Dirichlet kernel
E466248
The Dirichlet kernel is a trigonometric polynomial that arises in Fourier series as the summation kernel for partial sums, playing a key role in analyzing convergence properties.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Dirichlet kernel canonical | 7 |
How this entity was disambiguated
This entity first appeared as the object of triple T4746238 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Dirichlet kernel Context triple: [Peter Gustav Lejeune Dirichlet, notableWork, Dirichlet kernel]
-
A.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
-
B.
Dirichlet 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.
-
C.
Szegő kernel
The Szegő kernel is a fundamental reproducing kernel in complex analysis and operator theory, associated with Hardy spaces on the boundary of a domain and central to the study of orthogonal polynomials and boundary behavior of analytic functions.
-
D.
Dirac delta function
The Dirac delta function is a mathematical construct used in physics and engineering to model an idealized point mass or point charge, being zero everywhere except at a single point where it is infinitely large yet integrates to one.
-
E.
Fourier inversion theorem
The Fourier inversion theorem is a fundamental result in harmonic analysis that guarantees, under suitable conditions, that a function can be exactly reconstructed from its Fourier transform.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Dirichlet kernel Target entity description: The Dirichlet kernel is a trigonometric polynomial that arises in Fourier series as the summation kernel for partial sums, playing a key role in analyzing convergence properties.
-
A.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
-
B.
Dirichlet 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.
-
C.
Szegő kernel
The Szegő kernel is a fundamental reproducing kernel in complex analysis and operator theory, associated with Hardy spaces on the boundary of a domain and central to the study of orthogonal polynomials and boundary behavior of analytic functions.
-
D.
Dirac delta function
The Dirac delta function is a mathematical construct used in physics and engineering to model an idealized point mass or point charge, being zero everywhere except at a single point where it is infinitely large yet integrates to one.
-
E.
Fourier inversion theorem
The Fourier inversion theorem is a fundamental result in harmonic analysis that guarantees, under suitable conditions, that a function can be exactly reconstructed from its Fourier transform.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical object
ⓘ
summability kernel ⓘ trigonometric polynomial ⓘ |
| alternativeForm | D_n(x) = sin((n+1/2)x) / sin(x/2) for x not multiple of 2π ⓘ |
| appearsIn | expression for nth partial sum of Fourier series ⓘ |
| boundedInL1 | false ⓘ |
| boundedInLInfinity | false ⓘ |
| codomain | real numbers ⓘ |
| context | Fourier series on the interval [-π,π] ⓘ |
| contrastWith | Fejér kernel ⓘ |
| definition |
D_n(x) = 1 + 2 sum_{k=1}^{n} cos(kx)
ⓘ
D_n(x) = sum_{k=-n}^{n} e^{ikx} ⓘ |
| degree | n in the trigonometric sense ⓘ |
| dependsOn | integer parameter n ⓘ |
| differenceFromFejerKernel | not positive and not an approximate identity in L1 ⓘ |
| domain | real line ⓘ |
| evenFunction | true ⓘ |
| field |
Fourier analysis
ⓘ
harmonic analysis ⓘ |
| generalizationOf |
Dirichlet kernels on compact groups
ⓘ
linked to:
Bochner–Riesz means
|
| growthNearZero | behaves like 2n+1 near x = 0 ⓘ |
| integralValue | ∫_{-π}^{π} D_n(x) dx = 2π ⓘ |
| L1NormBehavior | L^1 norm grows like O(log n) ⓘ |
| LInfinityNormBehavior | L^∞ norm grows like O(n) ⓘ |
| namedAfter |
Johann Peter Gustav Lejeune Dirichlet
ⓘ
linked to:
Peter Gustav Lejeune Dirichlet
|
| period | 2π-periodic function ⓘ |
| property | integral over one period equals 2π ⓘ |
| realValued | true ⓘ |
| relatedConcept |
Poisson kernel
ⓘ
approximate identity ⓘ convolution on the circle ⓘ |
| relation | S_n(f,x) = (1/2π) ∫_{-π}^{π} f(t) D_n(x-t) dt ⓘ |
| role |
summation kernel for partial sums of Fourier series
ⓘ
tool for studying convergence of Fourier series ⓘ |
| smoothness | real-analytic away from points where sin(x/2)=0 ⓘ |
| support | entire real line ⓘ |
| symbol | D_n ⓘ |
| symmetry | D_n(-x) = D_n(x) ⓘ |
| usedIn |
Fourier series
ⓘ
analysis of pointwise convergence of Fourier series ⓘ analysis of uniform convergence of Fourier series ⓘ study of partial sums of orthogonal expansions on the circle ⓘ |
| usedInProofOf |
Dirichlet’s convergence theorem for Fourier series of piecewise smooth functions
ⓘ
linked to:
Dirichlet theorem on Fourier series
|
| usedToShow |
Gibbs phenomenon near jump discontinuities
ⓘ
existence of continuous functions with divergent Fourier series at some points ⓘ non-uniform convergence of Fourier series on some function classes ⓘ |
| valueAtZero | D_n(0) = 2n+1 ⓘ |
| zeroSet | zeros at x = 2πk/(2n+1), k integer, excluding multiples of 2π ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Dirichlet kernel Description of subject: The Dirichlet kernel is a trigonometric polynomial that arises in Fourier series as the summation kernel for partial sums, playing a key role in analyzing convergence properties.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.