Here is a study problem for my final exam tomorrow (its in 9 hours... help!):
Find the quadratic or lower-order polynomial that is the best approximation to cos(pie X) (using the L^2 norm) on the interval [-1, 1].
Then repeat for a quartic or lower-order polynomial. Hints: (1) You need an orthonormal set of polynomials.
Thanks!
You either know what the first few polynomials of the orthonormal basis polynomials on the interval are (They are multiples of the Legendre polynomials, see here) or you should generate them using the Gram-Schmidt process.
let them be , , , ...
Then the best quadratic polynomial approximation to in the required sense is:
where denotes the usual inner-product on
CB