Fejer's Kernel Rep. of the Arith. Means of the Part. Sums of a F. Series

Fejer's Kernel Representation of the Arithmetic Means of the Partial Sums of a Fourier Series

Recall from the Dirichlet's Kernel Representation of the Partial Sums of a Fourier Series page that if $f \in L([0, 2\pi])$ is a $2\pi$-periodic function and if for each $n \in \mathbb{N}$, $\displaystyle{s_n(x) = \frac{a_0}{2} + \sum_{k=1}^{n} (a_n\cos nx + b_n \sin nx)}$ is the $n^{\mathrm{th}}$ partial sum of the Fourier series generated by $f$ then an integral representation for $s_n(x)$ is:

(1)

\begin{align} \quad s_n(x) = \frac{2}{\pi} \int_0^{\pi} \frac{f(x + t) + f(x - t)}{2} D_n(t) \: dt \end{align}

The function $D_n(t)$ is defined for each $n \in \mathbb{N}$ below and the collection of such functions is called Dirichlet's kernel::

(2)

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

We will now determine a nice integral representation for the arithmetic means $\sigma_n$ of the partial sums where for each $n \in \mathbb{N}$:

(3)

\begin{align} \quad \sigma_n(x) = \frac{s_0 + s_1 + ... + s_{n-1}}{n} \end{align}

Theorem 1: Let $f \in L([0, 2\pi])$ be a $2\pi$-periodic function and let $\displaystyle{s_n(x) = \frac{a_0}{2} + \sum_{k=1}^{n} (a_n \cos nx + b_n \sin nx}$ be the $n^{\mathrm{th}}$ partial sum of the Fourier series generated by $f$. Then for each $n \in \mathbb{N}$, the arithmetic mean $\sigma_n$ has an integral representation as $\displaystyle{\sigma_n(x) = \frac{1}{n \pi} \int_0^{\pi} \frac{f(x + t) + f(x - t)}{2} \frac{\sin^2 \left ( \frac{1}{2} nt \right )}{\sin^2 \left ( \frac{1}{2} t \right )} dt}$.

The collection of functions $F_n(t) = \frac{1}{n} \frac{\sin^2 \left ( \frac{1}{2} n t \right )}{\sin^2 \left ( \frac{1}{2} t \right )}$ for $n \in \mathbb{N}$ are often called Fejer's Kernel in which case the formula in the theorem above can be rewritten as:

(4)

\begin{align} \quad \sigma_n(x) = \frac{1}{\pi} \int_0^{\pi} \frac{f(x + t) + f(x - t)}{2} F_n(t) \: dt \end{align}

• Proof: For each $n \in \mathbb{N}$ we have that $\displaystyle{s_n(x) = \frac{2}{\pi} \int_0^{\pi} \frac{f(x + t) + f(x - t)}{2} D_n(t) \: dt}$ and so substituting this into the equation for $\sigma_n(x)$ gives us:

(5)

\begin{align} \quad \sigma_n(x) &= \frac{s_0 + s_1 + ... + s_{n-1}}{n} \\ &= \frac{1}{n} (s_0 + s_1 + ... + s_{n-1}) \\ &= \frac{1}{n} \left ( \frac{2}{\pi} \int_0^{\pi} \frac{f(x + t) + f(x - t)}{2} \left [ \sum_{k=0}^{n-1} D_k(t) \: dt \right ] \right ) \\ &= \frac{1}{n\pi} \int_0^{\pi} \frac{f(x + t) + f(x - t)}{2} \sum_{k=0}^{n-1} \frac{\sin \left ( \left ( k + \frac{1}{2} \right ) t \right )}{\sin \left ( \left ( \frac{1}{2} \right ) t \right )} \: dt \\ &= \frac{1}{n\pi} \int_0^{\pi} \frac{f(x + t) + f(x - t)}{2} \frac{\sin^2 \left ( \frac{1}{2} nt \right )}{\sin^2 \left ( \frac{1}{2} t \right )} \: dt \quad \blacksquare \end{align}