# 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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### Uses of trigonometry

The mathematical topics of Fourier series and Fourier transforms rely heavily on knowledge of trigonometric functions and find application in a number of areas, including statistics.

### Group delay and phase delay

group delayphase delaydelay distortion
In signal processing, group delay is the time delay of the amplitude envelopes of the various sinusoidal components of a signal through a device under test, and is a function of frequency for each component.

### Generalized Fourier series

Fourier coefficientsFourier expansiongeneralized
In mathematical analysis, many generalizations of Fourier series have proved to be useful.

Ganesh Prasad (15 November 1876 – 9 March 1935) was an Indian mathematician who specialised in the theory of potentials, theory of functions of a real variable, Fourier series and the theory of surfaces.

### Voronoi formula

In mathematics, a Voronoi formula is an equality involving Fourier coefficients of automorphic forms, with the coefficients twisted by additive characters on either side.

### Karhunen–Loève theorem

Karhunen–Loève transformKarhunen–LoèveKarhunen–Loève expansion
In the theory of stochastic processes, the Karhunen–Loève theorem (named after Kari Karhunen and Michel Loève), also known as the Kosambi–Karhunen–Loève theorem is a representation of a stochastic process as an infinite linear combination of orthogonal functions, analogous to a Fourier series representation of a function on a bounded interval.

### Vector space

vectorvector spacesvectors
They offer a framework for Fourier expansion, which is employed in image compression routines, and they provide an environment that can be used for solution techniques for partial differential equations.

### Sidon sequence

Sidon setB_h[g]-sequencesSidon set problem
Sidon introduced the concept in his investigations of Fourier series.

### Integral transform

integral operatorkernelintegral kernel
The precursor of the transforms were the Fourier series to express functions in finite intervals.

### Hilda Geiringer

Geiringer, HildaHilda Pollaczek-Geiringer
She received her Ph.D. from the University of Vienna in 1917 under the guidance of Wilhelm Wirtinger with a thesis entitled "Trigonometrische Doppelreihen" about Fourier series in two variables.

### Total variation

total variation normvariationTonelli plane variation
He used the new concept in order to prove a convergence theorem for Fourier series of discontinuous periodic functions whose variation is bounded.

### Monstrous moonshine

McKay–Thompson seriesmoonshine theoryConway–Norton conjecture
In 1978, John McKay found that the first few terms in the Fourier expansion of the normalized J-invariant,

### Regressive discrete Fourier series

In applied mathematics, the regressive discrete Fourier series (RDFS) is a generalization of the discrete Fourier transform where the Fourier series coefficients are computed in a least squares sense and the period is arbitrary, i.e., not necessarily equal to the length of the data.

Clenshaw–Curtis integration
Equivalently, they employ a change of variables and use a discrete cosine transform (DCT) approximation for the cosine series.

### Cusp form

cuspcusp formscusps
A cusp form is distinguished in the case of modular forms for the modular group by the vanishing of the constant coefficient a 0 in the Fourier series expansion (see q-expansion)

### Scientist

scientistsresearch scientistscience
Fourier founded a new branch of mathematics — infinite, periodic series — studied heat flow and infrared radiation, and discovered the greenhouse effect.

### Spectral method

spectral methodsFourier spectralFourier spectral methods
The idea is to write the solution of the differential equation as a sum of certain "basis functions" (for example, as a Fourier series which is a sum of sinusoids) and then to choose the coefficients in the sum in order to satisfy the differential equation as well as possible.

### Laplace transform

Laplaces-domainLaplace domain
Laplace also recognised that Joseph Fourier's method of Fourier series for solving the diffusion equation could only apply to a limited region of space because those solutions were periodic.

### Lp space

L'' ''p'' spaceL ''p'' spacesL'' ''p'' spaces
The Fourier transform for the real line (or, for periodic functions, see Fourier series), maps L p (R) to L q (R) (or L p (T) to ℓ q ) respectively, where 1 ≤ p ≤ 2 and 1/p + 1/q = 1.

### Dini test

Dini's testDini–Lipschitz test
In mathematics, the Dini and Dini–Lipschitz tests are highly precise tests that can be used to prove that the Fourier series of a function converges at a given point.

### Classical Electrodynamics (book)

Classical ElectrodynamicsClassical Electrodynamics'' (book)widely-used graduate text
The necessary mathematical methods include vector calculus, ordinary and partial differential equations, Fourier series, and some special functions (the Bessel functions and Legendre polynomials).

### Bochner–Riesz mean

Bochner–Riesz conjectureBochner-Riesz meanBochner–Riesz operator
The Bochner–Riesz mean is a summability method often used in harmonic analysis when considering convergence of Fourier series and Fourier integrals.

### Bounded variation

functions of bounded variationboundedunbounded variation
According to Boris Golubov, BV functions of a single variable were first introduced by Camille Jordan, in the paper dealing with the convergence of Fourier series.