Chapter 12.9, Problem 16P

Prove the least squares approximation property of Legendre polynomials [see (9.5) and (9.6)] as follows. Let $f(x)$ be the given function to be approximated. Let the functions $p_l(x)$ be the normalized Legendre polynomials, that is, $p_l(x)=\sqrt{\frac{2l+1}{2}}{P}_l(x)$ so that .

Show that the Legendre series for $f(x)$ as far as the $p_2(x)$ term is $f(x)=c_0{p}_0(x)+c_1{p}_1(x)+c_2{p}_2(x)$ with $c_1={\displaystyle {\int }_{-1}^{1}f}(x)p_l(x)dx$.

Write the quadratic polynomial satisfying the least squares condition as $b_0{p}_0(x)+$ $b_1{p}_1(x)+b_2{p}_2(x)$ (by Problem 5.14 any quadratic polynomial can be written in this form). The problem is to find $b_0,b_1,b_2$ so that is a minimum. Square the bracket and write $I$ as a sum of integrals of the individual terms. Show that some of the integrals are zero by orthogonality, some are 1 because the $p_l$

’s are normalized, and others are equal to the coefficients $c_l$.

Add and subtract $c_0^2+c_1^2+c_2^2$ and show that .

Now determine the values of the $b$

’s to make $I$ as small as possible. (Hint: The smallest value the square of a real number can have is zero.) Generalize the proof to polynomials of degree $n$.