trigonometric series

Question

It is known that the eigenvalues of Sturm liouville problem: $$ u''(x)+\lambda u(x)=0 \\ u(0)=u'(\pi)=0 $$ are $\sin\left(\left(\frac{1}{2}+n\right)x\right)$ for $n=0,1...$ If for example we expand the function $x$ with respect to these eigenfunctions we get a series $$ \sum_{n=0}^{\infty}\frac{8(-1)^n}{\pi(2n+1)^2} \, \sin\left(\left(\frac{1}{2}+n\right)x\right) $$ My question is : is there any connection of the latter series to some standard fourier expansion on any interval (via extension to other segment or something similar) ?

I thought that all trigonometric expansions are related to some fourier series

thank you

Answer 1

I'm not aware of any relation to standard trigonometric series. There are trigonometric series expansions that do not have evenly spaced eigenvalues, and those definitely are not going to be related. Fourier considered such an example in his original Treatise on Heat Conduction from 1807: $$ X''(x)+\lambda^2 X(x) = 0,\;\;\;\; 0 \le x \le r.\\ X(0)=0,\;\; \left(h-\frac{1}{r}\right)X(r)+X'(r)=0. $$ The constant $h > 0$ is given. The eigenvalues are the solutions $\lambda_1 < \lambda_2 < \lambda_3 < \cdots$ of $$ \frac{\sqrt{\lambda}r}{\tan(\sqrt{\lambda}r)}=1-hr. $$ If $1-hr \ne 0$, then the eigenvalue square roots $\sqrt{\lambda_n}$ are not evenly spaced, and they're solutions of a transcendental equation. The eigenfunctions $\{ \sin(\sqrt{\lambda_n}x) \}$ are an orthogonal basis of $L^2[0,r]$.