Chebyshev Polynomials

The Chebyshev polynomials are two sequences of orthogonal polynomials related to the cosine and sine functions, usually written as two families denoted by the letters T and U. The Chebyshev polynomials of the first kind satisfy a definition built from the cosine of n times an angle, and the Chebyshev polynomials of the second kind satisfy a related identity built from the sine of the angle and of n plus one times the angle. Among polynomials of a given degree with a fixed leading coefficient, the first kind Chebyshev polynomials have the largest possible leading coefficient whose absolute value on the interval from negative one to one stays bounded by one, and they are extremal for many other properties as well. In 1952 Cornelius Lanczos showed they are important in approximation theory for solving linear systems, and the roots of the first kind polynomials, called Chebyshev nodes, are used as matching points for polynomial interpolation that minimizes Runge's phenomenon and leads directly to the method of Clenshaw-Curtis quadrature. The polynomials are named after Pafnuty Chebyshev, whose name is also transliterated Tchebycheff, Tchebyshev or Tschebyschow, which is why the letter T is used for the first kind.