We have $\begin{align} S_{n}(\phi)(x) &=\sum_{k=-n}^n e^{ikx} \hat{\phi}(k) \\ &= \sum_{k=-n}^n \frac{1}{2\pi} \int_{\pi}^\pi e^{ik(x-z)}\phi(z) \, dz \\ &= \sum_{l=0}^{2n} \frac{1}{2\pi} \int_{-\pi}^\pi e^{i(l-n)(x-z)} \phi(z) \, dz \\ &= \frac{1}{2\pi} \int_{\pi}^\pi e^{-in(x-z)} \sum_{l=0}^{2n} e^{il(x-z)} \phi(z) \, dz \end{align}$Since $\sum_{l=0}^{2n} e^{il(x-z)}$ is a geometric series, we have $\begin{align} S_{n}(\phi)(x) &= \frac{1}{2\pi} \int_{\pi}^\pi e^{-in(x-z)} \left( \frac{1-e^{i(x-z)(2n+1)}}{1-e^{i(k-z)}} \right) \phi(z) \, dz \\ &= \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{\left( \sin\left( n+\frac{1}{2} \right) \sin(x-z) \right)}{\sin\left( \frac{1}{2}(x-z) \right)} \phi(z) \\ &= \int_{-\pi}^\pi K_{n} (x-z) \phi(x) \, dz \end{align}$where $K_{n}(\theta):=\frac{\left( \sin\left( n+\frac{1}{2} \right) \sin(\theta) \right)}{\sin\left( \frac{1}{2}(\theta) \right)}$ is known as the [[Dirichlet Kernel|Dirichlet kernel]]