8.3 - Chebyshev Polynomials 8.3 - Chebyshev Polynomials Chebyshev polynomials Definition Chebyshev polynomial of degree n ≥= 0 is defined as Tn(x) = cos (n arccos x) ; x 2 [−1; 1]; or, in a more instructive form, Tn(x) = cos nθ ; x = cos θ ; θ 2 [0; π] : Recursive relation of Chebyshev polynomials T0(x) = 1 ;T1(x) = x ; Tn+1(x) = 2xTn(x) − Tn−1(x) ; n ≥ 1 : Thus 2 3 4 2 T2(x) = 2x − 1 ;T3(x) = 4x − 3x ; T4(x) = 8x − 8x + 1 ··· n−1 Tn(x) is a polynomial of degree n with leading coefficient 2 for n ≥ 1. 8.3 - Chebyshev Polynomials Orthogonality Chebyshev polynomials are orthogonal w.r.t. weight function w(x) = p 1 . 1−x2 Namely, Z 1 T (x)T (x) 0 if m 6= n np m dx = (1) 2 π −1 1 − x 2 if n = m for each n ≥ 1 Theorem (Roots of Chebyshev polynomials) The roots of Tn(x) of degree n ≥ 1 has n simple zeros in [−1; 1] at 2k−1 x¯k = cos 2n π ; for each k = 1; 2 ··· n : Moreover, Tn(x) assumes its absolute extrema at 0 kπ 0 k x¯k = cos n ; with Tn(¯xk) = (−1) ; for each k = 0; 1; ··· n : 0 −1 k For k = 0; ··· n, Tn(¯xk) = cos n cos (cos(kπ=n)) = cos(kπ) = (−1) : 8.3 - Chebyshev Polynomials Definition A monic polynomial is a polynomial with leading coefficient 1. The monic Chebyshev polynomial T~n(x) is defined by dividing Tn(x) by 2n−1; n ≥ 1. Hence, 1 T~ (x) = 1; T~ (x) = T (x) ; for each n ≥ 1 0 n 2n−1 n They satisfy the following recurrence relations ~ ~ 1 ~ T2(x) = xT1(x) − 2 T0(x) ~ ~ 1 ~ Tn+1(x) = xTn(x) − 4 Tn−1(x) for each n ≥ 2 The location of the zeros and extrema of T~n(x) coincides with those of Tn(x), however the extrema values are ~ 0 (−1)k 0 kπ Tn(¯xk) = 2n−1 ; atx ¯k = cos n ; k = 0; ··· n: 8.3 - Chebyshev Polynomials Definition Q~ Let n denote the set of all monic polynomials of degree n. Theorem (Min-Max Theorem) The monic Chebyshev polynomials T~n(x), have the property that 1 ~ Y~ = maxx2[−1;1]jTn(x)j ≤ maxx2[−1;1]jPn(x)j ; for all Pn(x) 2 : 2n−1 n Moreover, equality occurs only if Pn ≡ T~n. 8.3 - Chebyshev Polynomials Diagrams of monic Chebyshev polynomials 8.3 - Chebyshev Polynomials Optimal node placement in Lagrange interpolation If x0; x1; ··· xn are distinct points in the interval [−1; 1] and f 2 Cn+1[−1; 1], and P (x) the nth degree interpolating Lagrange polynomial, then 8x 2 [−1; 1], 9ξ(x) 2 (−1; 1) so that n f (n+1)(ξ(x)) Y f(x) − P (x) = (x − x ) (n + 1)! k k=0 Qn We place the nodes in a way to minimize the maximum k=0(x − xk). Qn Since k=0(x − xk) is a monic polynomial of degree (n + 1), the min-max is obtained when the nodes are chosen so that n Y 2k + 1 (x − x ) = T~ (x) ; i:e: x = cos π k n+1 k 2(n + 1) k=0 for k = 0; ··· ; n. Min-Max theorem implies that 1 Qn 2n = maxx2[−1;1]j(x − x¯1) ··· (x − x¯n+1)j ≤ maxx2[−1;1] k=0 j(x − xk)j 8.3 - Chebyshev Polynomials Theorem Suppose that P (x) is the interpolating polynomial of degree at most n with nodes at the zeros of Tn+1(x). Then 1 (n+1) maxx2[−1;1]jf(x) − P (x)j ≤ 2n(n+1)! maxx2[−1;1]jf (x)j ; for each f 2 Cn+1[−1; 1] . 1 Extending to any interval: The transformation x~ = 2 [(b − a)x + (a + b)] transforms the nodes xk in [−1; 1] into the corresponding nodes x~k in [a; b]. 8.3 - Chebyshev Polynomials.