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.
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