Dini test for convergence of Fourier series
E877690
The Dini test for convergence of Fourier series is a classical criterion in harmonic analysis that gives sufficient conditions, involving the behavior of a function near a point, to ensure the pointwise convergence of its Fourier series there.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
convergence test
ⓘ
criterion for pointwise convergence ⓘ theorem in harmonic analysis ⓘ |
| appearsIn | classical textbooks on Fourier analysis ⓘ |
| appliesTo |
Fourier series
NERFINISHED
ⓘ
trigonometric Fourier series ⓘ |
| assumptionOnPoint | x₀ is a Lebesgue point under the Dini condition ⓘ |
| concerns |
behavior of a function near a point
ⓘ
pointwise convergence of Fourier series ⓘ |
| conclusion |
Fourier series converges at x₀
ⓘ
limit equals (f(x₀+0)+f(x₀−0))/2 ⓘ |
| conditionType | sufficient but not necessary ⓘ |
| contrastsWith | global convergence criteria for Fourier series ⓘ |
| domain |
2π-periodic functions
ⓘ
integrable functions on an interval ⓘ |
| ensures |
convergence of the Fourier series to the average of one-sided limits
ⓘ
summability of Fourier series at a point under Dini condition ⓘ |
| field |
Fourier analysis
ⓘ
harmonic analysis ⓘ |
| focusesOn | local behavior of f near the convergence point ⓘ |
| generalizes | Jordan criterion for piecewise monotone functions NERFINISHED ⓘ |
| gives | sufficient conditions for convergence ⓘ |
| guarantees | convergence to midpoint of jump at a Dini point ⓘ |
| historicalPeriod | late 19th century ⓘ |
| involves |
control of oscillation of f near the point
ⓘ
singular integral estimates ⓘ |
| isPartOf | local convergence theory of Fourier series ⓘ |
| languageOfFormulation | real analysis ⓘ |
| mathematicalSubjectClassification | 42A20 ⓘ |
| namedAfter | Ulisse Dini NERFINISHED ⓘ |
| relatedTo |
Dini–Lipschitz test
NERFINISHED
ⓘ
Dirichlet test for Fourier series NERFINISHED ⓘ Jordan test for convergence of Fourier series NERFINISHED ⓘ |
| requires |
existence of one-sided limits at the point
ⓘ
integral condition on the modulus of continuity near the point ⓘ |
| strength |
stronger than mere existence of one-sided limits
ⓘ
weaker than uniform continuity assumptions ⓘ |
| typicalAssumptionOnFunction | f is 2π-periodic and integrable on [-π,π] GENERATED ⓘ |
| typicalCondition |
integral from 0 to δ of |f(x₀+t)+f(x₀−t)−2s|/t dt is finite for some δ>0
ⓘ
integral from 0 to δ of ω(f;x₀,t)/t dt is finite, where ω is a local modulus of continuity ⓘ |
| typicalFunctionSpace | L¹([-π,π]) ⓘ |
| typicalPointNotation | x₀ GENERATED ⓘ |
| typicalUseCase | functions with jump discontinuities ⓘ |
| usedIn |
analysis of Gibbs phenomenon near discontinuities
ⓘ
classical theory of Fourier series ⓘ study of boundary behavior of harmonic functions ⓘ |
| uses | Dirichlet kernel NERFINISHED ⓘ |
Referenced by (1)
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