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 The Chebyshev polynomials are two sequences of polynomials, denoted Tn(x) and Un(x). They are defined as follows. By the double angle formula, is a polynomial in cos(θ), so define T2(x) = 2x2 − 1. The other Tn(x) are defined similarly, using cos(nθ) = Tn(cos(θ)). Similarly, define the other sequence by sin(nθ) = Un−1(cos(θ)) sin(θ), where we have used de Moivre's formula to note that sin(nθ) is sin(θ) times a polynomial in cos(θ). For instance, gives U2(x) = 4x2 − 1. The Tn(x) and Un(x) are called Chebyshev polynomials of the first and second kind respectively. The Tn(x) are orthogonal with respect to the inner product and Un(x) are orthogonal with respect to a different product. This follows from the fact that the Chebyshev polynomials solve the Chebyshev differential equations which are Sturm–Liouville differential equations. It is a general feature of such differential equations that there is a distinguished orthonormal set of solutions. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval [−1, 1] is bounded by 1. They are also the extremal polynomials for many other properties. Chebyshev polynomials are important in approximation theory because the roots of Tn(x), which are also called Chebyshev nodes, are used as nodes in polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the polynomial of best approximation to a continuous function under the maximum norm. This approximation leads directly to the method of Clenshaw–Curtis quadrature. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German). (en)

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 P. K. (en)
 René F. (en)
 Roderick S. C. (en)
 Roelof (en)
 Tom H. (en)

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 Wong (en)
 Koekoek (en)
 Koornwinder (en)
 Swarttouw (en)
 Suetin (en)

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 : (en)
 Let's assume that is a polynomial of degree with leading coefficient 1 with maximal absolute value on the interval less than .
Define
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Because at extreme points of we have
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From the intermediate value theorem, has at least roots. However, this is impossible, as is a polynomial of degree , so the fundamental theorem of algebra implies it has at most roots. (en)
 The second derivative of the Chebyshev polynomial of the first kind is
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which, if evaluated as shown above, poses a problem because it is indeterminate at . Since the function is a polynomial, the derivatives must exist for all real numbers, so the taking to limit on the expression above should yield the desired value:
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where only is considered for now. Factoring the denominator:
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Since the limit as a whole must exist, the limit of the numerator and denominator must independently exist, and
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The denominator limits to zero, which implies that the numerator must be limiting to zero, i.e. which will be useful later on. Since the numerator and denominator are both limiting to zero, L'Hôpital's rule applies:
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The proof for is similar, with the fact that being important. (en)

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 Proof (en)
 Orthogonal Polynomials (en)
 Chebyshev polynomials (en)
 Chebyshev polynomial[s] of the first kind (en)

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 ChebyshevPolynomialoftheFirstKind (en)

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 The Chebyshev polynomials are two sequences of polynomials, denoted Tn(x) and Un(x). They are defined as follows. By the double angle formula, is a polynomial in cos(θ), so define T2(x) = 2x2 − 1. The other Tn(x) are defined similarly, using cos(nθ) = Tn(cos(θ)). Similarly, define the other sequence by sin(nθ) = Un−1(cos(θ)) sin(θ), where we have used de Moivre's formula to note that sin(nθ) is sin(θ) times a polynomial in cos(θ). For instance, gives U2(x) = 4x2 − 1. The Tn(x) and Un(x) are called Chebyshev polynomials of the first and second kind respectively. (en)

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 Chebyshev polynomials (en)

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