I am trying to understand the equations below. What seems to be confusing me is the use of two variables $n, m$ in the delta function: $\delta_{n,m}$. I understand that d_n would just be the spoke at x=n, but I have no idea how to use m.
I hope the question makes sense. I'm rather new to this so excuse the potentially basic question.
This is actually the Kronecker delta, as @ShubhamJohri defined. You can prove these equations by writing each integrand as half a sum or difference. (Note these equations implicitly assume $m,\,n$ are integers.) For example, the second equation admits the proof$$\begin{align}\int_0^T\tfrac12\left[\cos\tfrac{\pi(n-m)t}{T}-\cos\tfrac{\pi(n+m)t}{T}\right]dt&=\tfrac{1}{2\pi}\left[\tfrac{\pi t}{T}\operatorname{sinc}\tfrac{\pi(n-m)t}{T}-\tfrac{T}{n+m}\sin\tfrac{\pi(n+m)t}{T}\right]_0^T\\&=\tfrac12T\delta_{nm}-0,\end{align}$$where $\operatorname{sinc}$ is the unnormalized sinc function. In particular, it helps to evaluate the leftmost contribution separately in the cases $m=n,\,m\ne n$.