Dirichlet's Kernel

# Dirichlet's Kernel

 Definition: Dirichlet's Kernel is the collection of functions $D_n$ where for each $n \in \mathbb{N}$, $\displaystyle{D_n(t) = \frac{1}{2} + \sum_{k=1}^{n} \cos k t}$.

An alternative formula for the functions $D_n$ in Dirichlet's Kernel is given for each $n \in \mathbb{N}$ piecewise by:

(1)
\begin{align} \quad D_n(t) = \left\{\begin{matrix} n + \frac{1}{2} & \mathrm{if} \: t = 2m\pi, m \in \mathbb{Z} \\ \frac{\sin \left ( \left ( n + \frac{1}{2} \right ) t \right )}{2 \sin \left ( \frac{t}{2} \right )} & \mathrm{if} \: t \neq 2m\pi, m \in \mathbb{Z}\\ \end{matrix}\right. \end{align}

To verify this we see that for $t \neq 2m\pi$, $m \in \mathbb{Z}$ (so that we're not multiplying both sides by $0$) that:

(2)
\begin{align} \quad D_n(t) & = \frac{1}{2} + \sum_{k=1}^n \cos kt \\ \quad 2 \sin \left ( \frac{t}{2} \right ) D_n(t) & = \sin \left ( \frac{t}{2} \right ) + \sum_{k=1}^{n} 2 \cos kt \sin \left ( \frac{t}{2} \right ) \\ \end{align}

We will use the following trigonometric identity:

(3)
\begin{align} \quad 2\cos a \sin b = \sin (a + b) - \sin (a - b) \end{align}

Hence we have that:

(4)
\begin{align} \quad 2 \sin \left ( \frac{t}{2} \right ) D_n(t) &= \sin \left ( \frac{t}{2} \right ) + \sum_{k=1}^{n} [\sin \left ( kt + \frac{t}{2} \right ) - \sin \left ( kt - \frac{t}{2} \right )] \\ &= \sin \left ( \frac{t}{2} \right ) + \sin \left ( nt + \frac{t}{2} \right ) - \sin \left ( t - \frac{t}{2} \right ) \\ &= \sin \left ( \frac{t}{2} \right ) + \sin \left ( \left ( n + \frac{1}{2} \right ) t \right ) - \sin \left ( \frac{t}{2} \right ) \\ &= \sin \left ( \left ( n + \frac{1}{2} \right ) t \right ) \end{align}

Therefore:

(5)
\begin{align} \quad D_n(t) &= \frac{\sin \left ( \left ( n + \frac{1}{2} \right ) t \right )}{2 \sin \left ( \frac{t}{2} \right )} \end{align}

And for points $t = 2m\pi$ where $m \in \mathbb{Z}$ we define $D_n(t) = n + \frac{1}{2}$ so that $D_n$ is defined for all $t \in \mathbb{R}$.

The graphs of $D_1$ (red), $D_2$ (yellow), $D_3$ (green), and $D_4$ (blue) are given below:

We will now prove some elementary properties of the functions in Dirichlet's kernel.

 Theorem 1: For all $n \in \mathbb{N}$: a) $D_n$ is an even function. b) $D_n$ is $2\pi$-periodic. c) $\displaystyle{\int_0^{2\pi} D_n(t) \: dt = \pi}$.
• Proof of a) The functions $\cos t$, $\cos 2t$, …, $\cos nt$ are even for all $n \in \mathbb{N}$, so the sum, $\displaystyle{\sum_{k=1}^{n} \cos kt}$ is also even. Moreover, any vertical translation of a function is even, so indeed $D_n$ is even. $\blacksquare$
• Proof of b) Each of $\cos t$, $\cos 2t$, …, $\cos nt$ are $2\pi$-periodic and any vertical translation of a $2\pi$-periodic function is $2\pi$-periodic so $D_n$ is $2\pi$-periodic. $\blacksquare$
• Proof of c) For each $n \in \mathbb{N}$ we have that:
(6)
\begin{align} \quad \int_0^{2\pi} D_n(t) \: dt &= \int_0^{2\pi} \left ( \frac{1}{2} + \sum_{k=1}^n \cos kt \right ) \: dt \\ &= \int_0^{2\pi} \int_0^{2\pi} \left ( \frac{1}{2} + \cos t + \cos 2t + ... + \cos nt \right ) \: dt \\ &= \left [ \frac{1}{2}t + \sin t + \frac{1}{2} \sin 2t + ... + \frac{1}{n} \sin nt \right ]_{0}^{2\pi} \\ &= \left (\frac{1}{2} \cdot 2 \pi + \sin 2 \pi + \frac{1}{2} \sin 4 \pi + ... + \frac{1}{n} \sin 2n\pi \right ) - \left ( \frac{1}{2} \cdot 0 + \sin 0 + \frac{1}{2} \sin 0 + ... + \frac{1}{n} \sin 0 \right ) \\ &= \pi \quad \blacksquare \end{align}