Chapter 2. Legendre Polynomials Examples Show that each function set is orthogonal in the given interval...
Please explain the solution and write clearly for nu, ber 25. Thanks. 25. Approximate the following functions f(x) as a linear combination of the first four Legendre polynomials over the interval [-1,1]: Lo(x) = 1, Li(x) = x, L2(x) = x2-1. L3(x) = x3-3x/5. (a) f(x) = X4 (b) f(x) = k (c) f(x) =-1: x < 0, = 1: x 0 Example 8. Approximating e by Legendre Polynomials Let us use the first four Legendre polynomials Lo(x) 1, Li(x)...
(a) Check that {1, 2} is an orthogonal set with the weight function w(x) = x2 on the interval (-2,2). (b) Find a quadratic polynomial p(x) = 32 + ax + b that is orthogonal to the functions in the set, with the same weighted inner product. (c) Is this set complete, as an orthogonal set with the weighted inner product?