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Concept# Chebyshev polynomials

Summary

The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and . They can be defined in several equivalent ways, one of which starts with trigonometric functions:
The Chebyshev polynomials of the first kind are defined by
Similarly, the Chebyshev polynomials of the second kind are defined by
That these expressions define polynomials in may not be obvious at first sight, but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain and , which are respectively a polynomial in and a polynomial in multiplied by . Hence and .
An important and convenient property of the Tn(x) is that they are orthogonal with respect to the inner product:
and Un(x) are orthogonal with respect to another, analogous inner product, given below.
The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval 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 matching points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the best polynomial approximation to a continuous function under the maximum norm, also called the "minimax" criterion. 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).

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