Dirichlet conditions

E259780

criterion for Fourier series convergence mathematical concept set of sufficient conditions

Dirichlet conditions are a set of sufficient criteria on a function—such as piecewise continuity and having a finite number of extrema and discontinuities on an interval—that guarantee the convergence of its Fourier series representation.

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All labels observed (2)

Label	Occurrences
Dirichlet boundary conditions	1
Dirichlet conditions canonical	1

Statements (47)

Predicate	Object
instanceOf	criterion for Fourier series convergence ⓘ mathematical concept ⓘ set of sufficient conditions ⓘ
appliesTo	Fourier series ⓘ functions on a finite interval ⓘ real-valued functions ⓘ
appliesToRepresentation	trigonometric Fourier series ⓘ
areNotNecessaryFor	Fourier series convergence ⓘ
areSufficientFor	pointwise convergence of Fourier series at most points ⓘ
assume	finite number of jump discontinuities per period ⓘ no infinite discontinuities on the interval ⓘ
assumption	function is periodic or extended periodically ⓘ
category	convergence criteria ⓘ sufficient conditions in analysis ⓘ
clarifies	when Fourier series representation is valid ⓘ
contrastWith	Carleson theorem on almost-everywhere convergence ⓘ Lebesgue conditions for convergence ⓘ
ensure	no pathological behavior that prevents Fourier convergence ⓘ
field	Fourier analysis ⓘ harmonic analysis ⓘ mathematical analysis ⓘ
guaranteeThat	Fourier series converges to the function value at points of continuity ⓘ Fourier series converges to the midpoint of left and right limits at jump discontinuities ⓘ
historicalContext	introduced in 19th-century analysis ⓘ
imply	Fourier coefficients are well-defined ⓘ Fourier series converges at every point where one-sided limits exist ⓘ
namedAfter	Peter Gustav Lejeune Dirichlet ⓘ surface form: Johann Peter Gustav Lejeune Dirichlet
purpose	to guarantee convergence of Fourier series ⓘ
relatedTo	Dirichlet kernel ⓘ Dirichlet theorem on Fourier series ⓘ Gibbs phenomenon ⓘ
requirement	function has a finite number of discontinuities on any given period ⓘ function has a finite number of maxima and minima on any given period ⓘ function is absolutely integrable over a period ⓘ function is piecewise continuous on the interval ⓘ function is piecewise smooth on the interval ⓘ
scope	functions defined on closed and bounded intervals ⓘ
typicalStatement	function is bounded on the interval ⓘ on any period the function has a finite number of discontinuities and extrema ⓘ
usedBy	engineers ⓘ mathematicians ⓘ physicists ⓘ
usedIn	heat equation analysis ⓘ signal processing theory ⓘ solution of partial differential equations by separation of variables ⓘ theory of Fourier series ⓘ wave equation analysis ⓘ

Referenced by (2)

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

Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe → relatedConcept → Dirichlet conditions ⓘ

Can one hear the shape of a drum? → mainConcept → Dirichlet conditions ⓘ

this entity surface form: Dirichlet boundary conditions