Let $D = \frac{d^2}{dx^2}$ and $V$ be the set of functions that are infinitely differentiable, real, and 2$\pi$-periodic.
I've found the following about $V$ and $D$:
- $D$ is symmetric: for any two functions $f,g \in V, \langle Df, g \rangle = \langle f, Dg \rangle$.
- $\{1, \sin(x), \cos(x), \sin(2x), \cos(2x), ... \} \in V$
- If $f \in V$, then $f' \in V$.
What does it mean for "the symmetric differential operator $D : V \to V$ to have an orthogonal eigenbasis"?
I know that the second bullet I found is a set which is orthogonal, but I'm not sure how to form an eigenbasis if I don't know what sort of functions are in V. So I'm not sure what to do with that piece of information. I know that orthogonal set means that for any two eigen functions $f,g$ in the basis, $\langle f, g \rangle = 0.$ I'm not sure how to relate this knowledge to $D$. I believe this is somehow related to Fourier series convergence theorem.
Any help is greatly appreciated!
An eigenbasis is a basis of vectors which are eigenvectors of the differentiation operator $D$. So if you think the set $S=\{1,\sin(x),...\}$ could be the orthogonal eigenbasis you're looking for, then you should check that $D\sin(3x)=\lambda \sin(3x)$ for some scalar $\lambda$ (and similarly for all the other functions in $S$). It also needs to be a basis, which means linearly independent (this is related to orthogonality) and spanning. Here, spanning is equivalent to showing if $h$ is any function in $V$ (note that this is the entire vector space) and $\langle f,h\rangle=0$ for all $f\in S$, then $h$ must be identically 0 (this part should make use of Fourier convergence). Now, try filling in the details.