I've been asked to check the uniform convergence of the following function sequence on the real line: $$ f_{N}(t)=\sum_{n=-N}^{n=N}\sin(n) \,\frac{\sin(\pi t-\pi n)}{\pi t-\pi n} $$ It is asked in a course of Fourier Analysis. I've managed to prove it converges pointiwse to $ \sin(t) $ using the fourier series of $\sin(tx)$ and substituting $x=1$.
However, I have failed to prove/disprove uniform convergence. Using wolfram alpha I could see it does not uniformly converge. I took $N=20$ and $ x=20\pi + \pi /2 $, and the sum turned out to be roughly zero where it should have been roughly $1$.
[1] I assume that the term in which $t=n$,if it exists, equals $\sin t$...[2] Observe that $\sin (\pi t- \pi n)=(-1)^n \sin \pi t$ for integer $n$...[3] Assume uniform convergence. Then for some $N > |t|$ we have $|f_{N+1}(t)-f_N(t)| < 1/2$ for all $t$. But for $N>|t|$, we have $|f_{N+1}(t)-f_N(t)|=|[(2N+2) \sin \pi t ]/[\pi (t^2-(N+1)^2]|$ which for $t=N+1+d \ne N+1$ is $| [\sin \pi d)/ \pi d].[(2N+2)/(2N+2+d)]|$ which is arbitrarily close to $1$ if $|d|$ is small enough.Contradiction, so convergence is not uniform. (Question: Is convergence uniform on every bounded set? I dk.)