2. (a) Briey describe how orthogonal polynomials can be used to fi nd the nodes of Gaussian quadra-ture rules for a weighted integral ∫ a b f ( x ) w ( x ) d x . \int_{a}^bf(x)w(x)\mathrm{d}x. ∫abf(x)w(x)dx.(b) Using your described approach to find the nodes, and then the method of undetermined coefficients to find the weights, derive the Gauss quadrature rule of the form, ∫ − 1 1 f ( x ) ( 1 − x 2 ) 1 / 2 d x ≈ w 0 f ( x 0 ) + w 1 f ( x 1 ) . \int_{-1}^{1}f(x)(1-x^{2})^{1/2}\,\mathrm{d}x\approx w_{0}f(x_{0})+w_{1}f(x_{1}). ∫−11f(x)(1−x2)1/2dx≈w0f(x0)+w1f(x1).You may use the following facts without proof: ϕ 0 ( x ) = 1 , ϕ 1 ( x ) = x , ϕ 2 ( x ) = x t , ϕ 2 ( x ) = x 2 − 1 / 4. \phi_{0}(x)=1, \phi_{1}(x)=x, \phi_{2}(x)= x t, \phi_{2}(x)=x^{2}-1/4. ϕ0(x)=1,ϕ1(x)=x,ϕ2(x)=xt,ϕ2(x)=x2−1/4. are orthogonal polynomials w.r.t. the weight function w ( x ) = ( 1 − x 2 ) 1 / 2 w(x)=(1-x^{2})^{1/2} w(x)=(1−x2)1/2 on ( − 1 , 1 ) (-1,1) (−1,1), and ∫ − 1 1 ( 1 − x 2 ) 1 / 2 d x = π / 2. \int_{-1}^1(1-x^{2})^{1/2}\mathrm{d}x=\pi/2. ∫−11(1−x2)1/2dx=π/2.
Ans:
4. Let f ∈ [ − 1 , 1 ] f\in [-1,1] f∈[−1,1] and let p n p_n pn be the best weighted L 2 L_2 L2 approximation to f f f from polynomials of degree at most n n n with respect to the weight function w ( x ) = ( 1 − x 2 ) − 1 / 2 w(x)=(1-x^{2})^{-1/2} w(x)=(1−x2)−1/2 on ( − 1 ; 1 ) . (-1; 1). (−1;1).
