Chebyshev polynomials of the first kind

E697760

orthogonal polynomials

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.

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All labels observed (3)

Label	Occurrences
Chebyshev polynomials	5
Chebyshev polynomials of the first kind canonical	1
Chebyshev polynomials of the first kind for \lambda = 0 (limit case)	1

How this entity was disambiguated

This entity first appeared as the object of triple T7871817 — 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: Chebyshev polynomials of the first kind
Context triple: [Jacobi polynomials, generalizes, Chebyshev polynomials of the first kind]

A. Bernstein polynomials
Bernstein polynomials are a family of polynomials used in approximation theory that provide a constructive proof of the Weierstrass approximation theorem by uniformly approximating continuous functions on a closed interval.
B. Jacobi polynomials
Jacobi polynomials are a family of classical orthogonal polynomials depending on two parameters, widely used in approximation theory, numerical analysis, and solutions of differential equations.
C. Chebyshev functions
Chebyshev functions are arithmetic functions in number theory that encode information about the distribution of prime numbers and play a key role in analytic approaches to the prime number theorem.
D. Bernoulli polynomials
Bernoulli polynomials are a sequence of polynomials deeply connected to number theory and analysis, appearing in the study of special functions, series expansions, and the evaluation of sums of powers of integers.
E. Hermite
Hermite is a French surname most famously associated with the 19th-century mathematician Charles Hermite, known for his contributions to number theory, algebra, and analysis.
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: Chebyshev polynomials of the first kind
Target entity description: 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.

A. Bernstein polynomials
Bernstein polynomials are a family of polynomials used in approximation theory that provide a constructive proof of the Weierstrass approximation theorem by uniformly approximating continuous functions on a closed interval.
B. Jacobi polynomials
Jacobi polynomials are a family of classical orthogonal polynomials depending on two parameters, widely used in approximation theory, numerical analysis, and solutions of differential equations.
C. Chebyshev functions
Chebyshev functions are arithmetic functions in number theory that encode information about the distribution of prime numbers and play a key role in analytic approaches to the prime number theorem.
D. Bernoulli polynomials
Bernoulli polynomials are a sequence of polynomials deeply connected to number theory and analysis, appearing in the study of special functions, series expansions, and the evaluation of sums of powers of integers.
E. Hermite
Hermite is a French surname most famously associated with the 19th-century mathematician Charles Hermite, known for his contributions to number theory, algebra, and analysis.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	orthogonal polynomials ⓘ
belongsTo	Askey scheme of hypergeometric orthogonal polynomials ⓘ
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 ⓘ
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 ⓘ
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 ⓘ linked to: Chebyshev alternation theorem 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 ⓘ

How these facts were elicited