本文主要是介绍MATH36022 Numerical Analysis 2 Approximation of Functions – Week 3 Exercises，希望对大家解决编程问题提供一定的参考价值，需要的开发者们随着小编来一起学习吧！
- Show that the Chebyshev polynomials are orthogonal on ( − 1 , 1 ) (−1, 1) (−1,1) with respect to the weight function ( 1 − x 2 ) − 1 / 2 (1 − x^2)^{−1/2} (1−x2)−1/2.
Ans: T n ( x ) = cos ( n arccos x ) T_n(x)= \cos(n\arccos x) Tn(x)=cos(narccosx)
∫ − 1 1 T n ( x ) T m ( x ) ( 1 − x 2 ) − 1 / 2 d x \int^1_{-1}T_n(x)T_m(x)(1 − x^2)^{−1/2} \textrm{d}x ∫−11Tn(x)Tm(x)(1−x2)−1/2dx = ∫ π 0 cos n θ cos m θ 1 sin θ d cos θ =\int^0_{\pi}\cos n\theta \cos m\theta \frac{1}{\sin\theta} \textrm{d}\cos\theta =∫π0cosnθcosmθsinθ1dcosθ = ∫ 0 π cos n θ cos m θ d θ =\int_0^{\pi}\cos n\theta \cos m\theta \textrm{d}\theta =∫0πcosnθcosmθdθ = ∫ 0 π cos n θ cos m θ d θ
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