Consider the following linear mapping from C[-pi,pi] into itself:
L(f)=integral from -pi to pi of G(x),h(y),f(y)dy for any function f(x) in C[-pi,pi]. Here G(x), H(x) are given continuous functions. Find a function f such that L*f=lambda*f for some lambda and find the value of lambda. This is a generalization of the notion for particular case G(x)=cosx,H(x)=x^2. Hint Look for f(x)=aG(s). Explain why this assumption is reasonable.
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Consider L(f(x)) again. That is:
First of all, we can take G(x) out of this integral, because it does not depend on y. Therefore we have:
Ok, now let's consider the integral part. It is useful, as you will see, but not necessary to consider this fact that the integral part is actually an inner product. An inner product should have the following properties:
1- <x, x> >= 0 and <x, x> = 0 if and only if x=0
2- <y, x> = <y, x>
3- <cx, y> = c<x, y>
4- <x+y, z> = <x, z> + <y, z>
I just show you the first feature in this case and you can show the next ones. We actually want to show that:
= < H, f > ...
A relation between functions is investigated. The solution is detailed and well presented.