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

Prasad, Ganesh
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 quadrature

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)


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.