Dirichlet theorem on Fourier series

E898518

mathematical theorem result in Fourier analysis

The Dirichlet theorem on Fourier series gives conditions under which a periodic function can be represented by a convergent Fourier series, specifying how and where the series converges to the function.

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Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in Fourier analysis ⓘ
appliesTo	2π-periodic functions ⓘ periodic functions of real variable ⓘ
assumes	discontinuities are finite (no infinite jumps) ⓘ function has a finite number of discontinuities on a period ⓘ function has a finite number of maxima and minima on a period ⓘ function is 2π-periodic ⓘ
clarifies	behavior of Fourier series at discontinuities ⓘ conditions for convergence of Fourier series ⓘ
concerns	pointwise convergence of Fourier series ⓘ representation of periodic functions by trigonometric series ⓘ
concludes	Fourier series converges at each point of continuity ⓘ Fourier series converges at each point of discontinuity to the midpoint of the left and right limits ⓘ Fourier series converges to (f(x+0)+f(x-0))/2 at jump discontinuities ⓘ Fourier series converges to f(x) where f is continuous ⓘ
doesNotGuarantee	absolute convergence of Fourier series ⓘ uniform convergence ⓘ
field	Fourier analysis NERFINISHED ⓘ harmonic analysis ⓘ mathematical analysis ⓘ
givesConditionOn	piecewise continuously differentiable functions ⓘ piecewise monotone functions ⓘ piecewise smooth functions ⓘ
hasApplicationIn	heat equation ⓘ partial differential equations ⓘ signal processing ⓘ vibrating string problem ⓘ
historicalPeriod	19th century ⓘ
implies	Fourier series of a function of bounded variation converges at every point ⓘ Fourier series of a function of bounded variation converges to the average of one-sided limits at each point ⓘ Fourier series of a piecewise smooth periodic function converges everywhere ⓘ Fourier series represents the function almost everywhere under its hypotheses ⓘ
isTaughtIn	Fourier analysis courses ⓘ advanced calculus courses ⓘ real analysis courses ⓘ
namedAfter	Johann Peter Gustav Lejeune Dirichlet NERFINISHED ⓘ
oftenStatedFor	functions of bounded variation ⓘ
relatedTo	Dirichlet kernel NERFINISHED ⓘ Gibbs phenomenon NERFINISHED ⓘ pointwise convergence of trigonometric series ⓘ uniform convergence of Fourier series ⓘ
strongerThan	basic convergence results for continuous periodic functions with piecewise continuous derivative ⓘ
typeOfConvergence	pointwise convergence ⓘ
uses	Dirichlet kernel NERFINISHED ⓘ
weakerThan	Carleson theorem on almost everywhere convergence of Fourier series NERFINISHED ⓘ results assuming higher smoothness such as C^1 or C^2 periodic functions ⓘ

Referenced by (1)

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Dirichlet conditions → relatedTo → Dirichlet theorem on Fourier series ⓘ