题号:14631    题型:解答题    来源:2024年丘成桐大学生数学竞赛(计算与应用数学)-无答案
Consider the Chebyshev polynomial of the first kind
$$
T_n(x)=\cos (n \theta), \quad x=\cos (\theta), \quad x \in[-1,1] .
$$

The Chebyshev polynomials of the second kind are defined a $\mathrm{s}$
$$
U_n(x)=\frac{1}{n+1} T^{\prime}(x), \quad n \geq 0 .
$$
(a) Derive a recursive formula for computing $U_n(x)$ for all $n \geq 0$.
(b) Show that the Chebyshev polynomials of the second kind are orthogonal with respect to the inner product
$$
\langle f, g\rangle=\int_{-1}^1 f(x) g(x) \sqrt{1-x^2} \mathrm{~d} x .
$$
(c) Derive the 2-point Gaussian Quadrature rule for the integr al
$$
\int_{-1}^1 f(x) \sqrt{1-x^2} \mathrm{~d} x=\sum_{j=1}^3 w_j f\left(x_j\right) .
$$
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