1 Chebyshev Polynomials: w (x) = p on [−1; 1] 1 − x2 Orthogonal Polynomials: Gram-Schmidt process Thm: The set of polynomial functions fφ0; ··· ; φng defined below on [a; b] is orthogonal with respect to the weight function w. φ0 (x) = 1; φ1 (x) = x − B1; and for k ≥ 2 φk (x) = (x − Bk ) φk−1 (x) − Ck φk−2 (x) ; with R b 2 a x w (x) φj−1 (x) d x Bj = ; j = 1; 2; ··· ; n; R b 2 a w (x) φj−1 (x) d x R b a x w (x) φj−1 (x) φj−2 (x) d x Cj = ; j = 2; 3; ··· ; n: R b 2 a w (x) φj−2 (x) d x Orthogonal Polynomials: Gram-Schmidt process Thm: The set of polynomial functions fφ0; ··· ; φng defined below on [a; b] is orthogonal with respect to the weight function w. φ0 (x) = 1; φ1 (x) = x − B1; and for k ≥ 2 φk (x) = (x − Bk ) φk−1 (x) − Ck φk−2 (x) ; with R b 2 a x w (x) φj−1 (x) d x Bj = ; j = 1; 2; ··· ; n; R b 2 a w (x) φj−1 (x) d x R b a x w (x) φj−1 (x) φj−2 (x) d x Cj = ; j = 2; 3; ··· ; n: R b 2 a w (x) φj−2 (x) d x 1 Chebyshev Polynomials: w (x) = p on [−1; 1] 1 − x2 I Let x = cos (θ) ; θ 2 [0; π]. Then φj (x) = cos (jθ) ; j = 0; 1: I Induction hypothesis with x = cos (θ) ; θ 2 [0; π]: cos(jθ) φj (x) = 2j−1 for j = 2; ··· ; n − 1 . I By Gram-Schmidt, for n ≥ 2 φn (x) = (x − Bn) φn−1 (x) − Cn φn−2 (x) ; with R 1 2 −1 x w (x) φn−1 (x) d x Bn = ; R 1 2 −1 w (x) φn−1 (x) d x R 1 −1 x w (x) φn−1 (x) φn−2 (x) d x Cn = : R 1 2 −1 w (x) φn−2 (x) x8.3 Chebyshev Polynomials/Power Series Economization Chebyshev: Gram-Schmidt for orthogonal polynomial functions fφ ; ··· ; φ g on [−1; 1] with weight function w (x) = p 1 . 0 n 1−x2 R 1 p x d x −1 1−x2 I φ0 (x) = 1; φ1 (x) = x − B1, with B1 = 1 = 0: R p 1 d x −1 1−x2 I By Gram-Schmidt, for n ≥ 2 φn (x) = (x − Bn) φn−1 (x) − Cn φn−2 (x) ; with R 1 2 −1 x w (x) φn−1 (x) d x Bn = ; R 1 2 −1 w (x) φn−1 (x) d x R 1 −1 x w (x) φn−1 (x) φn−2 (x) d x Cn = : R 1 2 −1 w (x) φn−2 (x) x8.3 Chebyshev Polynomials/Power Series Economization Chebyshev: Gram-Schmidt for orthogonal polynomial functions fφ ; ··· ; φ g on [−1; 1] with weight function w (x) = p 1 . 0 n 1−x2 R 1 p x d x −1 1−x2 I φ0 (x) = 1; φ1 (x) = x − B1, with B1 = 1 = 0: R p 1 d x −1 1−x2 I Let x = cos (θ) ; θ 2 [0; π]. Then φj (x) = cos (jθ) ; j = 0; 1: I Induction hypothesis with x = cos (θ) ; θ 2 [0; π]: cos(jθ) φj (x) = 2j−1 for j = 2; ··· ; n − 1 . x8.3 Chebyshev Polynomials/Power Series Economization Chebyshev: Gram-Schmidt for orthogonal polynomial functions fφ ; ··· ; φ g on [−1; 1] with weight function w (x) = p 1 . 0 n 1−x2 R 1 p x d x −1 1−x2 I φ0 (x) = 1; φ1 (x) = x − B1, with B1 = 1 = 0: R p 1 d x −1 1−x2 I Let x = cos (θ) ; θ 2 [0; π]. Then φj (x) = cos (jθ) ; j = 0; 1: I Induction hypothesis with x = cos (θ) ; θ 2 [0; π]: cos(jθ) φj (x) = 2j−1 for j = 2; ··· ; n − 1 . I By Gram-Schmidt, for n ≥ 2 φn (x) = (x − Bn) φn−1 (x) − Cn φn−2 (x) ; with R 1 2 −1 x w (x) φn−1 (x) d x Bn = ; R 1 2 −1 w (x) φn−1 (x) d x R 1 −1 x w (x) φn−1 (x) φn−2 (x) d x Cn = : R 1 2 −1 w (x) φn−2 (x) 1 cos (θ) cos2 ((n − 1)θ) = cos (θ) (1 + cos (2(n − 1)θ)) 2 1 1 = cos (θ) + (cos ((2n − 3)θ) + cos ((2n − 1)θ)) ; so 2 2 Z π cos (θ) cos2 ((n − 1)θ) d θ 0 Z π 1 1 = cos (θ) + (cos ((2n − 3)θ) + cos ((2n − 1)θ)) d θ = 0; Bn = 0: 0 2 2 Chebyshev Polynomials: Compute Bn cos(jθ) Let x = cos (θ) ; θ 2 [0; π]. Then φj (x) = 2j−1 , 1 1 d x = −sin (θ) d θ; w (x) = p = : 1 − x2 sin (θ) 2 R π cos(θ)cos ((n−1)θ) sin (θ) d θ B = 0 sin(θ) n R π cos2((n−1)θ) 0 sin(θ) sin (θ) d θ R π cos (θ) cos2 ((n − 1)θ) d θ = 0 ; where R π 2 0 cos ((n − 1)θ) d θ Z π cos (θ) cos2 ((n − 1)θ) d θ 0 Z π 1 1 = cos (θ) + (cos ((2n − 3)θ) + cos ((2n − 1)θ)) d θ = 0; Bn = 0: 0 2 2 Chebyshev Polynomials: Compute Bn cos(jθ) Let x = cos (θ) ; θ 2 [0; π]. Then φj (x) = 2j−1 , 1 1 d x = −sin (θ) d θ; w (x) = p = : 1 − x2 sin (θ) 2 R π cos(θ)cos ((n−1)θ) sin (θ) d θ B = 0 sin(θ) n R π cos2((n−1)θ) 0 sin(θ) sin (θ) d θ R π cos (θ) cos2 ((n − 1)θ) d θ = 0 ; where R π 2 0 cos ((n − 1)θ) d θ 1 cos (θ) cos2 ((n − 1)θ) = cos (θ) (1 + cos (2(n − 1)θ)) 2 1 1 = cos (θ) + (cos ((2n − 3)θ) + cos ((2n − 1)θ)) ; so 2 2 Bn = 0: Chebyshev Polynomials: Compute Bn cos(jθ) Let x = cos (θ) ; θ 2 [0; π]. Then φj (x) = 2j−1 , 1 1 d x = −sin (θ) d θ; w (x) = p = : 1 − x2 sin (θ) 2 R π cos(θ)cos ((n−1)θ) sin (θ) d θ B = 0 sin(θ) n R π cos2((n−1)θ) 0 sin(θ) sin (θ) d θ R π cos (θ) cos2 ((n − 1)θ) d θ = 0 ; where R π 2 0 cos ((n − 1)θ) d θ 1 cos (θ) cos2 ((n − 1)θ) = cos (θ) (1 + cos (2(n − 1)θ)) 2 1 1 = cos (θ) + (cos ((2n − 3)θ) + cos ((2n − 1)θ)) ; so 2 2 Z π cos (θ) cos2 ((n − 1)θ) d θ 0 Z π 1 1 = cos (θ) + (cos ((2n − 3)θ) + cos ((2n − 1)θ)) d θ = 0; 0 2 2 Chebyshev Polynomials: Compute Bn cos(jθ) Let x = cos (θ) ; θ 2 [0; π]. Then φj (x) = 2j−1 , 1 1 d x = −sin (θ) d θ; w (x) = p = : 1 − x2 sin (θ) 2 R π cos(θ)cos ((n−1)θ) sin (θ) d θ B = 0 sin(θ) n R π cos2((n−1)θ) 0 sin(θ) sin (θ) d θ R π cos (θ) cos2 ((n − 1)θ) d θ = 0 ; where R π 2 0 cos ((n − 1)θ) d θ 1 cos (θ) cos2 ((n − 1)θ) = cos (θ) (1 + cos (2(n − 1)θ)) 2 1 1 = cos (θ) + (cos ((2n − 3)θ) + cos ((2n − 1)θ)) ; so 2 2 Z π cos (θ) cos2 ((n − 1)θ) d θ 0 Z π 1 1 = cos (θ) + (cos ((2n − 3)θ) + cos ((2n − 1)θ)) d θ = 0; Bn = 0: 0 2 2 cos (θ) cos ((n − 1)θ) cos ((n − 2)θ) 1 = cos ((n − 1)θ)(cos ((n − 1)θ) + cos ((n − 3)θ)) 2 1 = (1 + cos (2(n − 1)θ) + cos (2(n − 2)θ) + cos (2θ)) ; so 4 Z π π 2 ; for n = 2, cos (θ) cos ((n − 1)θ) cos ((n − 2)θ) d θ = π 0 4 ; for n > 2. Chebyshev Polynomials: Compute Cn (I) cos(jθ) Let x = cos (θ) ; θ 2 [0; π]. Then φj (x) = 2j−1 , 1 1 d x = −sin (θ) d θ; w (x) = p = : 1 − x2 sin (θ) R π cos(θ)cos((n−1)θ) cos((n−2)θ) sin (θ) d θ C = 0 sin(θ) n R π cos2((n−2)θ) 2 0 sin(θ) sin (θ) d θ R π cos (θ) cos ((n − 1)θ) cos ((n − 2)θ) d θ = 0 ; where R π 2 2 0 cos ((n − 2)θ) d θ Z π π 2 ; for n = 2, cos (θ) cos ((n − 1)θ) cos ((n − 2)θ) d θ = π 0 4 ; for n > 2. Chebyshev Polynomials: Compute Cn (I) cos(jθ) Let x = cos (θ) ; θ 2 [0; π]. Then φj (x) = 2j−1 , 1 1 d x = −sin (θ) d θ; w (x) = p = : 1 − x2 sin (θ) R π cos(θ)cos((n−1)θ) cos((n−2)θ) sin (θ) d θ C = 0 sin(θ) n R π cos2((n−2)θ) 2 0 sin(θ) sin (θ) d θ R π cos (θ) cos ((n − 1)θ) cos ((n − 2)θ) d θ = 0 ; where R π 2 2 0 cos ((n − 2)θ) d θ cos (θ) cos ((n − 1)θ) cos ((n − 2)θ) 1 = cos ((n − 1)θ)(cos ((n − 1)θ) + cos ((n − 3)θ)) 2 1 = (1 + cos (2(n − 1)θ) + cos (2(n − 2)θ) + cos (2θ)) ; so 4 Chebyshev Polynomials: Compute Cn (I) cos(jθ) Let x = cos (θ) ; θ 2 [0; π].