In

Fourier_series_square_wave_circles_animation.gif, link=, 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=, 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.

_{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
$$\backslash begin\; a\_k\&=\backslash int\_^1\backslash varphi(y)\backslash cos(2k+1)\backslash frac\backslash ,dy\; \backslash \backslash \; \&=\; \backslash int\_^1\backslash left(a\backslash cos\backslash frac\backslash cos(2k+1)\backslash frac+a\text{'}\backslash cos\; 3\backslash frac\backslash cos(2k+1)\backslash frac+\backslash cdots\backslash right)\backslash ,dy\; \backslash end$$
can be carried out term-by-term. But all terms involving $\backslash cos(2j+1)\backslash frac\; \backslash cos(2k+1)\backslash frac$ for vanish when integrated from −1 to 1, leaving only the ''k''th term.
In these few lines, which are close to the modern

If $f$ belongs to $L^2(P)$ (an interval of length $P$) then: $\backslash sum\_^\backslash infty\; \backslash Bigl,\; F;\; href="/html/ALL/s/.html"\; ;"title="">$

^{th} derivative is continuous.
* If $f\; \backslash in\; C^1(\backslash mathbb)$, then the Fourier coefficients $\backslash widehat;\; href="/html/ALL/s/.html"\; ;"title="">$

^{2}(''G''), where ''G'' is a compact group, in such a way that the Fourier transform carries

_{''mn''} is the

^{''ix''} transforms a Fourier series into a Laurent series, or conversely. This is used in the ''q''-series expansion of the ''j''-invariant.
*

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...

, a Fourier series () is a periodic function
A periodic function is a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logica ...

composed of harmonically related sinusoids
A capillary is a small blood vessel
The blood vessels are the components of the circulatory system that transport blood throughout the human body. These vessels transport blood cells, nutrients, and oxygen to the tissues of the body. They also ...

combined by a weighted summation. With appropriate weights, one cycle (or ''period'') of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). As such, the summation is a synthesis of another function. The discrete-time Fourier transform
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

is an example of Fourier series
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used i ...

. The process of deriving weights that describe a given function is a form of Fourier analysis
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis ...

. For functions on unbounded intervals, the analysis and synthesis analogies are Fourier transform#REDIRECT Fourier transform
In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...

and inverse transform.
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 at the time Fourier completed his original work. Fourier originally defined the Fourier series for real-valued functions of real arguments, and using the sine and cosine functions as the basis set for the decomposition. Many other Fourier-related transforms have since been defined, extending the initial idea to other applications. This general area of inquiry is now sometimes called harmonic analysis
Harmonic analysis is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathe ...

. A Fourier series, however, can be used only for periodic functions, or for functions on a bounded (compact) interval.
Definition

Consider a real-valued function, $s(x),$ that isintegrableIn mathematics, integrability is a property of certain dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in a Manifold, geometric ...

on an interval of length $P,$ which will be the period
Period may refer to:
Common uses
* Era, a length or span of time
* Full stop (or period), a punctuation mark
Arts, entertainment, and media
* Period (music), a concept in musical composition
* Period, a descriptor for a historical or period drama ...

of the Fourier series. The correlation function:$$\backslash Chi\_f(\backslash tau)\; \backslash triangleq\; \backslash int\_\; s(x)\backslash cdot\; \backslash cos\backslash left(\; 2\backslash pi\; f\; (x-\backslash tau)\; \backslash right)\backslash ,\; dx\; ;\; \backslash quad\; \backslash tau\; \backslash in\; \backslash left;\; href="/html/ALL/s/0,\backslash \_2\backslash pi\_/\_f\_\backslash right.html"\; ;"title="0,\backslash \; 2\backslash pi\; /\; f\; \backslash right">0,\backslash \; 2\backslash pi\; /\; f\; \backslash right$$matched filter
In signal processing
Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Signal pr ...

, with ''template'' $\backslash cos(2\backslash pi\; f\; x).$ Its peak value is a relative measure of the presence of frequency $f$ in function $s.$ The analysis process determines, for certain key frequencies, the maximum correlation and the corresponding phase offset, $(\backslash tau\; f).$ The synthesis process (the actual Fourier series), in terms of parameters to be determined by analysis, is:
*In general, integer $N$ is theoretically infinite. Even so, the series might not converge or exactly equate to $s(x)$ at all values of $x$ (such as a single-point discontinuity) in the analysis interval. For the "well-behaved" functions typical of physical processes, equality is customarily assumed.
*Integer $n,$ used as an index, is also the number of cycles of the $n$-th harmonic in interval $P.$ Therefore, the length of a cycle, in the units of $x,$ is $P/n.$ The corresponding harmonic frequency is $n/P,$ so the $n$-th harmonic is $\backslash cos\backslash left(2\backslash pi\; x\; \backslash tfrac\backslash right).$ Some texts define $P\; =\; 2\backslash pi$ to simplify the argument of the sinusoid functions at the expense of generality.
*Clearly 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 ($N$).
Rather than computationally intensive cross-correlation, Fourier analysis customarily exploits a trigonometric identity:
:$A\_n\backslash cdot\; \backslash cos\backslash left(\backslash tfracnx-\backslash varphi\_n\backslash right)\; \backslash \; \backslash equiv\; \backslash \; \backslash underbrace\_\backslash cdot\; \backslash cos\backslash left(\backslash tfrac\; nx\backslash right)\; +\; \backslash underbrace\_\backslash cdot\; \backslash sin\backslash left(\backslash tfrac\; nx\backslash right),$
where parameters $a\_n$ and $b\_n$ replace $A\_n$ and $\backslash varphi\_n,$ and can be found by evaluating the cross-correlation at only two values of phase:
Then: $A\_n\; =\; \backslash sqrt$ and $\backslash varphi\_n\; =\; \backslash operatorname(b\_n,a\_n)$ (see Atan2
The function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automati ...

) or more directly:
And note that $a\_0$ and $b\_0$ can be reduced to $a\_0\; =\; \backslash frac\; \backslash int\_P\; s(x)\; \backslash ,\; dx$ and $b\_0\; =\; 0.$
Another applicable identity is Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematics, mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex number, complex exponential function. Euler's ...

. Here, complex conjugation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

is denoted by an asterisk:
:$\backslash begin\; \backslash cos\backslash left(\; \backslash tfracnx\; -\; \backslash varphi\_n\; \backslash right)\; \&\backslash equiv\; \backslash tfrace^\; \&\; +\; \backslash tfrace^\backslash \backslash \; \&=\backslash left(\backslash tfrac\; e^\backslash right)\; \backslash cdot\; e^\; \&+\backslash left(\backslash tfrac\; e^\backslash right)^*\; \backslash cdot\; e^.\; \backslash end$
Therefore, with definitions:
:$c\_n\; \backslash triangleq\; \backslash left\backslash \backslash quad\; =\backslash quad\; \backslash frac\backslash int\_P\; s(x)\backslash cdot\; e^\backslash \; dx,$
the final result is:
This is the customary form for generalizing to .
Example

Consider a sawtooth function: :$s(x)\; =\; \backslash frac,\; \backslash quad\; \backslash mathrm\; -\backslash pi\; <\; x\; <\; \backslash pi,$ :$s(x\; +\; 2\backslash pi\; k)\; =\; s(x),\; \backslash quad\; \backslash mathrm\; -\backslash pi\; <\; x\; <\; \backslash pi\; \backslash text\; k\; \backslash in\; \backslash mathbb.$ In this case, the Fourier coefficients are given by :$\backslash begin\; a\_n\; \&\; =\; \backslash frac\backslash int\_^s(x)\; \backslash cos(nx)\backslash ,dx\; =\; 0,\; \backslash quad\; n\; \backslash ge\; 0.\; \backslash \backslash $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: When $x=\backslash pi$, the Fourier series converges to 0, which is the half-sum of the left- and right-limit of ''s'' at $x=\backslash pi$. This is a particular instance of the Dirichlet theorem for Fourier series. This example leads to a solution of theBasel problem
The Basel problem is a problem in mathematical analysis
Analysis is the branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebr ...

.
Convergence

Inengineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more specializ ...

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 $s$ 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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

on the interval $;\; href="/html/ALL/s/\_0,x\_0+P.html"\; ;"title="\_0,x\_0+P">\_0,x\_0+P$Convergence of Fourier seriesIn mathematics, the question of whether the Fourier series of a periodic function convergent series, converges to the given function (mathematics), function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. ...

. 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.
Complex-valued functions

If $s(x)$ is a complex-valued function of a real variable $x,$ 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: :$c\_\; =\; \backslash frac\backslash int\_P\; \backslash operatorname\backslash \backslash cdot\; e^\backslash \; dx$ and $c\_\; =\; \backslash frac\backslash int\_P\; \backslash operatorname\backslash \backslash cdot\; e^\backslash \; dx$ :$s\_(x)\; =\; \backslash sum\_^N\; c\_\backslash cdot\; e^\; +\; i\backslash cdot\; \backslash sum\_^N\; c\_\backslash cdot\; e^\; =\backslash sum\_^N\; \backslash left(c\_+i\backslash cdot\; c\_\backslash right)\; \backslash cdot\; e^.$ Defining $c\_n\; \backslash triangleq\; c\_+i\backslash cdot\; c\_$ yields: This is identical to except $c\_n$ and $c\_$ are no longer complex conjugates. The formula for $c\_n$ is also unchanged: :$\backslash begin\; c\_n\; \&=\; \backslash frac\backslash int\_\; \backslash operatorname\backslash \backslash cdot\; e^\backslash \; dx\; +\; i\backslash cdot\; \backslash frac\; \backslash int\_\; \backslash operatorname\backslash \backslash cdot\; e^\backslash \; dx\backslash \backslash $&= \frac \int_ \left(\operatorname\ +i\cdot \operatorname\\right)\cdot e^\ dx \ = \ \frac\int_ s(x)\cdot e^\ dx. \endOther 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 $\backslash hat;\; href="/html/ALL/s/.html"\; ;"title="">$&= \sum_^\infty S cdot e^ && \scriptstyle \mathsf \end In engineering, particularly when the variable $x$ represents time, the coefficient sequence is called afrequency domain
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or Signal (information theory), signals with respect to frequency, rather than time. Put simply, a time-dom ...

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
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

:
:$S(f)\; \backslash \; \backslash triangleq\; \backslash \; \backslash sum\_^\backslash infty\; S;\; href="/html/ALL/s/.html"\; ;"title="">$
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
Unit may refer to:
Arts and entertainment
* UNIT
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action ...

. The "teeth" of the comb are spaced at multiples (i.e. harmonics
A harmonic is any member of the harmonic series
Harmonic series may refer to either of two related concepts:
*Harmonic series (mathematics)
*Harmonic series (music)
{{Disambig .... The term is employed in various disciplines, including music ...

) of $1/P$, which is called the fundamental frequency
The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency
Frequency is the number of occurrences of a repeating event per unit of time
A unit of time is any particular time
Time is th ...

. $s\_(x)$ can be recovered from this representation by an inverse Fourier transformIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

:
:$\backslash begin\; \backslash mathcal^\backslash \; \&=\; \backslash int\_^\backslash infty\; \backslash left(\; \backslash sum\_^\backslash infty\; S;\; href="/html/ALL/s/.html"\; ;"title="">$&= \sum_^\infty S cdot \int_^\infty \delta\left(f-\frac\right) e^\,df, \\&= \sum_^\infty S cdot e^ \ \ \triangleq \ s_\infty(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.
History

The Fourier series is named in honor ofJean-Baptiste Joseph Fourier
Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such t ...

(1768–1830), who made important contributions to the study of trigonometric series
In mathematics, a trigonometric series is a series of the form:
: \frac+\displaystyle\sum_^(A_ \cos + B_ \sin).
It is called a Fourier series if the terms A_ and B_ have the form:
:A_=\frac\displaystyle\int^_0\! f(x) \cos \,dx\qquad (n=0,1,2, ...

, after preliminary investigations by Leonhard Euler
Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

, Jean le Rond 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 ''Enc ...

, and Daniel Bernoulli
Daniel Bernoulli Fellows of the Royal Society, FRS (; – 27 March 1782) was a Swiss people, Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for ...

. Fourier introduced the series for the purpose of solving the heat equation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

-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
French (french: français(e), link=no) may refer to:
* Something of, from, or related to France
France (), officially the French Republic (french: link=no, République française), is a transcontinental country
This is a list of co ...

. 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.
The heat equation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

is a partial differential equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

. 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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

or cosine
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 al ...

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 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'' and ''y'' would be ...

) 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
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

and integral
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

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
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study ...

and Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics ...

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 sinusoid
A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave. It is named after the function sine, of which it is the graph. It occurs often in both pure and applied mathemat ...

s. 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 which use electricity, electronics
The field of electronics is a branch of physics and electrical enginee ...

, vibration
Vibration is a mechanical phenomenon whereby oscillation
Oscillation is the repetitive variation, typically in time
Time is the indefinite continued sequence, progress of existence and event (philosophy), events that occur in an apparentl ...

analysis, acoustics
Acoustics is a branch of physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other wo ...

, 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
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetis ...

, image processing
Digital image processing is the use of a digital computer
A computer is a machine
A machine is a man-made device that uses power to apply forces and control movement to perform an action. Machines can be driven by animals and people
...

, quantum mechanics
Quantum mechanics is a fundamental theory
A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...

, econometrics
Econometrics is the application of Statistics, statistical methods to economic data in order to give Empirical evidence, empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," ''The New Palgrave: A Dictionary of Econ ...

, shell theory, etc.
Beginnings

Joseph Fourier wrote: This immediately gives any coefficient ''aformalism
Formalism may refer to:
* Form (disambiguation)
* Formal (disambiguation)
* Legal formalism, legal positivist view that the substantive justice of a law is a question for the legislature rather than the judiciary
* Formalism (linguistics)
* Scienti ...

used in Fourier series, Fourier revolutionized both mathematics and physics. Although similar trigonometric series were previously used by Euler
Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

, d'Alembert
Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician
A mechanician is an engineer or a scientist working in the field of mechanics, or in a related or sub-field: engineering or com ...

, Daniel Bernoulli
Daniel Bernoulli Fellows of the Royal Society, FRS (; – 27 March 1782) was a Swiss people, Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for ...

and Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician
This is a List of German mathematician
A mathematician is someone who uses an extensive knowledge of m ...

, 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), ''Convergence'' (book series), edited by Ruth Nanda Anshen
*Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics:
**A four-par ...

, function space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s, and harmonic analysis
Harmonic analysis is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathe ...

.
When Fourier submitted a later competition essay in 1811, the committee (which included Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaRoman_Forum.html" ;"title="Curia Julia in the Roman Forum">Curia Julia in the Roman Forum
A senate is a deliberative assembly, often the upper house or Debating chamber, chamber of a bicame ...

, Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath
A polymath ( el, πολυμαθής, ', "having learned much"; Latin
Latin (, or , ) is a classical language belonging to the I ...

, Malus
''Malus'' ( or ) is a genus
Genus /ˈdʒiː.nəs/ (plural genera /ˈdʒen.ər.ə/) is a taxonomic rank
In biological classification
In biology
Biology is the natural science that studies life and living organisms, including t ...

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
Rigour (British English
British English (BrE) is the standard dialect of the English language
English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval England, wh ...

''.
Fourier's motivation

The Fourier series expansion of the sawtooth function (above) looks more complicated than the simple formula $s(x)=x/\; \backslash pi$, so it is not immediately apparent why one would need the Fourier series. While there are many applications, Fourier's motivation was in solving theheat equation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

. For example, consider a metal plate in the shape of a square whose sides measure $\backslash pi$ meters, with coordinates $(x,y)\; \backslash in;\; href="/html/ALL/s/,\backslash pi.html"\; ;"title=",\backslash pi">,\backslash pi$hyperbolic sine
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

function. This solution of the heat equation is obtained by multiplying each term of by $\backslash sinh(ny)/\backslash sinh(n\backslash pi)$. 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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

. This method of solving the heat problem was made possible by Fourier's work.
Complex Fourier series animation

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).Other applications

Another application of this Fourier series is to solve theBasel problem
The Basel problem is a problem in mathematical analysis
Analysis is the branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebr ...

by using Parseval's theorem. The example generalizes and one may compute (2''n''), for any positive integer ''n''.
Table of common Fourier series

Some common pairs of periodic functions and their Fourier Series coefficients are shown in the table below. The following notation applies: *$f(x)$ designates a periodic function defined on $0\; <\; x\; \backslash le\; P$. *$a\_0,\; a\_n,\; b\_n$ designate the Fourier Series coefficients (sine-cosine form) of the periodic function $f$ as defined in .Table of basic properties

This table shows some mathematical operations in the time domain and the corresponding effect in the Fourier series coefficients. Notation: *Complex conjugation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

is denoted by an asterisk.
*$f(x),g(x)$ designate $P$-periodic functions or functions defined only for $x\; \backslash in;\; href="/html/ALL/s/,P.html"\; ;"title=",P">,P$.
* $F;\; href="/html/ALL/s/.html"\; ;"title="">$Other properties

Symmetry

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: :$\backslash begin\; \backslash text\; \&\; f\; \&\; =\; \&\; f\_\; \&\; +\; \&\; f\_\; \&\; +\; \&\; i\; f\_\; \&\; +\; \&\backslash underbrace\; \backslash \backslash \; \&\backslash Bigg\backslash Updownarrow\backslash mathcal\; \&\; \&\backslash Bigg\backslash Updownarrow\backslash mathcal\; \&\; \&\backslash \; \backslash \; \backslash Bigg\backslash Updownarrow\backslash mathcal\; \&\; \&\backslash \; \backslash \; \backslash Bigg\backslash Updownarrow\backslash mathcal\; \&\; \&\backslash \; \backslash \; \backslash Bigg\backslash Updownarrow\backslash mathcal\backslash \backslash \; \backslash text\; \&\; F\; \&\; =\; \&\; F\_\; \&\; +\; \&\; \backslash overbrace\; \&\; +\; \&i\backslash \; F\_\; \&\; +\; \&\; F\_\; \backslash end$ From this, various relationships are apparent, for example: *The transform of a real-valued function () is the even symmetric function . Conversely, an even-symmetric transform implies a real-valued time-domain. *The transform of an imaginary-valued function () is the odd symmetric function , and the converse is true. *The transform of an even-symmetric function () is the real-valued function , and the converse is true. *The transform of an odd-symmetric function () is the imaginary-valued function , and the converse is true.Riemann–Lebesgue lemma

If $f$ isintegrableIn mathematics, integrability is a property of certain dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in a Manifold, geometric ...

, $\backslash lim\_\; F;\; href="/html/ALL/s/.html"\; ;"title="">$, $\backslash lim\_\; a\_n=0$ and $\backslash lim\_\; b\_n=0.$ This result is known as the Riemann–Lebesgue lemma
In mathematics, the Riemann–Lebesgue lemma, named after Bernhard Riemann and Henri Lebesgue, states that the Fourier transform or Laplace transform of an Lp space, ''L''1 function vanishes at infinity. It is of importance in harmonic analysis and ...

.

Plancherel's theorem
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

If $c\_0,\backslash ,\; c\_,\backslash ,\; c\_,\; \backslash ldots$ are coefficients and $\backslash sum\_^\backslash infty\; ,\; c\_n,\; ^2\; <\; \backslash infty$ then there is a unique function $f\backslash in\; L^2(P)$ such that $F;\; href="/html/ALL/s/.html"\; ;"title="">$ for every $n$.
Convolution theorems

Given $P$-periodic functions, $f\_$ and $g\_$ with Fourier series coefficients $F;\; href="/html/ALL/s/.html"\; ;"title="">$doubly infinite
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

sequence $\backslash left\; \backslash \_$ in $c\_0(\backslash mathbb)$ is the sequence of Fourier coefficients of a function in $L^1(;\; href="/html/ALL/s/,2\backslash pi.html"\; ;"title=",2\backslash pi">,2\backslash pi$ if and only if it is a convolution of two sequences in $\backslash ell^2(\backslash mathbb)$. See
Derivative property

We say that $f$ belongs to $C^k(\backslash mathbb)$ if $f$ is a 2-periodic function on $\backslash mathbb$ which is $k$ times differentiable, and its ''k''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 anycompact group
of center 0 and radius 1 in the complex plane is a compact Lie group with complex multiplication.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structu ...

. Typical examples include those classical group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

s that are compact. This generalizes the Fourier transform to all spaces of the form ''L''convolution
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

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.
Riemannian manifolds

If the domain is not a group, then there is no intrinsically defined convolution. However, if $X$ is acompact
Compact as used in politics may refer broadly to a pact
A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations
International relations (IR), international affairs (IA) or internationa ...

Riemannian manifold
In differential geometry
Differential geometry is a mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a c ...

, it has a Laplace–Beltrami operator
In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian manifold, Riemannian and pseudo-Riemannian manifol ...

. The Laplace–Beltrami operator is the differential operator that corresponds to Laplace operator
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

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 $;\; href="/html/ALL/s/\backslash pi,\backslash pi.html"\; ;"title="\backslash pi,\backslash pi">\backslash pi,\backslash pi$spherical harmonics
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

.
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 $L^1(G)$ or $L^2(G)$, where $G$ is an LCA group. If $G$ is compact, one also obtains a Fourier series, which converges similarly to the $;\; href="/html/ALL/s/\backslash pi,\backslash pi.html"\; ;"title="\backslash pi,\backslash pi">\backslash pi,\backslash pi$Fourier integral
In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequency, such as the expression o ...

. This generalization yields the usual Fourier transform#REDIRECT Fourier transform
In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...

when the underlying locally compact Abelian group is $\backslash mathbb$.
Extensions

Fourier series on a square

We can also define the Fourier series for functions of two variables $x$ and $y$ in the square $;\; href="/html/ALL/s/\backslash pi,\backslash pi.html"\; ;"title="\backslash pi,\backslash pi">\backslash pi,\backslash pi$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 inimage compression
Image compression is a type of data compression
In signal processing
Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image proc ...

. In particular, the jpeg
JPEG ( ) is a commonly used method of lossy compression
In information technology, lossy compression or irreversible compression is the class of data encoding methods that uses inexact approximations and partial data discarding to represe ...

image compression standard uses the two-dimensional discrete cosine transform
A discrete cosine transform (DCT) expresses a finite sequence of data points
In statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics ...

, which is a Fourier-related transform
This is a list of linear transformation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, chan ...

using only the cosine basis functions.
For two-dimensional arrays with a staggered appearance, half of the Fourier series coefficients disappear, due to additional symmetry.
Fourier series of Bravais-lattice-periodic-function

A three-dimensionalBravais lattice
In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of Translation operator (quantum mechanics)#Discrete Translational Symmetry, discrete translation operations described in t ...

is defined as the set of vectors of the form:
$$\backslash mathbf\; =\; n\_1\backslash mathbf\_1\; +\; n\_2\backslash mathbf\_2\; +\; n\_3\backslash mathbf\_3$$
where $n\_i$ are integers and $\backslash mathbf\_i$ are three linearly independent vectors. Assuming we have some function, $f(\backslash mathbf)$, such that it obeys the condition of periodicity for any Bravais lattice vector $\backslash mathbf$, $f(\backslash mathbf)\; =\; f(\backslash mathbf+\backslash mathbf)$, 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
300px, Isosurface of the square modulus of a Bloch state in a silicon lattice
In condensed matter physics
Condensed matter physics is the field of physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), ...

. First, we may write any arbitrary position vector $\backslash mathbf$ in the coordinate-system of the lattice:
$$\backslash mathbf\; =\; x\_1\backslash frac+\; x\_2\backslash frac+\; x\_3\backslash frac,$$
where $a\_i\; \backslash triangleq\; ,\; \backslash mathbf\_i,\; ,$ meaning that $a\_i$ is defined to be the magnitude of $\backslash mathbf\_i$, so $\backslash hat\; =\; \backslash frac$ is the unit vector directed along $\backslash mathbf\_i$.
Thus we can define a new function,
$$g(x\_1,x\_2,x\_3)\; \backslash triangleq\; f(\backslash mathbf)\; =\; f\; \backslash left\; (x\_1\backslash frac+x\_2\backslash frac+x\_3\backslash frac\; \backslash right\; ).$$
This new function, $g(x\_1,x\_2,x\_3)$, is now a function of three-variables, each of which has periodicity $a\_1$, $a\_2$, and $a\_3$ respectively:
$$g(x\_1,x\_2,x\_3)\; =\; g(x\_1+a\_1,x\_2,x\_3)\; =\; g(x\_1,x\_2+a\_2,x\_3)\; =\; g(x\_1,x\_2,x\_3+a\_3).$$
This enables us to build up a set of Fourier coefficients, each being indexed by three independent integers $m\_1,m\_2,m\_3$. In what follows, we use function notation to denote these coefficients, where previously we used subscripts. If we write a series for $g$ on the interval $\backslash left;\; href="/html/ALL/s/0,\_a\_1\backslash right\_.html"\; ;"title="0,\; a\_1\backslash right\; ">0,\; a\_1\backslash right$Jacobian determinant
In vector calculus
Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes ...

:
$$\backslash begin\; \backslash dfrac\; \&\; \backslash dfrac\; \&\; \backslash dfrac\; \backslash \backslash ;\; href="/html/ALL/s/2pt.html"\; ;"title="2pt">2pt$$
which after some calculation and applying some non-trivial cross-product identities can be shown to be equal to:
$$\backslash frac$$
(it may be advantageous for the sake of simplifying calculations, to work in such a Cartesian coordinate system, in which it just so happens that $\backslash mathbf\_1$ is parallel to the ''x'' axis, $\backslash mathbf\_2$ lies in the ''xy''-plane, and $\backslash mathbf\_3$ has components of all three axes). The denominator is exactly the volume of the primitive unit cell which is enclosed by the three primitive-vectors $\backslash mathbf\_1$, $\backslash mathbf\_2$ and $\backslash mathbf\_3$. In particular, we now know that
$$dx\_1\; \backslash ,\; dx\_2\; \backslash ,\; dx\_3\; =\; \backslash frac\; \backslash cdot\; dx\; \backslash ,\; dy\; \backslash ,\; dz.$$
We can write now $h(\backslash mathbf)$ as an integral with the traditional coordinate system over the volume of the primitive cell, instead of with the $x\_1$, $x\_2$ and $x\_3$ variables:
$$h(\backslash mathbf)\; =\; \backslash frac\backslash int\_\; d\backslash mathbf\; f(\backslash mathbf)\backslash cdot\; e^$$
writing $d\backslash mathbf$ for the volume element $dx\; \backslash ,\; dy\; \backslash ,\; dz$; and where $C$ is the primitive unit cell, thus, $\backslash mathbf\_1\backslash cdot(\backslash mathbf\_2\; \backslash times\; \backslash mathbf\_3)$ is the volume of the primitive unit cell.
Hilbert space interpretation

In the language ofHilbert space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

s, the set of functions $\backslash left\backslash $ is an orthonormal basisIn linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of u ...

for the space $L^2(;\; href="/html/ALL/s/\backslash pi,\backslash pi.html"\; ;"title="\backslash pi,\backslash pi">\backslash pi,\backslash pi$ of square-integrable functions on $;\; href="/html/ALL/s/\backslash pi,\backslash pi.html"\; ;"title="\backslash pi,\backslash pi">\backslash pi,\backslash pi$inner product
In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...

given for any two elements $f$ and $g$ by:
:$\backslash langle\; f,\backslash ,\; g\; \backslash rangle\; \backslash ;\backslash triangleq\; \backslash ;\; \backslash frac\backslash int\_^\; f(x)g^*(x)\backslash ,dx,$ where $g^(x)$ is the complex conjugate of $g(x).$
The basic Fourier series result for Hilbert spaces can be written as
:$f=\backslash sum\_^\backslash infty\; \backslash langle\; f,e\_n\; \backslash rangle\; \backslash ,\; e\_n.$
This corresponds exactly to the complex exponential formulation given above. The version with sines and cosines is also justified with the Hilbert space interpretation. Indeed, the sines and cosines form an orthogonal set
In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms. Two elements ''u'' and ''v'' of a vector space with bilinear form ''B'' are orthogonal when . Depending on the bilin ...

:
$$\backslash int\_^\; \backslash cos(mx)\backslash ,\; \backslash cos(nx)\backslash ,\; dx\; =\; \backslash frac\backslash int\_^\; \backslash cos((n-m)x)+\backslash cos((n+m)x)\backslash ,\; dx\; =\; \backslash pi\; \backslash delta\_,\; \backslash quad\; m,\; n\; \backslash ge\; 1,$$
$$\backslash int\_^\; \backslash sin(mx)\backslash ,\; \backslash sin(nx)\backslash ,\; dx\; =\; \backslash frac\backslash int\_^\; \backslash cos((n-m)x)-\backslash cos((n+m)x)\backslash ,\; dx\; =\; \backslash pi\; \backslash delta\_,\; \backslash quad\; m,\; n\; \backslash ge\; 1$$
(where ''δ''Kronecker delta
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

), and
$$\backslash int\_^\; \backslash cos(mx)\backslash ,\; \backslash sin(nx)\backslash ,\; dx\; =\; \backslash frac\backslash int\_^\; \backslash sin((n+m)x)+\backslash sin((n-m)x)\backslash ,\; dx\; =\; 0;$$
furthermore, the sines and cosines are orthogonal to the constant function $1$. An ''orthonormal basis'' for $L^2(;\; href="/html/ALL/s/\backslash pi,\backslash pi.html"\; ;"title="\backslash pi,\backslash pi">\backslash pi,\backslash pi$ consisting of real functions is formed by the functions $1$ and $\backslash sqrt\; \backslash cos\; (nx)$, $\backslash sqrt\; \backslash sin\; (nx)$ with ''n'' = 1, 2, …. The density of their span is a consequence of the Stone–Weierstrass theoremIn mathematical analysis, the Weierstrass approximation theorem states that every continuous function
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struct ...

, but follows also from the properties of classical kernels like the .
Approximation and convergence of Fourier series

Recalling : $$s\_(x)\; =\; \backslash sum\_^N\; S;\; href="/html/ALL/s/.html"\; ;"title="">$$ it is atrigonometric polynomialIn the mathematical subfields of numerical analysis
(c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/602 + 10/603 = 1.41421296...
Numeri ...

of degree $N$, generally: $$p\_(x)=\backslash sum\_^N\; p;\; href="/html/ALL/s/.html"\; ;"title="">$$
Least squares property

Parseval's theorem implies that:Convergence

Because of the least squares property, and because of the completeness of the Fourier basis, we obtain an elementary convergence result. We have already mentioned that if $s$ is continuously differentiable, then $(i\backslash cdot\; n)\; S;\; href="/html/ALL/s/.html"\; ;"title="">$Cauchy–Schwarz inequality
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

, that $s\_$ is absolutely summable. The sum of this series is a continuous function, equal to $s$, since the Fourier series converges in the mean to $s$:
This result can be proven easily if $s$ is further assumed to be $C^2$, since in that case $n^2S;\; href="/html/ALL/s/.html"\; ;"title="">$Hölder condition
In mathematics, a real or complex-valued function ''f'' on ''d''-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants ''C'', α>0, such that
: , f(x) - f(y) , \leq C\, x ...

of order $\backslash alpha\; >\; 1/2$. In the absolutely summable case, the inequality:
:$\backslash sup\_x\; ,\; s(x)\; -\; s\_(x),\; \backslash le\; \backslash sum\_\; ,\; S;\; href="/html/ALL/s/.html"\; ;"title="">$ proves uniform convergence.
Many other results concerning the convergence of Fourier seriesIn mathematics, the question of whether the Fourier series of a periodic function convergent series, converges to the given function (mathematics), function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. ...

are known, ranging from the moderately simple result that the series converges at $x$ if $s$ is differentiable at $x$, to Lennart Carleson
Lennart Axel Edvard Carleson (born 18 March 1928) is a Sweden, Swedish mathematician, known as a leader in the field of harmonic analysis. One of his most noted accomplishments is his proof of Lusin's conjecture. He was awarded the Abel Prize in ...

's much more sophisticated result that the Fourier series of an $L^2$ function actually converges almost everywhere
In measure theory
Measure is a fundamental concept of . Measures provide a mathematical abstraction for common notions like , /, , , of events, and — after — . These seemingly distinct concepts are innately very similar and may, in many ...

.
These theorems, and informal variations of them that don't specify the convergence conditions, are sometimes referred to generically as "Fourier's theorem" or "the Fourier theorem".
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. Theuniform boundedness principle
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

yields a simple non-constructive proof of this fact.
In 1922, Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovie ...

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 .
See also

* ATS theorem *Dirichlet kernelIn mathematical analysis, the Dirichlet kernel, named after the Germany, German mathematician Peter Gustav Lejeune Dirichlet, is the collection of functions defined as
:D_n(x)=\frac\sum_^n
e^=\frac\left(1+2\sum_^n\cos(kx)\right)=\frac,
where is any ...

* Discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discret ...

* Fast Fourier transform
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in t ...

* Fejér's theoremIn mathematics, Fejér's theorem,Leopold FejérUntersuchungen über Fouriersche Reihen ''Mathematische Annalen, Math. Annalen''vol. 58 1904, 51-69. named after Hungary, Hungarian mathematician Lipót Fejér, states that if ''f'':R → C is ...

* Fourier analysis
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis ...

* Fourier sine and cosine series
* Fourier transform#REDIRECT Fourier transform
In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...

* Gibbs phenomenonIn mathematics, the Gibbs phenomenon, discovered by Available on-line at:National Chiao Tung University: Open Course Ware: Hewitt & Hewitt, 1979. and rediscovered by , is the peculiar manner in which the Fourier series of a piecewise continuously ...

* Laurent series
(Holomorphic functions are analytic, analytic).
In mathematics, the Laurent series of a complex function ''f''(''z'') is a representation of that function as a power series which includes terms of negative degree. It may be used to express compl ...

– the substitution ''q'' = ''e''Least-squares spectral analysis
Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum
The power spectrum S_(f) of a time series x(t) describes the distribution of power
Power typically refers to:
* Power (physics)
In physics, power is t ...

* Multidimensional transformIn mathematical analysis and applications, multidimensional transforms are used to analyze the frequency content of signals in a domain of two or more dimensions.
Multidimensional Fourier transform
One of the more popular multidimensional trans ...

* Spectral theoryIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

* Sturm–Liouville theoryIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

* Residue Theorem
In complex analysis
of the function
.
Hue represents the argument, brightness the magnitude.
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigat ...

integrals of f singularities, poles
Notes

References

Further reading

* * 2003 unabridged republication of the 1878 English translation by Alexander Freeman of Fourier's work ''Théorie Analytique de la Chaleur'', originally published in 1822. * * * *Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...

, ''Development of mathematics in the 19th century''. Mathsci Press Brookline, Mass, 1979. 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