Chebyshev polynomials information

The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as $T_{n}(x)$ and $U_{n}(x)$ . They can be defined in several equivalent ways, one of which starts with trigonometric functions:

The Chebyshev polynomials of the first kind $T_{n}$ are defined by:

T_{n}(\cos \theta )=\cos(n\theta ).

Similarly, the Chebyshev polynomials of the second kind $U_{n}$ are defined by:

U_{n}(\cos \theta )\sin \theta =\sin {\big (}(n+1)\theta {\big )}.

That these expressions define polynomials in $\cos \theta$ may not be obvious at first sight but follows by rewriting $\cos(n\theta )$ and $\sin {\big (}(n+1)\theta {\big )}$ using de Moivre's formula or by using the angle sum formulas for $\cos$ and $\sin$ repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain $T_{2}(\cos \theta )=\cos(2\theta )=2\cos ^{2}\theta -1$ and $U_{1}(\cos \theta )\sin \theta =\sin(2\theta )=2\cos \theta \sin \theta$ , which are respectively a polynomial in $\cos \theta$ and a polynomial in $\cos \theta$ multiplied by $\sin \theta$ . Hence $T_{2}(x)=2x^{2}-1$ and $U_{1}(x)=2x$ .

An important and convenient property of the $T n (x)$ is that they are orthogonal with respect to the inner product:

\langle f,g\rangle =\int _{-1}^{1}f(x)\,g(x)\,{\frac {\mathrm {d} x}{\sqrt {1-x^{2}}}},

and

U n (x)

are orthogonal with respect to another, analogous inner product, given below.

The Chebyshev polynomials $T n$ 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.^[1]

In 1952, Cornelius Lanczos showed that the Chebyshev polynomials are important in approximation theory for the solution of linear systems;^[2] the roots of $T n (x)$ , 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.^[3] The letter $T$ is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).

^ Rivlin, Theodore J. (1974). "Chapter 2, Extremal properties". The Chebyshev Polynomials. Pure and Applied Mathematics (1st ed.). New York-London-Sydney: Wiley-Interscience [John Wiley & Sons]. pp. 56–123. ISBN 978-047172470-4.
^ Lanczos, C. (1952). "Solution of systems of linear equations by minimized iterations". Journal of Research of the National Bureau of Standards. 49 (1): 33. doi:10.6028/jres.049.006.
^ Chebyshev polynomials were first presented in Chebyshev, P. L. (1854). "Théorie des mécanismes connus sous le nom de parallélogrammes". Mémoires des Savants étrangers présentés à l'Académie de Saint-Pétersbourg (in French). 7: 539–586.

[1] Rivlin, Theodore J. (1974). "Chapter 2, Extremal properties". The Chebyshev Polynomials. Pure and Applied Mathematics (1st ed.). New York-London-Sydney: Wiley-Interscience [John Wiley & Sons]. pp. 56–123. ISBN 978-047172470-4.

[2] Lanczos, C. (1952). "Solution of systems of linear equations by minimized iterations". Journal of Research of the National Bureau of Standards. 49 (1): 33. doi:10.6028/jres.049.006.

[3] Chebyshev polynomials were first presented in Chebyshev, P. L. (1854). "Théorie des mécanismes connus sous le nom de parallélogrammes". Mémoires des Savants étrangers présentés à l'Académie de Saint-Pétersbourg (in French). 7: 539–586.

Chebyshev polynomials information

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