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Section 15.2 Legendre Expansions

Legendre functions have many remarkable properties.

Figure 15.2 below shows the \(m\)th order Legendre expansion of a given function.

Figure 15.2. The \(m\)th order Legendre expansion of a given function.

Figure 15.3 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 unnormalized Legendre polynomials.

Figure 15.3. Varying the individual Legendre coefficients, using unnormalized Legendre polynomials.

Figure 15.4 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 

Figure 15.4. Varying the individual Legendre coefficients, using normalized Legendre polynomials.
Since the normalization involves square roots, the discrete coefficients allowed in Figure 15.4 are not sufficient to exactly match the given function. How close can you get?