Section 15.2 Legendre Expansions
Legendre functions have many remarkable properties.
Figure 15.2.1 below shows the \(m\)th order Legendre expansion of a given function.
Figure 15.2.2 below allows you to see the effect of varying the Legendre coefficients individually while trying to guess the Legendre expansion of a given function, using unnormalized Legendre polynomials.
Figure 15.2.3 below allows you to see the effect of varying the Legendre coefficients individually while trying to guess the Legendre expansion of a given function, using normalized Legendre polynomials. 1
Since the normalization involves square roots, the discrete coefficients allowed in Figure 15.2.3 are not sufficient to exactly match the given function. How close can you get?