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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
 1 
en.wikipedia.org/wiki/Fej%C3%A9r_kernel
, which is built out of the same complex exponentials that appear in classical Fourier series.

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}
Every good proof hinges on a set of technical lemmas. Here are those that we need to prove that Fourier series converge.

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.

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*}
\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.