Definition 6.2.1.
The Fejér kernel \(K_m:[-\pi, \pi]\to \C\) is given by
\begin{equation}
K_m(t) = \frac{1}{m+1} \sum_{j=0}^m \sum_{n=-j}^j e^{int}.\tag{6.2.1}
\end{equation}
Section 6.2 The Fejér kernel
Our goal is to establish that classical Fourier series converge, so we need a tool that involves the terms of such series. We’re going to be constructing functions by convolution with a function called the Fejér kernel, which is built out of the same complex exponentials that appear in classical Fourier series.
1
en.wikipedia.org/wiki/Fej%C3%A9r_kernelEvery good proof hinges on a set of technical lemmas. Here are those that we need to prove that Fourier series converge.
Lemma 6.2.2.
For any \(t \in \R\) that is not an integer multiple of \(2\pi\text{,}\)
\begin{equation}
K_m(t) = \frac{1}{m+1} \frac{\sin^2(\frac{(m+1)t}{2})}{\sin^2(\frac{t}{2})}.\tag{6.2.2}
\end{equation}
Proof.
Let \(z = e^{it}\) and \(\cc{z} = e^{-it}\) for \(t \notin 2\pi \mathbb{Z}\text{.}\) Then \(z \neq 1\) and
\begin{align*}
\sum_{n=-j}^je^{int} \amp= \cc{z}^j(1 + z + z^2 + \cdots + z^{2j})\\
\amp= \cc{z}^j \left(\frac{1 - z^{2j + 1}}{1 - z}\right)\\
\amp = \frac{\cc{z}^j - z^{j+1}}{1-z}.
\end{align*}
Then, by (6.2.1), we can write
\begin{equation*}
(m+1) K_m(t) = \sum_{j=0}^m \frac{\cc{z}^j - z^{j+1}}{1-z}.
\end{equation*}
Proceed by elbow grease. We calculate
\begin{align*}
(m+1) K_m(t) \amp= \sum_{j=0}^m \frac{\cc{z}^j - z^{j+1}}{1-z}\\
\amp= \frac{1}{1-z} \sum_{j=0}^m (\cc{z}^j - z^{j+1})\\
\amp= \frac{1}{1-z} (\sum_{j=0}^m \cc{z}^j - \sum_{j=0}^m z^{j+1})\\
\amp= \frac{1}{1-z} (\frac{1-\cc{z}^{m+1}}{1-\cc{z}} - \frac{z(1 - z^{m+1})}{1-z}\\
\amp= \frac{1}{1-z} \left(\frac{1-\cc{z}^{m+1}}{1-\cc{z}} - \frac{z(1 - z^{m+1})}{1-z}\frac{\cc{z}}{\cc{z}}\right)\\
\amp= \frac{1}{1-z} \left(\frac{1-\cc{z}^{m+1}}{1-\cc{z}} + \frac{1 - z^{m+1}}{1-\cc{z}}\right)\\
\amp= \frac{1}{(1-z)(1-\cc{z})} \left(-\cc{z}^{m+1} - z^{m+1}+ 2\right)\\
\amp= \frac{1}{\abs{1-z}^2} \left(-\cc{z}^{m+1} - z^{m+1}+ 2\right).
\end{align*}
Now, we are ready to convert these piles of \(z = e^{it}\) into sines. First, note that
\begin{equation*}
\abs{1 - z} = \abs{1 - e^{it}} = \abs{e^{it/2}}\abs{e^{-it} - e^{it}} = 2\abs{\sin(t/2)},
\end{equation*}
and so \(\abs{1 - z}^2 = 4\sin^2(t/2)\text{.}\) For the term in the numerator, notice that
\begin{align*}
z^{m+1} - 2 + \cc{z}^{m+1} \amp= e^{i(m+1)t} - 2 + e^{-i(m+1)t}\\
\amp= (e^{i(m+1)t/2} - e^{-i(m+1)t/2})^2\\
\amp= -4 \sin^2((m+1)t/2).
\end{align*}
Then putting the pieces together, we conclude that
\begin{align*}
(m+1) K_m(t) \amp= \frac{1}{\abs{1-z}^2} \left(-\cc{z}^{m+1} - z^{m+1}+ 2\right)\\
\amp= \frac{4\sin^2((m+1)t/2)}{4\sin^2(t/2)},
\end{align*}
and so
\begin{equation*}
K_m(t) = \frac{1}{m+1} \frac{\sin^2((m+1)t/2)}{\sin^2(t/2)}.
\end{equation*}
We can now establish some of the good properties of the Fejér kernel that make it useful in applications involving Fourier series. Essentially, we want to show that \(K_m\) has properties similar to the function \(g_\delta\) that we explored in the previous section.
Lemma 6.2.3.
The function \(K_m\) satisfies the following properties:
- \begin{equation*} K_m(t) \geq 0, \hspace{1cm} \forall t \in \R, m \in \mathbb{Z}^{\geq 0}\text{;} \end{equation*}
- \begin{equation*} \int_{-\pi}^{\pi} K_m(t)\, dt = 2\pi, \hspace{1cm} \forall m \in \mathbb{Z}^{\geq 0}; \end{equation*}
- For all \(\delta\) with \(0 \lt \delta \lt \pi\text{,}\)\begin{equation*} \int_{-\pi}^{-\delta} + \int_{\delta}^{\pi} K_m(t) \, dt \to 0 \text{ as } m \to \infty. \end{equation*}
Proof.
Property 1 follows directly from the form of \(K_m\) given in (6.2.2) for any \(t\) not an integer multiple of \(2\pi\) and at those points by continuity.
To see property 2, observe that
\begin{equation*}
\int_{-\pi}^\pi e^{int}\, dt = \begin{cases} 2\pi \amp \text{ if } n = 0 \\ 0 \amp \text{ if } n \neq 0 \end{cases}.
\end{equation*}
Since \(K_m(t) = \frac{1}{m+1} \sum_{j=0}^m \sum_{n=-j}^j e^{int}\text{,}\) expanding the sum and integrating gives
\begin{equation*}
\int_{-\pi}^\pi K_m(t) \, dt = \frac{1}{m+1} (m+1) 2\pi = 2\pi.
\end{equation*}
The third property leverages the characterization of \(K_m\) in terms of sine functions. Suppose \(0 \lt \delta \lt \pi\text{.}\) If \(-\pi \lt t \lt \delta\) or \(\delta \lt t \lt \pi\text{,}\) then
\begin{equation*}
0 \lt \sin^2(t/2) \lt \sin^2(\delta/2).
\end{equation*}
By Lemma 6.2.2,
\begin{align*}
K_m(t) \amp= \frac{1}{m+1} \frac{\sin^2((m+1)t/2)}{\sin^2(t/2)}\\
\amp\leq \frac{1}{m+1} \frac{1}{\sin^2(t/2)}\\
\amp\leq \frac{1}{m+1} \frac{1}{\sin^2(\delta/2)}\\
\amp= \frac{1}{m+1} \csc^2(\delta/2),
\end{align*}
which notably does not depend on \(t\text{.}\) Then we can estimate the integral by
\begin{equation*}
0 \leq \int_{-\pi}^{-\delta} + \int_\delta^\pi K_m(t) \leq 2\pi \frac{1}{m+1} \csc^2(\delta/2),
\end{equation*}
and for any fixed \(\delta\text{,}\)
\begin{equation*}
2\pi \frac{1}{m+1} \csc^2(\delta/2) \to 0, \text{ as } m \to \infty,
\end{equation*}
which establishes property 3.
The content of these Lemmas is that the kernel \(K_m\) has the properties that will make it well-behaved in a convolution.
