Fourier series

Fourier coefficientFourier expansionFourier coefficientsFourier modesFourierFourier decompositioncosine seriesFourier theoremFourier's theoremHilbert space
In mathematics, a Fourier series is a periodic function composed of harmonically related sinusoids, combined by a weighted summation.wikipedia
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Fourier analysis

FourierFourier synthesisanalyse the output wave into its constituent harmonics
The process of deriving the weights that describe a given function is a form of Fourier analysis.
Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.

Joseph Fourier

FourierJean Baptiste Joseph FourierJean-Baptiste Joseph Fourier
The Fourier series is named in honour of Jean-Baptiste Joseph Fourier (1768–1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli.
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 harmonic analysis, and their applications to problems of heat transfer and vibrations.

Discrete-time Fourier transform

convolution theoremDFTDTFT § Properties
The discrete-time Fourier transform is an example of Fourier series.
, for all integers n, is a Fourier series, which produces a periodic function of a frequency variable.

Peter Gustav Lejeune Dirichlet

DirichletLejeune DirichletGustav Dirichlet
Later, Peter Gustav Lejeune Dirichlet and Bernhard Riemann expressed Fourier's results with greater precision and formality.
Johann Peter Gustav Lejeune Dirichlet (13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a function.

Bernhard Riemann

RiemannGeorg Friedrich Bernhard RiemannRiemann, Bernhard
Later, Peter Gustav Lejeune Dirichlet and Bernhard Riemann expressed Fourier's results with greater precision and formality.
In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series.

Series (mathematics)

infinite seriesseriespartial sum
In mathematics, a Fourier series is a periodic function composed of harmonically related sinusoids, combined by a weighted summation.
:The most important example of a trigonometric series is the Fourier series of a function.

Periodic function

periodicperiodperiodicity
In mathematics, a Fourier series is a periodic function composed of harmonically related sinusoids, combined by a weighted summation.
The subject of Fourier series investigates the idea that an 'arbitrary' periodic function is a sum of trigonometric functions with matching periods.

Sine

sine functionsinnatural sines
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.
The Fourier series for this correction

Convergence of Fourier series

classic harmonic analysisdivergesmore about absolute convergence of Fourier series
See Convergence of Fourier series.
In mathematics, the question of whether the Fourier series of a periodic function converges to the given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics.

Eigenfunction

eigenfunctionseigenfunction expansioneigensolutions
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.
:for example through a Fourier expansion of f(t).

List of important publications in mathematics

Publications in topologyList of publications in mathematicsMémoire sur la propagation de la chaleur dans les corps solides
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.
Introduced Fourier analysis, specifically Fourier series.

Partial differential equation

partial differential equationsPDEPDEs
The heat equation is a partial differential equation.
If the domain is finite or periodic, an infinite sum of solutions such as a Fourier series is appropriate, but an integral of solutions such as a Fourier integral is generally required for infinite domains.

Harmonic analysis

abstract harmonic analysisFourier theoryharmonic
In what sense that is actually true is a somewhat subtle issue and the attempts over many years to clarify this idea have led to important discoveries in the theories of convergence, function spaces, and harmonic analysis.
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis).

Trigonometric functions

cosinetrigonometric functiontangent
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.
can be expressed as a sum of sine waves or cosine waves in a Fourier series.

Parseval's theorem

Another application of this Fourier series is to solve the Basel problem by using Parseval's theorem.
It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later applied to the Fourier series.

Carleson's theorem

Lusin's conjectureCarleson theoremCarleson-Hunt theorem
If a function is square-integrable on the interval [x_0,x_0+P], then the Fourier series converges to the function at almost every point.
Carleson's theorem is a fundamental result in mathematical analysis establishing the pointwise (Lebesgue) almost everywhere convergence of Fourier series of L 2 functions, proved by.

Dirac comb

Sampling functionimpulse traininfinite impulse train
Another commonly used frequency domain representation uses the Fourier series coefficients to modulate a Dirac comb:
Because the Dirac comb function is periodic, it can be represented as a Fourier series:

Trigonometric series

Trigonometrical seriestrigonometric
The Fourier series is named in honour of Jean-Baptiste Joseph Fourier (1768–1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli.
It is called a Fourier series if the terms A_{n} and B_{n} have the form:

Deferent and epicycle

epicyclesdeferentdeferents and epicycles
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.
This is because epicycles can be represented as a complex Fourier series; so, with a large number of epicycles, very complicated paths can be represented in the complex plane.

Fundamental frequency

fundamentalfundamental tonefundamental frequencies
The "teeth" of the comb are spaced at multiples (i.e. harmonics) of 1/P, which is called the fundamental frequency.
Waveforms can be represented by Fourier series.

Dirichlet conditions

conditionsDirichlet theoremDirichlet's condition for Fourier series
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.
In mathematics, the Dirichlet conditions are sufficient conditions for a real-valued, periodic function f to be equal to the sum of its Fourier series at each point where f is continuous.

Even and odd functions

even functionodd functioneven
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.
They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series.

List of Fourier-related transforms

Fourier-related transformFourier-related transforms
Many other Fourier-related transforms have since been defined, extending the initial idea to other applications.

Spherical harmonics

spherical harmonicspherical functionsLaplace series
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.
The spherical harmonics are a complete set of orthogonal functions on the sphere, and thus may be used to represent functions defined on the surface of a sphere, just as circular functions (sines and cosines) are used to represent functions on a circle via Fourier series.

Frequency domain

frequency-domainFourier spaceFourier domain
In engineering, particularly when the variable x represents time, the coefficient sequence is called a frequency domain representation.