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.
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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
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 NERFINISHED ⓘ |
| 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 ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.