Pointwise Convergence of Fourier Series Review

Pointwise Convergence of Fourier Series Review

We will now review some of the recent material regarding pointwise convergence of Fourier series.

• Recall from the Periodic Functions page that a function $f$ is said to be $p$-Periodic ($p > 0$) if the following equality holds for all $x \in D(f)$:
(1)
\begin{align} \quad f(x + p) = f(x) \end{align}
• We noted that the trigonometric functions $\sin x$, $\cos x$, and $\tan x$ are all $2\pi$-periodic functions. We also looked at some examples of functions with different periods.
• On the Dirichlet's Kernel page we looked at a special collection of functions known as Dirichlet's Kernel where for each $n \in \mathbb{N}$, the function $D_n$ is defined as:
(2)
\begin{align} \quad D_n(x) = \frac{1}{2} + \sum_{k=1}^{n}\cos kt \end{align}
(3)
\begin{align} \quad D_n(x) = \left\{\begin{matrix} n + \frac{1}{2} & \mathrm{if} \: x = 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} \: x \neq 2m \pi, m \in \mathbb{Z} \\ \end{matrix}\right. \end{align}
• The graphs of the first four functions $D_1$, $D_2$, $D_3$, and $D_4$ of Dirichlet's kernel are graphed below:
• We then looked at some important properties of the functions in Dirichlet's Kernel. We first noted that for all $n \in \mathbb{N}$, $D_n$ is an even function since each $D_n$ is a finite sum of even functions. Also, each $D_n$ is $2\pi$-periodic. Lastly, we noted that for each $n \in \mathbb{N}$:
(4)
\begin{align} \quad \int_0^{2\pi} D_n(t) \: dt = \pi \end{align}
• On the Dirichlet's Kernel Representation of the Partial Sums of a Fourier Series we began to investigate the usefulness of Dirichlet's kernel. We proved a very important result which said that if $f \in L([0, 2\pi])$ is a $2\pi$-periodic function, then if for each $n \in \mathbb{N}$ we let $s_n$ be the $n^{\mathrm{th}}$ partial sum of the Fourier series of $f$ generated by the trigonometric system, then we can actually obtain an integral representation for $s_n(x)$, namely:
(5)
\begin{align} \quad s_n(x) = \frac{2}{\pi} \int_0^{2\pi} \frac{f(x + t) + f(x - t)}{2} D_n(t) \: dt \end{align}
• On the The Riemann Localization Theorem page we used this result to prove a strong result regarding convergence of a Fourier series. We proved that if $f \in L([0, 2\pi])$ is a $2\pi$-periodic function then the Fourier series of $f$ generated by the trigonometric system converges at a point $x$ if and only if there exists a $b \in \mathbb{R}$ with $0 < b \leq \pi$ for which the following limit exists:
(6)
\begin{align} \quad \lim_{n \to \infty} \frac{2}{\pi} \int_0^{b} \frac{f(x + t) + f(x - t)}{2} \frac{\sin \left (\left ( n + \frac{1}{2} \right ) t \right )}{t} \: dt \end{align}
• If this limit does exist for some $x$, then the Fourier series of $f$ generated by the trigonometric series will converge at $x$ to this limit.
(7)
\begin{align} \quad g(t) = \frac{f(x + t) + f(x - t)}{2} \end{align}
• So by Jordan's Theorem we obtain Jordan's test for convergence which says that if $f \in L([0, 2\pi])$ is a $2\pi$-periodic function and if $f$ is of bounded variation on $[x - \delta, x + \delta]$ for some $\delta$ with $0 < \delta \leq b$ then the Fourier series of $f$ generated by the trigonometric system will converge at $x$ to $\displaystyle{g(0+) = \lim_{t \to 0+} \frac{f(x + t) + f(x - t)}{2}}$.
• Also, by Dini's Theorem we obtain Dini's test for convergence which says that if $f \in L([0, 2\pi])$ is a $2\pi$-periodic function and $g(0+)$ exists and Lebesgue integral $\displaystyle{\int_0^{\delta} \frac{g(t) - g(0+)}{t} \: dt}$ exists for some $\delta$ with $0 < \delta < \pi$, then the Fourier series of $f$ generated by the trignonometric system will converge at $x$ to $\displaystyle{g(0+) = \lim_{t \to 0+} \frac{f(x + t) + f(x - t)}{2}}$.