(1)

\begin{align*} D_{n}\left(-x\right) & =\sum_{k=-n}^{n}e^{-ikx}\\ & =\sum_{k=-n}^{n}e^{ikx}\\ & =D_{n}\left(x\right) \end{align*}

(2)

\begin{align*} D_{n}\left(x+2\pi\right) & =\sum_{k=-n}^{n}e^{-ik\left(x+2\pi\right)}\\ & =\sum_{k=-n}^{n}e^{-ikx}\\ & =D_{n}\left(x\right) \end{align*}

(3)

\begin{align*} \int_{0}^{\pi}D_{n}\left(x\right)dx & =\int_{0}^{\pi}\left\{ 1+2\sum_{k=1}^{n}\cos\left(kx\right)\right\} dx\\ & =\left[x+2\sum_{k=1}^{n}\frac{1}{k}\sin\left(kx\right)\right]_{0}^{\pi}\\ & =\pi \end{align*}

(4)

櫛型関数のフーリエ級数展開より、
\begin{align*} \lim_{n\rightarrow\infty}D_{n}\left(x\right) & =\sum_{k=-\infty}^{\infty}e^{ikx}\\ & =2\pi\cdot\frac{1}{2\pi}\sum_{k=-\infty}^{\infty}e^{2\pi i\frac{k}{2\pi}x}\\ & =2\pi\mathrm{comb}_{2\pi}\left(x\right) \end{align*} となるので与式は成り立つ。