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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 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 ...

periodic function
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 ...

sinusoids
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 is
integrableIn 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 ...

period
of the Fourier series. The correlation function:\Chi_f(\tau) \triangleq \int_ s(x)\cdot \cos\left( 2\pi f (x-\tau) \right)\, dx ; \quad \tau \in \left 0,\ 2\pi / f \right/math>is essentially a
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'' \cos(2\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, (\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 \cos\left(2\pi x \tfrac\right). Some texts define P = 2\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\cdot \cos\left(\tfracnx-\varphi_n\right) \ \equiv \ \underbrace_\cdot \cos\left(\tfrac nx\right) + \underbrace_\cdot \sin\left(\tfrac nx\right), where parameters a_n and b_n replace A_n and \varphi_n, and can be found by evaluating the cross-correlation at only two values of phase: Then: A_n = \sqrt and \varphi_n = \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 ...

Atan2
) or more directly: And note that a_0 and b_0 can be reduced to  a_0 = \frac \int_P s(x) \, 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 ...

Euler's formula
. 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: : \begin \cos\left( \tfracnx - \varphi_n \right) &\equiv \tfrace^ & + \tfrace^\\ &=\left(\tfrac e^\right) \cdot e^ &+\left(\tfrac e^\right)^* \cdot e^. \end Therefore, with definitions: :c_n \triangleq \left\\quad =\quad \frac\int_P s(x)\cdot e^\ dx, the final result is: This is the customary form for generalizing to .


Example

Consider a sawtooth function: :s(x) = \frac, \quad \mathrm -\pi < x < \pi, :s(x + 2\pi k) = s(x), \quad \mathrm -\pi < x < \pi \text k \in \mathbb. In this case, the Fourier coefficients are given by :\begin a_n & = \frac\int_^s(x) \cos(nx)\,dx = 0, \quad n \ge 0. \\
pt
pt
b_n & = \frac\int_^s(x) \sin(nx)\, dx\\
pt
pt
&= -\frac\cos(n\pi) + \frac\sin(n\pi)\\
pt
pt
&= \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=\pi, the Fourier series converges to 0, which is the half-sum of the left- and right-limit of ''s'' at x=\pi. This is a particular instance of the Dirichlet theorem for Fourier series. This example leads to a solution of the
Basel 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

In
engineering 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 ...

engineering
applications, the Fourier series is generally presumed to converge almost everywhere (the exceptions being at discrete discontinuities) since the functions encountered in engineering are better-behaved than the functions that mathematicians can provide as counter-examples to this presumption. In particular, if 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 _0,x_0+P/math>, then the Fourier series converges to the function at almost every point. Convergence of Fourier series also depends on the finite number of maxima and minima in a function which is popularly known as one of the Dirichlet's condition for Fourier series. See
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. 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.


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_ = \frac\int_P \operatorname\\cdot e^\ dx     and     c_ = \frac\int_P \operatorname\\cdot e^\ dx : s_(x) = \sum_^N c_\cdot e^ + i\cdot \sum_^N c_\cdot e^ =\sum_^N \left(c_+i\cdot c_\right) \cdot e^. Defining c_n \triangleq c_+i\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: : \begin c_n &= \frac\int_ \operatorname\\cdot e^\ dx + i\cdot \frac \int_ \operatorname\\cdot e^\ dx\\
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&= \frac \int_ \left(\operatorname\ +i\cdot \operatorname\\right)\cdot e^\ dx \ = \ \frac\int_ s(x)\cdot e^\ dx. \end


Other 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 \hat /math> or S /math>, and functional notation often replaces subscripting: :\begin s_\infty(x) &= \sum_^\infty \hat cdot e^ \\
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pt
&= \sum_^\infty S cdot e^ && \scriptstyle \mathsf \end In engineering, particularly when the variable x represents time, the coefficient sequence is called a
frequency 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 ...

Dirac comb
: :S(f) \ \triangleq \ \sum_^\infty S 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 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 ...

hertz
. 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 ...

harmonics
) 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 ...
: :\begin \mathcal^\ &= \int_^\infty \left( \sum_^\infty S cdot \delta \left(f-\frac\right)\right) e^\,df, \\
pt
pt
&= \sum_^\infty S cdot \int_^\infty \delta\left(f-\frac\right) e^\,df, \\
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pt
&= \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 of
Jean-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 ) ...

Leonhard Euler
,
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). ...

sine
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 ...

cosine
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 ...

integral
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 ...

Peter Gustav Lejeune Dirichlet
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 ...

sinusoid
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 ...

electrical engineering
,
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 ...

vibration
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 ...

optics
,
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 ...

signal processing
,
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 ...

econometrics
, shell theory, etc.


Beginnings

Joseph Fourier wrote: This immediately gives any coefficient ''ak'' 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 \begin a_k&=\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''th term. In these few lines, which are close to the modern
formalism 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 ) ...

Euler
,
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 ...

d'Alembert
,
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 ...

Gauss
, Fourier believed that such trigonometric series could represent any arbitrary function. In what sense that is actually true is a somewhat subtle issue and the attempts over many years to clarify this idea have led to important discoveries in the theories of
convergence 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 ...

Lagrange
,
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 ...

Laplace
,
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/ \pi, so it is not immediately apparent why one would need the Fourier series. While there are many applications, Fourier's motivation was in solving the
heat equation 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 \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 period of time has elapsed) is given by : T(x,y) = 2\sum_^\infty \frac \sin(nx) . Here, sinh is the
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 \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 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 the
Basel 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 \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 \in ,P. * F G /math> designate the Fourier series coefficients (exponential form) of f and g as defined in equation .


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: : \begin \text & f & = & f_ & + & f_ & + & i f_ & + &\underbrace \\ &\Bigg\Updownarrow\mathcal & &\Bigg\Updownarrow\mathcal & &\ \ \Bigg\Updownarrow\mathcal & &\ \ \Bigg\Updownarrow\mathcal & &\ \ \Bigg\Updownarrow\mathcal\\ \text & F & = & F_ & + & \overbrace & + &i\ F_ & + & F_ \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 is
integrableIn 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 ...
, \lim_ F 0, \lim_ a_n=0 and \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 ...
.


Parseval's theorem

If f belongs to L^2(P) (an interval of length P) then: \sum_^\infty \Bigl, F Bigr, ^2 = \frac\int_ , f(x), ^2 \, dx.


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,\, c_,\, c_, \ldots are coefficients and \sum_^\infty , c_n, ^2 < \infty then there is a unique function f\in L^2(P) such that F = c_n for every n.


Convolution theorems

Given P-periodic functions, f_ and g_ with Fourier series coefficients F /math> and G n \in \mathbb, *The pointwise product: h_(x) \triangleq f_(x)\cdot g_(x) is also P-periodic, and its Fourier series coefficients are given by the discrete convolution of the F and G sequences: H = \ *The periodic convolution: h_(x) \triangleq \int_ f_(\tau)\cdot g_(x-\tau)\, d\tau is also P-periodic, with Fourier series coefficients: H = P \cdot F cdot G *A
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 \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

We say that f belongs to C^k(\mathbb) if f is a 2-periodic function on \mathbb which is k times differentiable, and its ''k''th derivative is continuous. * If f \in C^1(\mathbb), then the Fourier coefficients \widehat /math> of the derivative f' can be expressed in terms of the Fourier coefficients \widehat /math> of the function f, via the formula \widehat = in \widehat /math>. * If f \in C^k(\mathbb), then \widehat = (in)^k \widehat /math>. In particular, since for a fixed k\geq 1 we have \widehat to 0 as n\to\infty, it follows that , n, ^k\widehat /math> tends to zero, which means that the Fourier coefficients converge to zero faster than the ''k''th power of ''n'' for any k\geq 1.


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 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''2(''G''), where ''G'' is a compact group, in such a way that the Fourier transform carries
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 a
compact 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 \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 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 ...

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) 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 \pi,\pi/math> case, but if G is noncompact, one obtains instead a
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 \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 \pi,\pitimes \pi,\pi/math>: \begin f(x,y) & = \sum_ c_e^e^,\\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 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 ...

jpeg
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-dimensional
Bravais 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: \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 vectors. Assuming we have some function, f(\mathbf), such that it obeys the condition of periodicity for any Bravais lattice vector \mathbf, f(\mathbf) = f(\mathbf+\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 \mathbf in the coordinate-system of the lattice: \mathbf = x_1\frac+ x_2\frac+ x_3\frac, where a_i \triangleq , \mathbf_i, , meaning that a_i is defined to be the magnitude of \mathbf_i, so \hat = \frac is the unit vector directed along \mathbf_i. Thus we can define a new function, g(x_1,x_2,x_3) \triangleq f(\mathbf) = f \left (x_1\frac+x_2\frac+x_3\frac \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 \left 0, a_1\right /math> for x_1, we can define the following: h^\mathrm(m_1, x_2, x_3) \triangleq \frac\int_0^ g(x_1, x_2, x_3)\cdot e^\, dx_1 And then we can write: g(x_1, x_2, x_3)=\sum_^\infty h^\mathrm(m_1, x_2, x_3) \cdot e^ Further defining: \begin h^\mathrm(m_1, m_2, x_3) & \triangleq \frac\int_0^ h^\mathrm(m_1, x_2, x_3)\cdot e^\, dx_2 \\ 2pt& = \frac\int_0^ dx_2 \frac\int_0^ dx_1 g(x_1, x_2, x_3)\cdot e^ \end We can write g once again as: g(x_1, x_2, x_3)=\sum_^\infty \sum_^\infty h^\mathrm(m_1, m_2, x_3) \cdot e^ \cdot e^ Finally applying the same for the third coordinate, we define: \begin h^\mathrm(m_1, m_2, m_3) & \triangleq \frac\int_0^ h^\mathrm(m_1, m_2, x_3)\cdot e^\, dx_3 \\ 2pt& = \frac\int_0^ dx_3 \frac\int_0^ dx_2 \frac\int_0^ dx_1 g(x_1, x_2, x_3)\cdot e^ \end We write g as: g(x_1, x_2, x_3)=\sum_^\infty \sum_^\infty \sum_^\infty h^\mathrm(m_1, m_2, m_3) \cdot e^ \cdot e^\cdot e^ Re-arranging: g(x_1, x_2, x_3)=\sum_ h^\mathrm(m_1, m_2, m_3) \cdot e^. Now, every ''reciprocal'' lattice vector can be written (but does not mean that it is the only way of writing) as \mathbf = m_1\mathbf_1 + m_2\mathbf_2 + m_3\mathbf_3, where m_i are integers and \mathbf_i are reciprocal lattice vectors to satisfy \mathbf \cdot \mathbf=2\pi\delta_ (\delta_ = 1 for i = j, and \delta_ = 0 for i \neq j). Then for any arbitrary reciprocal lattice vector \mathbf and arbitrary position vector \mathbf in the original Bravais lattice space, their scalar product is: \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 ). So it is clear that in our expansion of g(x_1,x_2,x_3) = f(\mathbf), the sum is actually over reciprocal lattice vectors: f(\mathbf)=\sum_ h(\mathbf) \cdot e^, where h(\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)\cdot e^. Assuming \mathbf = (x,y,z) = x_1\frac+x_2\frac+x_3\frac, we can solve this system of three linear equations for x, y, and z in terms of x_1, x_2 and x_3 in order to calculate the volume element in the original cartesian coordinate system. Once we have x, y, and z in terms of x_1, x_2 and x_3, we can calculate the
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 ...
: \begin \dfrac & \dfrac & \dfrac \\ 2pt\dfrac & \dfrac & \dfrac \\ 2pt\dfrac & \dfrac & \dfrac \end which after some calculation and applying some non-trivial cross-product identities can be shown to be equal to: \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 \mathbf_1 is parallel to the ''x'' axis, \mathbf_2 lies in the ''xy''-plane, and \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 \mathbf_1, \mathbf_2 and \mathbf_3. In particular, we now know that dx_1 \, dx_2 \, dx_3 = \frac \cdot dx \, dy \, dz. We can write now h(\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(\mathbf) = \frac\int_ d\mathbf f(\mathbf)\cdot e^ writing d\mathbf for the volume element dx \, dy \, dz; and where C is the primitive unit cell, thus, \mathbf_1\cdot(\mathbf_2 \times \mathbf_3) is the volume of the primitive unit cell.


Hilbert space interpretation

In the language of
Hilbert 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 \left\ 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( \pi,\pi of square-integrable functions on \pi,\pi/math>. This space is actually a Hilbert space with an
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: :\langle f,\, g \rangle \;\triangleq \; \frac\int_^ f(x)g^*(x)\,dx,     where g^(x) is the complex conjugate of g(x). The basic Fourier series result for Hilbert spaces can be written as :f=\sum_^\infty \langle f,e_n \rangle \, 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 ...
: \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
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 \int_^ \cos(mx)\, \sin(nx)\, dx = \frac\int_^ \sin((n+m)x)+\sin((n-m)x)\, dx = 0; furthermore, the sines and cosines are orthogonal to the constant function 1. An ''orthonormal basis'' for L^2( \pi,\pi consisting of real functions is formed by the functions 1 and \sqrt \cos (nx), \sqrt \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) = \sum_^N S e^, it is a
trigonometric 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)=\sum_^N p e^.


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\cdot n) S /math>  is the ''n''th Fourier coefficient of the derivative s'. It follows, essentially from the
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 /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 ''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 \alpha > 1/2. In the absolutely summable case, the inequality: :\sup_x , s(x) - s_(x), \le \sum_ , S   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. The
uniform 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 ...

Gibbs phenomenon
*
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 ...

Laurent series
– the substitution ''q'' = ''e''''ix'' transforms a Fourier series into a Laurent series, or conversely. This is used in the ''q''-series expansion of the ''j''-invariant. *
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