given

P

-periodic functions,

s_P

and

r_P

with Fourier series coefficients

S /math> and R n \in \mathbb, *The pointwise product: h_P(x) \triangleq s_P(x)\cdot r_P(x) is also P -periodic, and its Fourier series coefficients are given by the discrete convolution of the S and R sequences: H = \ *The periodic convolution : h_P(x) \triangleq \int_ s_P(\tau)\cdot r_P(x-\tau)\, d\tau is also P -periodic, with Fourier series coefficients: H = P \cdot S cdot R *A doubly infinite sequence \left \_in c_0(\mathbb) is the sequence of Fourier coefficients of a function in L^1(,2\pi if and only if it is a convolution of two sequences in \ell^2(\mathbb) . See

Derivative property

s

is a 2-periodic function on

\mathbb

which is

k

times differentiable, and its

k^

derivative is continuous, then

s

belongs to the

C^k(\mathbb)

. * If

s \in C^k(\mathbb)

, then the Fourier coefficients of the

k^

derivative of

s

can be expressed in terms of the Fourier coefficients

\widehat /math> of s, via the formula \widehat = (in)^k \widehat In particular, since for any fixed k\geq 1 we have \widehat to 0 as n\to\infty, it follows that, n, ^k\widehat /math> tends to zero, i.e., the Fourier coefficients converge to zero faster than the k^power of, 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 group In mathematics, the classical groups are defined as the special linear groups over the reals \mathbb, the complex numbers \mathbb and the quaternions \mathbb together with special automorphism groups of Bilinear form#Symmetric, skew-symmetric an ...

s that are compact. This generalizes the Fourier transform to all spaces of the form ''L''²(''G''), where ''G'' is a compact group, in such a way that the Fourier transform carries

convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...

s to pointwise products. The Fourier series exists and converges in similar ways to the case. An alternative extension to compact groups is the Peter–Weyl theorem, which proves results about representations of compact groups analogous to those about finite groups. F orbital

Riemannian manifolds

If the domain is not a group, then there is no intrinsically defined convolution. However, if

X

is a

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

, it has a Laplace–Beltrami operator. The Laplace–Beltrami operator is the differential operator that corresponds to

Laplace operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a Scalar field, scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \ ...

for the Riemannian manifold

X

. Then, by analogy, one can consider heat equations on

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

L^2(X)

, where

X

is a Riemannian manifold. The Fourier series converges in ways similar to the

\pi,\pi /math> case. A typical example is to take 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

L^1(G)

L^2(G)

, where

G

is an LCA group. If

G

is compact, one also obtains a Fourier series, which converges similarly to the

\pi,\pi /math> case, but if G is noncompact, one obtains instead a Fourier integral . This generalization yields the usual

when the underlying locally compact Abelian group is

\mathbb

Extensions

Fourier-Stieltjes series

Let

F(x)

be a function of

bounded variation In mathematical analysis, a function of bounded variation, also known as ' function, is a real number, real-valued function (mathematics), function whose total variation is bounded (finite): the graph of a function having this property is well beh ...

defined on the closed interval

subseteq\mathbb

. The Fourier series whose coefficients are given by

c_n = \frac\int_0^P \ e^\,dF(x), \quad \forall n\in\mathbb,

is called the ''Fourier-Stieltjes series''. The space of functions of bounded variation

BV

is a subspace of

L^1

. As any

F \in BV

defines a Radon measure (i.e. a locally finite

Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. ...

\mathbb

), this definition can be extended as follows. Consider the space

M

of all finite Borel measures on the real line; as such

L^1 \subset M

. If there is a measure

\mu \in M

such that the Fourier-Stieltjes coefficients are given by

c_n = \hat\mu(n)=\frac\int_0^P \ e^\,d\mu(x), \quad \forall n\in\mathbb,

then the series is called a Fourier-Stieltjes series. Likewise, the function

\hat\mu(n)

, where

\mu \in M

, is called a Fourier-Stieltjes transform. The question whether or not

\mu

exists for a given sequence of

c_n

forms the basis of the trigonometric moment problem. Furthermore,

M

is a strict subspace of the space of (tempered) distributions

\mathcal

, i.e.,

M \subset \mathcal

. If the Fourier coefficients are determined by a distribution

F \in \mathcal

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

\mathcal(\mathbb^n)

Fourier series on a square

We can also define the Fourier series for functions of two variables

x

and

y

in the square

\pi,\pi times \pi,\pi /math>: \begin
f(x,y) & = \sum_ c_e^e^,\\ pt c_ & = \frac \int_^\pi \int_^\pi f(x,y) e^e^\, dx \, dy.
\end 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 JPEG ( , short for Joint Photographic Experts Group and sometimes retroactively referred to as JPEG 1) is a commonly used method of lossy compression for digital images, particularly for those images produced by digital photography. The degr ...

image compression standard uses the two-dimensional

discrete cosine transform A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequency, frequencies. The DCT, first proposed by Nasir Ahmed (engineer), Nasir Ahmed in 1972, is a widely ...

, a discrete form of the

Fourier cosine transform Fourier may refer to: * Fourier (surname), French surname Mathematics *Fourier series, a weighted sum of sinusoids having a common period, the result of Fourier analysis of a periodic function * Fourier analysis, the description of functions as ...

, 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

\mathbf = n_1\mathbf_1 + n_2\mathbf_2 + n_3\mathbf_3

where

n_i

are integers and

\mathbf_i

are three linearly independent but not necessarily orthogonal vectors. Let us consider some function

f(\mathbf)

with the same periodicity as the Bravais lattice, ''i.e.''

f(\mathbf) = f(\mathbf+\mathbf)

for any lattice vector

\mathbf

. This situation frequently occurs in

solid-state physics Solid-state physics is the study of rigid matter, or solids, through methods such as solid-state chemistry, quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state phy ...

where

f(\mathbf)

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

f(\mathbf)

in a Fourier series, it is convenient to introduce an auxiliary function

g(x_1,x_2,x_3) \triangleq f(\mathbf) = f \left (x_1\frac+x_2\frac+x_3\frac \right ).

Both

f(\mathbf)

and

g(x_1,x_2,x_3)

contain essentially the same information. However, instead of the position vector

\mathbf

, the arguments of

g

are coordinates

x_

along the unit vectors

\mathbf_/

of the Bravais lattice, such that

g

is an ordinary periodic function in these variables,

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

g

as a multi-dimensional Fourier series, in complete analogy with the square-periodic function discussed in the previous section. Its Fourier coefficients are

\begin
c(m_1, m_2, m_3) 
 = \frac\int_0^ dx_3 \frac\int_0^ dx_2 \frac\int_0^ dx_1\, g(x_1, x_2, x_3)\, e^
\end,

where

m_1,m_2,m_3

are all integers.

c(m_1,m_2,m_3)

plays the same role as the coefficients

c_

in the previous section but in order to avoid double subscripts we note them as a function. Once we have these coefficients, the function

g

can be recovered via the Fourier series

g(x_1, x_2, x_3)=\sum_ \,c(m_1, m_2, m_3) \, e^.

We would now like to abandon the auxiliary coordinates

x_

and to return to the original position vector

\mathbf

. This can be achieved by means of the reciprocal lattice whose vectors

\mathbf_

are defined such that they are orthonormal (up to a factor

2\pi

) to the original Bravais vectors

\mathbf_

\mathbf_i\cdot\mathbf=2\pi\delta_,

with

\delta_

the

Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...

. With this, the scalar product between a reciprocal lattice vector

\mathbf

and an arbitrary position vector

\mathbf

written in the Bravais lattice basis becomes

\mathbf \cdot \mathbf = \left ( m_1\mathbf_1 + m_2\mathbf_2 + m_3\mathbf_3 \right ) \cdot \left (x_1\frac+ x_2\frac +x_3\frac \right ) = 2\pi \left( x_1\frac+x_2\frac+x_3\frac \right ),

which is exactly the expression occurring in the Fourier exponents. The Fourier series for

f(\mathbf) =g(x_1,x_2,x_3)

can therefore be rewritten as a sum over the all reciprocal lattice vectors

\mathbf= m_1\mathbf_1+m_2\mathbf_2+m_3\mathbf_3

f(\mathbf)=\sum_ c(\mathbf)\, e^,

and the coefficients are

c(\mathbf) = \frac \int_0^ dx_3 \, \frac\int_0^ dx_2 \, \frac\int_0^ dx_1 \, f\left(x_1\frac + x_2\frac + x_3\frac \right) e^.

The remaining task will be to convert this integral over lattice coordinates back into a volume integral. The relation between the lattice coordinates

x_

and the original cartesian coordinates

\mathbf = (x,y,z)

is a linear system of equations,

\mathbf = x_1\frac+x_2\frac+x_3\frac,

which, when written in matrix form,

\beginx\\y\\z\end =\mathbf\beginx_1\\x_2\\x_3\end =\begin\frac,\frac,\frac\end\beginx_1\\x_2\\x_3\end\,,

involves a constant matrix

\mathbf

whose columns are the unit vectors

\mathbf_j/a_j

of the Bravais lattice. When changing variables from

\mathbf

(x_1,x_2,x_3)

in an integral, the same matrix

\mathbf

appears as a

Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of component ...

\dfrac & \dfrac & \dfrac \end\,.

Its determinant

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

\Gamma

in both coordinate systems:

V_ = \int_ d^3 r = J \int_^ dx_1 \int_^ dx_2 \int_^ dx_3=J\, a_1 a_2 a_3

The unit cell being a

parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. Three equiva ...

, we have

V_=\mathbf_1\cdot(\mathbf_2 \times \mathbf_3)

and thus

d^3r=J dx_1 dx_2 dx_3 =\frac dx_1 dx_2 dx_3.

This allows us to write

c (\mathbf)

as the desired volume integral over the primitive unit cell

\Gamma

in ordinary cartesian coordinates:

c(\mathbf) = \frac\int_ d^3 r\, f(\mathbf)\cdot e^\,.

Hilbert space

As the trigonometric series is a special class of orthogonal system, Fourier series can naturally be defined in the context of

Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

s. For example, the space of square-integrable functions on

\pi,\pi /math> forms the Hilbert space L^2(\pi,\pi . Its

inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...

, defined for any two elements

f

and

g

, is given by:

\langle f, g \rangle = \frac\int_^ f(x)\overline\,dx.

This space is equipped with the

orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite Dimension (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit vec ...

\left\

. Then the (generalized) Fourier series expansion of

f \in L^(\pi,\pi

, given by

f(x) = \sum_^\infty c_n e^,

can be written as

f=\sum_^\infty \langle f,e_n \rangle \, e_n.

The sine-cosine form follows in a similar fashion. Indeed, the sines and cosines form an orthogonal set:

\int_^ \cos(mx)\, \cos(nx)\, dx = \frac\int_^ \cos((n-m)x)+\cos((n+m)x)\, dx = \pi \delta_, \quad m, n \ge 1,

\int_^ \sin(mx)\, \sin(nx)\, dx = \frac\int_^ \cos((n-m)x)-\cos((n+m)x)\, dx = \pi \delta_, \quad m, n \ge 1

(where ''δ''_''mn'' is the

), and

\int_^ \cos(mx)\, \sin(nx)\, dx = \frac\int_^ \sin((n+m)x)+\sin((n-m)x)\, dx = 0;

Hence, the set

\left\,

also forms an orthonormal basis for

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

engineering Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...

, 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

s

is continuous and the derivative of

s(x)

(which may not exist everywhere) is square integrable, then the Fourier series of

s

converges absolutely and uniformly to

s(x)

. If a function is square-integrable on the interval

_0,x_0+P /math>, then the Fourier series converges to the function

almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...

. It is possible to define Fourier coefficients for more general functions or distributions, in which case

pointwise convergence In mathematics, pointwise convergence is one of Modes of convergence (annotated index), various senses in which a sequence of function (mathematics), functions can Limit (mathematics), converge to a particular function. It is weaker than uniform co ...

often fails, and convergence in norm or weak convergence is usually studied. Fourier_series_square_wave_circles_animation.gif, link=//upload.wikimedia.org/wikipedia/commons/b/bd/Fourier_series_square_wave_circles_animation.svg, 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)Fourier_series_sawtooth_wave_circles_animation.gif, link=//upload.wikimedia.org/wikipedia/commons/1/1e/Fourier_series_sawtooth_wave_circles_animation.svg, 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_Fourier_Convergence.gif , Example of convergence to a somewhat arbitrary function. Note the development of the "ringing" ( Gibbs phenomenon) at the transitions to/from the vertical sections. 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 : :

s_N(x) = \sum_^N S e^,

is a trigonometric polynomial of degree

N

that can be generally expressed as: :

p_N(x)=\sum_^N p e^.

Parseval's theorem implies that:

Convergence theorems

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

s

is continuously differentiable, then

(i n) S /math> is the n^Fourier coefficient of the first derivative s' . Since s' is continuous, and therefore bounded, it is square-integrable and its Fourier coefficients are square-summable. Then, by the

Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is ...

, :

\left(\sum_, S \right)^2\le \sum_\frac1\cdot\sum_ , nS^2.

This means that

s

is absolutely summable. The sum of this series is a continuous function, equal to

s

, since the Fourier series converges in

L^1

s

: This result can be proven easily if

s

is further assumed to be

C^2

, since in that case

n^2S /math> tends to zero as n \rightarrow \infty . More generally, the Fourier series is absolutely summable, thus converges uniformly to s, provided that s satisfies a

Hölder condition In mathematics, a real or complex-valued function on -dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants , , such that , f(x) - f(y) , \leq C\, x - y\, ^ for all and in the do ...

of order

\alpha > 1/2

. In the absolutely summable case, the inequality: :

\sup_x , s(x) - s_N(x),  \le \sum_ , S

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

x

s

is differentiable at

x

, to more sophisticated results such as Carleson's theorem which states that the Fourier series of an

L^2

function converges

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 ,πby :

f(x) = \sum_^ \frac \sin\left \left( 2^ +1 \right) \frac\right

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

\sum_^\infty C_m \cos(mx).

The coefficients are: :

C_m=\frac 1\pi\sum_^ \frac \left\

As increases, the coefficients will be positive and increasing until they reach a value of about

C_m\approx 2/(n^2\pi)

m=2^/2

for some and then become negative (starting with a value around

-2/(n^2\pi)

) and getting smaller, before starting a new such wave. At

x=0

the Fourier series is simply the running sum of

C_m,

and this builds up to around :

\frac 1\sum_^\frac 2\sim\frac 1\ln 2^=\frac n\pi\ln 2

in the th 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.

Notes

References

Bibliography

* * * * * 2003 unabridged republication of the 1878 English translation by Alexander Freeman of Fourier's work ''Théorie Analytique de la Chaleur'', originally published in 1822. * * * * * * * Translated by M. Ackerman from '' Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert'', Springer, Berlin, 1928. * * * * * * * * The first edition was published in 1935.

External links

* * * * {{DEFAULTSORT:Fourier Series Joseph Fourier

History

Beginnings

Fourier's motivation

Other applications

Definition

Synthesis

Analysis

Example

Amplitude-phase form

Common notations

Table of common Fourier series

Table of basic transformation rules

Properties

Symmetry relations

Riemann–Lebesgue lemma

Parseval's theorem

Plancherel's theorem

Convolution theorems

Derivative property

Compact groups

Riemannian manifolds

Locally compact Abelian groups

Extensions

Fourier-Stieltjes series

Fourier series on a square

Fourier series of a Bravais-lattice-periodic function

Hilbert space

Fourier theorem proving convergence of Fourier series

Least squares property

Convergence theorems

Divergence

See also

Notes

References

Bibliography

External links