Minimization Problem (using Projection)

[Dwolfson]

Homework Statement

Minimize ||cos(2x) - f(x)|| where f(x) is a a function in the span of {(1,sin(x),cos(x)}

Where the inner produect is defined (1/pi)(integral from -pi to pi of f(x)g(x) dx)

Homework Equations

I found f(x) to be zero. Is this correct I am uneasy about this solution.

The Attempt at a Solution

My solution I took the inner product of cos(2x) with each of the elements of the set {(1,sin(x),cos(x)}

Knowing that the projection of cos(2x) onto this span would give me the smallest norm.

I found that the projection of cos(2x) onto this set is 0.. Because given this inner product definition cos(2x) is orthogonal to {(1,sin(x),cos(x)}.

I am uneasy about this answer because intuitively it does not seem that f(x)=0 would minimize the norm of ||cos(2x)-f(x)||

Thanks.

 

[henry_m]

I think that's right! The integrals you have to do well known from the orthogonality of Fourier components.

If you draw the graphs on top of each other maybe it will be a bit more intuitive. It should be clear that 1 shouldn't overlap, since the average of cos(2x) is 0. As for sin and cos, the different periodicity means that for everywhere the functions agree, there is somewhere else half a phase later where they disagree just as much so they shouldn't overlap. Hope that makes a little sense...