I am trying to approximate sign$(K_n(t))$, where $$K_n(t)=\frac{\text{sin}((n+\frac{1}{2})t)}{\text{sin}(\frac{1}{2}t)}$$ by a sequence of functions from $$P=\lbrace f\in C([-\pi,\pi];\mathbb{R}):f(-\pi)=f(\pi)\rbrace$$ with $\operatorname{sign}(t)=1$ for $t>0$, $\operatorname{sign}(t)=-1$ for $t<0$, and $\operatorname{sign}(0)=0$.
I am using old undergraduate lecture notes to prove a result about Fourier series, but the sequence of functions used for this was given in a separate document and is now unobtainable.
Any ideas as to what sort of sequence of functions I should be trying would be much appreciated.
EDIT: I need a sequence $f_k\in P$ with $||f_k-K_n||_\infty\to 0$ as $k\to \infty$, $n$ fixed.
You will not get uniform convergence for $sign(K_n)$, simply because uniform convergence preserves continuity (the $f_k$ are continuous and $sign(K_n)$ is not).
On the other hand $K_n\in P$, so there is nothing to approximate.