Fourier series

E482288
mathematical concept representation of periodic functions series expansion

A Fourier series is a way of representing a periodic function as an infinite sum of sines and cosines with appropriately chosen coefficients.

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Observed surface forms (1)

Surface form Occurrences
Fourier series expansions 1

Statements (51)

Predicate Object
instanceOf mathematical concept ⓘ
representation of periodic functions ⓘ
series expansion ⓘ
appliesTo piecewise smooth functions ⓘ
square-integrable periodic functions ⓘ
assumesProperty periodicity of function ⓘ
basisProperty orthogonality of sines and cosines ⓘ
convergesInSense L2 norm ⓘ
mean square ⓘ
pointwise almost everywhere ⓘ
convergesUnderCondition Dirichlet conditions NERFINISHED ⓘ
expandsInDomain interval [0,2π] ⓘ
interval [−π,π] ⓘ
field Fourier analysis ⓘ
harmonic analysis ⓘ
mathematical analysis ⓘ
signal processing ⓘ
generalizesTo Fourier series on groups ⓘ
Fourier transform NERFINISHED ⓘ
discrete-time Fourier series ⓘ
hasComponent Fourier coefficients ⓘ
Fourier cosine series NERFINISHED ⓘ
Fourier sine series NERFINISHED ⓘ
complex Fourier series ⓘ
hasFormulaType complex Fourier series ⓘ
real Fourier series ⓘ
introducedBy Joseph Fourier NERFINISHED ⓘ
introducedInWork Théorie analytique de la chaleur NERFINISHED ⓘ
introducedInYear 1822 ⓘ
namedAfter Joseph Fourier NERFINISHED ⓘ
relatedConcept Fourier coefficients ⓘ
Gibbs phenomenon NERFINISHED ⓘ
Hilbert space NERFINISHED ⓘ
Parseval's identity NERFINISHED ⓘ
Plancherel theorem NERFINISHED ⓘ
orthogonal functions ⓘ
trigonometric polynomials ⓘ
represents periodic function ⓘ
usedFor acoustics ⓘ
filter design ⓘ
heat equation ⓘ
image processing ⓘ
quantum mechanics NERFINISHED ⓘ
signal decomposition ⓘ
solving partial differential equations ⓘ
spectral analysis ⓘ
vibration analysis ⓘ
wave equation ⓘ
usesBasisFunctions complex exponentials ⓘ
cosine functions ⓘ
sine functions ⓘ

Referenced by (8)

Full triples — surface form annotated when it differs from this entity's canonical label.

Dirichlet conditions → appliesTo → Fourier series ⓘ
Fourier inversion theorem → isRelatedTo → Fourier series ⓘ
Poisson summation formula → relatedTo → Fourier series ⓘ
Weierstrass M-test → relatedTo → Fourier series ⓘ
Riesz–Fischer theorem → relatesConcept → Fourier series ⓘ
Laplace equation → solutionMethod → Fourier series ⓘ
Arnaud Denjoy → studied → Fourier series ⓘ
Poisson summation formula → usedIn → Fourier series ⓘ
this entity surface form: Fourier series expansions