The Chebyshev polynomial of degree n is denoted Tn (x), and is given by the explicit formula This may look trigonometric at first glance (and there is in fact a close relation between the Chebyshev polynomials and the discrete Fourier transform); however () can | 190 Chapter5. Evaluation ofFunctions Chebyshev Approximation The Chebyshev polynomial of degree n is denoted Tn x and is given by the explicit formula Tn x cos n arccos x This may look trigonometric at first glance and there is in fact a close relation between the Chebyshev polynomials and the discrete Fourier transform however can be combined with trigonometric identities to yield explicit expressions for Tn x see Figure To x 1 Ti x x T2 x 2x2 - 1 Ts x 4x3 - 3x T4 x 8x4 8x2 1 Tn i x 2xT x - T -i x n 1. There also exist inverse formulas for the powers of x in terms of the Tn s see equations . The Chebyshev polynomials are orthogonal in the interval -1 1 over a weight 1 - x2 -1 2. In particular Ti x TiLx dx 2 x2 u r1 Ti x Li VT i j i j 0 i j 0 The polynomial Tn x has n zeros in the interval 1 1 and they are located at the points x cos k 1 2 . . . n In this same interval there are n 1 extrema maxima and minima located at Sample page from NUMERICAL RECIPES IN C THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 x cos k 0 1 . . . n At all of the maxima Tn x 1 while at all of the minima Tn x 1 it is precisely this property that makes the Chebyshev polynomials so useful in polynomial approximation of functions. ChebyshevApproximation 191 Figure . Chebyshev polynomials To x through Tg . Note that Tj has j roots in the interval 1 1 and that all the polynomials are bounded between 1. The Chebyshev polynomials satisfy a discrete orthogonality relation as well as the continuous one If xk k 1 . . m are the m zeros of Tm x given by and if i j m then m 0 i j XTi xk Tj xk m 2 i j 0 k i Im i j 0 It is not too difficult to combine equations and to prove the following theorem If f x is an arbitrary function in the interval -1 1 and if N coefficients cj j 0 . N - 1 are defined by 2 N cj nX f xk Tj xk k i 2 N N X f k 1 Ak - - CO N 1 cos Sample page from NUMERICAL RECIPES IN C
