Description

Linear Operators
Assignment 3
Exercise 3.1
Let Pn be the nth Legendre polynomial. Show that
.
(2Marks)
Exercise 3.2
The first three Legendre polynomials are
P0(x) = 1, P1(x) = x, .
i) Use these polynomials to find an approximation p(x) to the function f : [−1,1] → R, f(x) = ex. (2Marks)
ii) Plot in one graph the functions ex, p(x) found above and the Taylor series at x0 = 0, ex ≈ 1 + x + x2/2. Comment on the quality of the approximation.
(2Marks)
Exercise 3.3
In this exercise we use the scalar product

on C([a,b]) for any interval [a,b] ⊂ R
i) Show that the family of functions defined on the interval [−1,1] and given by

is an orthonormal system. (2Marks) ii) Show that if {en} is an orthonormal system in C([−1,1]), then defined by
is an orthonormal system in
(2Marks)