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In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivres formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and Chebyshev polynomials of the second kind which are denoted Un. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff (French) or Tschebyschow (German).
The Chebyshev polynomials Tn or Un are polynomials of degree n and the sequence of Chebyshev polynomials of either kind composes a polynomial sequence.
Chebyshev polynomials are important in approximation theory because the roots of the Chebyshev polynomials of the first kind, which are also called Chebyshev nodes, are used as nodes in polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runges 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.