have problem in understanding the fourier series and eigenfunction expansion

Question

Basically, I do not quite understand what does M look like and how does this textbook "easily verify the eigenvalues and eigenfunctions of M"

Answer 1

By definition, $M$ is the map $$ u \mapsto -u'' $$ defined on $W$. To check the eigenvectors and eigenvalues, observe that $$ M\left(\cos\left(\frac{n\pi x}{L}\right)\right) = -\left(\cos\left(\frac{n\pi x}{L} \right)\right)'' = \left(\frac{n \pi}{L}\right)^2 \cos\left(\frac{n\pi x}{L}\right) $$ Likewise, you can check that $$ M\left(\sin\left(\frac{n\pi x}{L}\right)\right) = \left(\frac{n \pi}{L}\right)^2 \sin\left(\frac{n\pi x}{L}\right) $$ In each case, we have verified that the function is an eigenvector with eigenvalue $\left(\frac{n\pi}{L}\right)^2$.

That these functions cover all of the eigenvectors of $M$ is a little less trivial.