Statements (51)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
orthogonal polynomials |
| gptkbp:alsoKnownAs |
Chebyshev polynomial of the first kind
|
| gptkbp:application |
gptkb:signal_processing
Fourier analysis numerical analysis solving differential equations approximation theory minimax approximation |
| gptkbp:definedIn |
real numbers
|
| gptkbp:degree |
n
|
| gptkbp:differential |
(1-x²)y'' - xy' + n²y = 0
|
| gptkbp:domain |
[-1,1]
|
| gptkbp:explicitFormula |
Tₙ(x) = cos(n arccos x)
|
| gptkbp:fifthPolynomial |
T₄(x) = 8x⁴ - 8x² + 1
|
| gptkbp:firstPolynomial |
T₀(x) = 1
|
| gptkbp:fourthPolynomial |
T₃(x) = 4x³ - 3x
|
| gptkbp:generatingFunction |
(1 - xt)/(1 - 2xt + t²)
|
| https://www.w3.org/2000/01/rdf-schema#label |
Chebyshev T
|
| gptkbp:maximumSpeed |
1
|
| gptkbp:minimumPressure |
-1
|
| gptkbp:namedAfter |
gptkb:Pafnuty_Chebyshev
|
| gptkbp:orthogonalWithRespectTo |
weight function (1-x²)^(-1/2) on [-1,1]
|
| gptkbp:par |
Tₙ(-x) = (-1)ⁿ Tₙ(x)
|
| gptkbp:recurrence |
T₀(x) = 1, T₁(x) = x, Tₙ₊₁(x) = 2xTₙ(x) - Tₙ₋₁(x)
|
| gptkbp:relatedTo |
gptkb:Chebyshev_polynomials_of_the_second_kind
|
| gptkbp:roots |
cos((2k-1)π/(2n)), k=1,...,n
|
| gptkbp:secondPolynomial |
T₁(x) = x
|
| gptkbp:sequence |
T₀(x), T₁(x), T₂(x), ...
|
| gptkbp:thirdPolynomial |
T₂(x) = 2x² - 1
|
| gptkbp:usedIn |
gptkb:machine_learning
computer graphics control theory data compression image processing interpolation numerical integration root-finding algorithms digital filter design polynomial approximation spectral methods signal reconstruction Chebyshev filter design fast Fourier transform (FFT) algorithms least squares fitting orthogonal expansions orthogonal transforms quadrature rules |
| gptkbp:zeros |
n distinct points in [-1,1]
|
| gptkbp:bfsParent |
gptkb:Chebyshev_(crater)
|
| gptkbp:bfsLayer |
7
|