Chebyshev polynomials of the first kind
E697760
Chebyshev polynomials of the first kind are a classical family of orthogonal polynomials on the interval [-1, 1] that play a central role in approximation theory, numerical analysis, and spectral methods.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Chebyshev polynomials | 1 |
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf | orthogonal polynomials ⓘ |
| belongsTo | Askey scheme of hypergeometric orthogonal polynomials NERFINISHED ⓘ |
| codomain | real numbers ⓘ |
| connectionToCosine | T_n(x) is the real part of (x + i\sqrt{1-x^2})^n ⓘ |
| definedOnInterval | [-1,1] ⓘ |
| degreeOfT_n | n ⓘ |
| denotedBy | T_n(x) ⓘ |
| domain | real variable x ⓘ |
| expansionProperty | any sufficiently smooth function on [-1,1] can be expanded in a Chebyshev series in T_n ⓘ |
| extremaLocation | x_k = \cos\left(\frac{k\pi}{n}\right), k=0,…,n ⓘ |
| formsBasisOf | polynomial space on [-1,1] with appropriate inner product ⓘ |
| generatingFunction | \sum_{n=0}^{\infty} T_n(x) t^n = \frac{1 - xt}{1 - 2xt + t^2} ⓘ |
| hasTrigonometricDefinition | T_n(\cos \theta) = \cos(n\theta) ⓘ |
| isSpecialCaseOf | Jacobi polynomials P_n^{(-1/2,-1/2)}(x) up to normalization ⓘ |
| leadingCoefficientOfT_n | 2^{n-1} for n ≥ 1 ⓘ |
| minimize | maximum deviation from zero on [-1,1] among monic polynomials of given degree (up to scaling) ⓘ |
| namedAfter | Pafnuty Chebyshev NERFINISHED ⓘ |
| orthogonalityNormalization |
\int_{-1}^1 T_n(x)^2 (1-x^2)^{-1/2} dx = \pi for n=0
ⓘ
\int_{-1}^1 T_n(x)^2 (1-x^2)^{-1/2} dx = \pi/2 for n ≥ 1 ⓘ |
| orthogonalityRelation | \int_{-1}^1 T_m(x) T_n(x) (1-x^2)^{-1/2} dx = 0 for m ≠ n ⓘ |
| orthogonalOnInterval | [-1,1] ⓘ |
| orthogonalWithRespectToWeight | w(x) = (1 - x^2)^(-1/2) ⓘ |
| parityProperty | T_n(-x) = (-1)^n T_n(x) ⓘ |
| rangeOnInterval | [-1,1] on [-1,1] ⓘ |
| relatedTo | Chebyshev polynomials of the second kind NERFINISHED ⓘ |
| satisfiesDifferentialEquation | (1 - x^2) y'' - x y' + n^2 y = 0 ⓘ |
| satisfiesRecurrence |
T_0(x) = 1
ⓘ
T_1(x) = x ⓘ T_{n+1}(x) = 2x T_n(x) - T_{n-1}(x) ⓘ |
| T_0(x) | 1 ⓘ |
| T_1(x) | x ⓘ |
| T_2(x) | 2x^2 - 1 ⓘ |
| T_3(x) | 4x^3 - 3x ⓘ |
| T_4(x) | 8x^4 - 8x^2 + 1 ⓘ |
| usedIn |
Chebyshev approximation
NERFINISHED
ⓘ
approximation theory ⓘ minimax approximation ⓘ numerical analysis ⓘ polynomial interpolation ⓘ solution of differential equations ⓘ spectral collocation methods ⓘ spectral methods ⓘ |
| valueAtMinusOne | T_n(-1) = (-1)^n ⓘ |
| valueAtOne | T_n(1) = 1 ⓘ |
| valueAtZero |
T_{2k+1}(0) = 0
ⓘ
T_{2k}(0) = (-1)^k ⓘ |
| zerosLocation | x_k = \cos\left(\frac{(2k-1)\pi}{2n}\right), k=1,…,n ⓘ |
Referenced by (2)
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Chebyshev polynomials