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A Fourier series () is a summation of
harmonically In music, harmony is the process by which individual sounds are joined together or composed into whole units or compositions. Often, the term harmony refers to simultaneously occurring frequencies, pitches ( tones, notes), or chords. However ...
related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''period''), the number of components, and their amplitudes and phase parameters. With appropriate choices, 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). The number of components is theoretically infinite, in which case the other parameters can be chosen to cause the series to converge to almost any ''well behaved'' periodic function (see
Pathological Pathology is the study of the causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when used in th ...
and Dirichlet–Jordan test). The components of a particular function are determined by ''analysis'' techniques described in this article. Sometimes the components are known first, and the unknown function is ''synthesized'' by a Fourier series. Such is the case of a
discrete-time Fourier transform In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term ''discrete-time'' refers to the ...
.
Convergence of Fourier series In mathematics, the question of whether the Fourier series of a periodic function converges to a given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. Convergence is not necessarily given in t ...
means that as more and more components from the series are summed, each successive ''partial Fourier series'' sum will better
approximate An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix '' ...
the function, and will equal the function with a potentially infinite number of components. The
mathematical proof A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proo ...
s for this may be collectively referred to as the ''Fourier Theorem'' (see ). The figures below illustrate some partial Fourier series results for the components of a square wave. File:SquareWaveFourierArrows,rotated,nocaption 20fps.gif, A square wave (represented as the blue dot) is approximated by its sixth partial sum (represented as the purple dot), formed by summing the first six terms (represented as arrows) of the square wave's Fourier series. Each arrow starts at the vertical sum of all the arrows to its left (i.e. the previous partial sum). File:Fourier Series.svg, The first four partial sums of the Fourier series for a square wave. As more harmonics are added, the partial sums ''converge to'' (become more and more like) the square wave. File:Fourier series and transform.gif, Function s_6(x) (in red) is a Fourier series sum of 6 harmonically related sine waves (in blue). Its Fourier transform S(f) is a frequency-domain representation that reveals the amplitudes of the summed sine waves. Another analysis technique (not covered here), suitable for both periodic and non-periodic functions, is the Fourier transform, which provides a frequency-continuum of component information. But when applied to a periodic function all components have zero amplitude, except at the harmonic frequencies. The inverse Fourier transform is a synthesis process (like the Fourier series), which converts the component information (often known as the
frequency domain In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a si ...
representation) back into its time domain representation. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
-valued functions of real arguments, and used the sine and cosine functions as the basis set for the decomposition. Many other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis.


Analysis process

This section describes the analysis process that derives the parameters of a Fourier series that approximates a known function, s(x).  An example of synthesizing an unknown function from known parameters is
discrete-time Fourier transform In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term ''discrete-time'' refers to the ...
.


Common forms

The Fourier series can be represented in different forms. The ''amplitude-phase'' form, ''sine-cosine'' form, and ''exponential'' form are commonly used and are expressed here for a
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
-valued
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
s(x). (See and for alternative forms). The number of terms summed, N, is a potentially infinite integer. 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 In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved. T ...
functions typical of physical processes, equality is customarily assumed, and the Dirichlet conditions provide sufficient conditions. The integer index, n, is also the number of cycles the n^ harmonic makes in the function's period P. Therefore: * The n^ harmonic's wavelength is \tfrac and in units of x. * The n^ harmonic's frequency is \tfrac and in reciprocal units of x.


Amplitude-phase form

The Fourier series in
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of ampl ...
- phase form is: *Its n^ harmonic is A_n \cdot \cos\left(\tfrac n x - \varphi_n\right). *A_n is the n^ harmonic's
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of ampl ...
and \varphi_n is its
phase shift In physics and mathematics, the phase of a periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is denoted \phi(t) and expressed in such a scale that it ...
. *The
fundamental frequency The fundamental frequency, often referred to simply as the ''fundamental'', is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. I ...
of s_(x) is the term for when n equals 1, and can be referred to as the 1^ harmonic. *\tfrac is sometimes called the 0^ harmonic or DC component. It is the mean value of s(x). 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). The coefficients A_n and \varphi_n can be understood and derived in terms of the cross-correlation between s(x) and a sinusoid at frequency \tfrac. For a general frequency f, and an analysis interval _0,x_0+P the cross-correlation function: is essentially a matched filter, with ''template'' \cos(2\pi f x). The maximum of \Chi_f(\tau) is a measure of the amplitude (A) of frequency f in the function s(x), and the value of \tau at the maximum determines the phase (\varphi) of that frequency. Figure 2 is an example, where s(x) is a square wave (not shown), and frequency f is the 4^ harmonic. It is also an example of deriving the maximum from just two samples, instead of searching the entire function. That is made possible by a trigonometric identity: Combining this with gives: \begin \Chi_n(\varphi) &= \tfrac \int_P s(x) \cdot \cos\left(\tfrac n x-\varphi \right)\, dx ; \quad \varphi \in , 2\pi\ &=\cos(\varphi)\cdot \underbrace_ +\sin(\varphi)\cdot \underbrace_\\ &=\cos(\varphi)\cdot a_n + \sin(\varphi)\cdot b_n \end which introduces the definitions of a_n and b_n.  And we note for later reference that a_0 and b_0 can be simplified: a_0 = \tfrac \int_P s(x) dx ~ ,\quad b_0 = 0 ~ . The derivative of \Chi_n(\varphi) is zero at the phase of maximum correlation.\Chi'_n(\varphi_n)=\sin(\varphi_n)\cdot a_n - \cos(\varphi_n)\cdot b_n = 0 \quad \longrightarrow\quad \tan(\varphi_n) = \frac \quad \longrightarrow\quad \varphi_n = \arctan(b_n, a_n) And the correlation peak value is: : \begin A_n \triangleq \Chi_n(\varphi_n)\ &= \cos(\varphi_n)\cdot a_n + \sin(\varphi_n)\cdot b_n \\ &=\frac\cdot a_n + \frac\cdot b_n =\frac &= \sqrt. \end Therefore a_n and b_n are the
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
of a vector with
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
A_n and \varphi_n.


Sine-cosine form

Substituting into gives: : In terms of the readily computed quantities, a_n and b_n, recall that: :\cos(\varphi_n) = a_n/A_n :\sin(\varphi_n) = b_n/A_n :A_0 = \sqrt = \sqrt = a_0 Therefore an alternative form of the Fourier series, using the Cartesian coordinates, is the sine-cosine form:


Exponential form

Another applicable identity is
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for a ...
: : \begin \cos\left( \tfrac P nx - \varphi_n \right) &\equiv \tfrac e^ + \tfrac e^ \\ pt& = \left(\tfrac e^\right) \cdot e^ +\left(\tfrac e^\right)^* \cdot e^ \end (Note: the ∗ denotes
complex conjugation In mathematics, the complex conjugate of a complex number is the number with an equal real part and an Imaginary number, imaginary part equal in magnitude but opposite in Sign (mathematics), sign. That is, (if a and b are real, then) the complex ...
.) Therefore, with definitions: :c_n \triangleq \left\ = \tfrac \int_P s(x) \cdot e^ \, dx, the final result is: This is the customary form for generalizing to . Negative values of n correspond to negative frequency (explained in ).


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. \\ ptb_n & = \frac\int_^s(x) \sin(nx)\, dx\\ pt&= -\frac\cos(n\pi) + \frac\sin(n\pi)\\ 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 with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 ...
.


Convergence

A proof that a Fourier series is a valid representation of any periodic function (that satisfies the Dirichlet conditions) is overviewed in . In engineering applications, the Fourier series is generally presumed to converge almost everywhere (the exceptions being at discrete discontinuities) since the functions encountered in engineering are better-behaved than the functions that mathematicians can provide as counter-examples to this presumption. In particular, if s is continuous and the derivative of s(x) (which may not exist everywhere) is square integrable, then the Fourier series of s converges absolutely and uniformly to s(x). If a function is square-integrable on the interval _0,x_0+P/math>, then the Fourier series converges to the function at almost every point. It is possible to define Fourier coefficients for more general functions or distributions, in such cases convergence in norm or weak convergence is usually of interest. 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 &= \tfrac\int_ \operatorname\\cdot e^\ dx + i\cdot \tfrac \int_ \operatorname\\cdot e^\ dx\\ pt&= \tfrac \int_ \left(\operatorname\ +i\cdot \operatorname\\right)\cdot e^\ dx \ = \ \tfrac\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^ \\ 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 signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a si ...
representation. Square brackets are often used to emphasize that the domain of this function is a discrete set of frequencies. Another commonly used frequency domain representation uses the Fourier series coefficients to modulate a
Dirac comb In mathematics, a Dirac comb (also known as shah function, impulse train or sampling function) is a periodic function with the formula \operatorname_(t) \ := \sum_^ \delta(t - k T) for some given period T. Here ''t'' is a real variable and t ...
: :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 "teeth" of the comb are spaced at multiples (i.e. harmonics) of \tfrac, which is called the
fundamental frequency The fundamental frequency, often referred to simply as the ''fundamental'', is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. I ...
. s_(x) can be recovered from this representation by an inverse Fourier transform: :\begin \mathcal^\ &= \int_^\infty \left( \sum_^\infty S cdot \delta \left(f-\frac\right)\right) e^\,df, \\ pt&= \sum_^\infty S cdot \int_^\infty \delta\left(f-\frac\right) e^\,df, \\ 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 and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and har ...
(1768–1830), who made important contributions to the study of
trigonometric series In mathematics, a trigonometric series is a infinite series of the form : \frac+\displaystyle\sum_^(A_ \cos + B_ \sin), an infinite version of a trigonometric polynomial. It is called the Fourier series of the integrable function f if the ter ...
, after preliminary investigations by
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
, Jean le Rond d'Alembert, and Daniel Bernoulli. Fourier introduced the series for the purpose of solving the heat equation 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, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. ...
-smooth) function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier in 1807, before the French Academy. Early ideas of decomposing a periodic function into the sum of simple oscillating functions date back to the 3rd century BC, when ancient astronomers proposed an empiric model of planetary motions, based on deferents and epicycles. The heat equation is a
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
. Prior to Fourier's work, no solution to the heat equation was known in the general case, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a sine or cosine 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) 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 type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
and integral in the early nineteenth century. Later, Peter Gustav Lejeune Dirichlet and
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
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, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ...
s. The Fourier series has many such applications in electrical engineering, vibration analysis,
acoustics Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
, optics,
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, d ...
,
image processing An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...
, quantum mechanics,
econometrics Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8� ...
, shell theory, etc.


Beginnings

Joseph Fourier wrote: This immediately gives any coefficient ''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^ 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) * Scien ...
used in Fourier series, Fourier revolutionized both mathematics and physics. Although similar trigonometric series were previously used by Euler,
d'Alembert Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the ''Encyclopédie ...
, Daniel Bernoulli and
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
, Fourier believed that such trigonometric series could represent any arbitrary function. In what sense that is actually true is a somewhat subtle issue and the attempts over many years to clarify this idea have led to important discoveries in the theories of
convergence Convergence may refer to: Arts and media Literature *''Convergence'' (book series), edited by Ruth Nanda Anshen * "Convergence" (comics), two separate story lines published by DC Comics: **A four-part crossover storyline that united the four Wei ...
, function spaces, and harmonic analysis. When Fourier submitted a later competition essay in 1811, the committee (which included Lagrange, Laplace,
Malus ''Malus'' ( or ) is a genus of about 30–55 species of small deciduous trees or shrubs in the family Rosaceae, including the domesticated orchard apple, crab apples, wild apples, and rainberries. The genus is native to the temperate zone of th ...
and Legendre, among others) concluded: ''...the manner in which the author arrives at these equations is not exempt of difficulties and...his analysis to integrate them still leaves something to be desired on the score of generality and even rigour''.


Fourier's motivation

The Fourier series expansion of the sawtooth function (above) looks more complicated than the simple formula s(x)=\tfrac, so it is not immediately apparent why one would need the Fourier series. While there are many applications, Fourier's motivation was in solving the heat equation. 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 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, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th r ...
. 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

The
discrete-time Fourier transform In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term ''discrete-time'' refers to the ...
is an example of a Fourier series. Another application is to solve the
Basel problem The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 ...
by using Parseval's theorem. The example generalizes and one may compute ζ(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. *s(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 s(x).


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, the complex conjugate of a complex number is the number with an equal real part and an Imaginary number, imaginary part equal in magnitude but opposite in Sign (mathematics), sign. That is, (if a and b are real, then) the complex ...
is denoted by an asterisk. *s(x),r(x) designate P-periodic functions or functions defined only for x \in ,P * S R /math> designate the Fourier series coefficients (exponential form) of s and r.


Symmetry properties

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 & s & = & s_ & + & s_ & + & i s_ & + & \underbrace \\ &\Bigg\Updownarrow\mathcal & &\Bigg\Updownarrow\mathcal & &\ \ \Bigg\Updownarrow\mathcal & &\ \ \Bigg\Updownarrow\mathcal & &\ \ \Bigg\Updownarrow\mathcal\\ \text & S & = & S_\text & + & \overbrace & + & i S_\text & + & S_\text \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.


Other properties


Riemann–Lebesgue lemma

If S is
integrable In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
, \lim_ S 0, \lim_ a_n=0 and \lim_ b_n=0. This result is known as the Riemann–Lebesgue lemma.


Parseval's theorem

If s belongs to L^2(P) (periodic over an interval of length P) then: \frac\int_ , s(x), ^2 \, dx = \sum_^\infty \Bigl, S Bigr, ^2


Hesham's identity

If s belongs to L^4(P) (periodic over an interval of length P), and S /math> is of a finite-length M then: for S \in \C, then \frac\int_ , s(x), ^4 \, dx = \sum_^ S \sum_^ S^* \Bigg \underset_S^*[mS[m-(k-l).html"_;"title=".html"_;"title="\underset_S^*[m">\underset_S^*[mS[m-(k-l)">.html"_;"title="\underset_S^*[m">\underset_S^*[mS[m-(k-l)+_\underset_S^*[m-(l-k).html" ;"title="">\underset_S^*[mS[m-(k-l).html" ;"title=".html" ;"title="\underset S^*[m">\underset S^*[mS[m-(k-l)">.html" ;"title="\underset S^*[m">\underset S^*[mS[m-(k-l)+ \underset S^*[m-(l-k)">">\underset_S^*[mS[m-(k-l).html" ;"title=".html" ;"title="\underset S^*[m">\underset S^*[mS[m-(k-l)">.html" ;"title="\underset S^*[m">\underset S^*[mS[m-(k-l)+ \underset S^*[m-(l-k)S \Bigg] and for S \in \R, then \frac\int_ , s(x), ^4 \, dx = \sum_^ S \sum_^ S \sum_^ S S k-l,


Plancherel's theorem

If c_0,\, c_,\, c_, \ldots are coefficients and \sum_^\infty , c_n, ^2 < \infty then there is a unique function s\in L^2(P) such that S = c_n for every n.


Convolution theorems

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


Derivative property

We say that s belongs to C^k(\mathbb) if s is a 2-periodic function on \mathbb which is k times differentiable, and its k^ derivative is continuous. * If s \in C^1(\mathbb), then the Fourier coefficients \widehat /math> of the derivative s' can be expressed in terms of the Fourier coefficients \widehat /math> of the function s, via the formula \widehat = in \widehat /math>. * If s \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 In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural ge ...
. Typical examples include those classical groups that are compact. This generalizes the Fourier transform to all spaces of the form ''L''2(''G''), where ''G'' is a compact group, in such a way that the Fourier transform carries
convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
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 In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Pete ...
, 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 or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
Riemannian manifold, 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 and pseudo-Riemannian manifolds. It is named af ...
. The Laplace–Beltrami operator is the differential operator that corresponds to Laplace operator for the Riemannian manifold X. Then, by analogy, one can consider heat equations on X. Since Fourier arrived at his basis by attempting to solve the heat equation, the natural generalization is to use the eigensolutions of the Laplace–Beltrami operator as a basis. This generalizes Fourier series to spaces of the type L^2(X), where X is a Riemannian manifold. The Fourier series converges in ways similar to the \pi,\pi/math> case. A typical example is to take X to be the sphere with the usual metric, in which case the Fourier basis consists of spherical harmonics.


Locally compact Abelian groups

The generalization to compact groups discussed above does not generalize to noncompact, nonabelian groups. However, there is a straightforward generalization to Locally Compact Abelian (LCA) groups. This generalizes the Fourier transform to L^1(G) 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. This generalization yields the usual Fourier transform 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^,\\ ptc_ & = \frac \int_^\pi \int_^\pi f(x,y) e^e^\, dx \, dy. \end Aside from being useful for solving partial differential equations such as the heat equation, one notable application of Fourier series on the square is in image compression. In particular, the jpeg image compression standard uses the two-dimensional discrete cosine transform, a discrete form of the Fourier cosine transform, which uses only cosine as the basis function. For two-dimensional arrays with a staggered appearance, half of the Fourier series coefficients disappear, due to additional symmetry.


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 discrete translation operations described in three dimensional space by : \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
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 In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who di ...
. 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 \\
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& = \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 PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
& = \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: \begin \dfrac & \dfrac & \dfrac \\
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\dfrac & \dfrac & \dfrac \\
2pt PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
\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 spaces, the set of functions \left\ is an orthonormal basis 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 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: \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, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
), 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 theorem, but follows also from the properties of classical kernels like the
Fejér kernel In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity. It is named after the Hungarian mathematician Lipót ...
.


Fourier theorem proving convergence of Fourier series

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''. The earlier s_(x) = \sum_^N S e^, is a
trigonometric polynomial In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more natural numbers. The c ...
of degree N that can be generally expressed as: p_(x)=\sum_^N p e^.


Least squares property

Parseval's theorem implies that:


Convergence theorems

Because of the least squares property, and because of the completeness of the Fourier basis, we obtain an elementary convergence result. We have already mentioned that if s is continuously differentiable, then (i\cdot n) S /math> is the n^ Fourier coefficient of the derivative s'. It follows, essentially from the
Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality fo ...
, 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 ...
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 series In mathematics, the question of whether the Fourier series of a periodic function converges to a given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. Convergence is not necessarily given in t ...
are known, ranging from the moderately simple result that the series converges at x if s is differentiable at x, to Lennart Carleson's much more sophisticated result that the Fourier series of an L^2 function actually converges
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
.


Divergence

Since Fourier series have such good convergence properties, many are often surprised by some of the negative results. For example, the Fourier series of a continuous ''T''-periodic function need not converge pointwise. The uniform boundedness principle yields a simple non-constructive proof of this fact. In 1922,
Andrey Kolmogorov 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 Sovi ...
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 In mathematics, the ATS theorem is the theorem on the approximation of a trigonometric sum by a shorter one. The application of the ATS theorem in certain problems of mathematical and theoretical physics can be very helpful. History of the proble ...
*
Dirichlet kernel In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of periodic functions defined as D_n(x)= \sum_^n e^ = \left(1+2\sum_^n\cos(kx)\right)=\frac, where is any nonn ...
*
Discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comp ...
* Fast Fourier transform *
Fejér's theorem In mathematics, Fejér's theorem,Leopold FejérUntersuchungen über Fouriersche Reihen ''Mathematische Annalen''vol. 58 1904, 51-69. named after Hungarian mathematician Lipót Fejér, states the following: Explanation of Fejér's Theorem's Exp ...
* Fourier analysis *
Fourier sine and cosine series In mathematics, particularly the field of calculus and Fourier analysis, the Fourier sine and cosine series are two mathematical series named after Joseph Fourier. Notation In this article, denotes a real valued function on \mathbb which is peri ...
* Fourier transform * Gibbs phenomenon * Half range Fourier series * 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, based on a least squares fit of sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the most used spectral method in science, generally ...
* Multidimensional transform *
Spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result ...
* Sturm–Liouville theory * Residue theorem integrals of ''f''(''z''), 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