Lecture 14 (Mar 22)

Expansions on other intervals

Given a periodic function $f(x)$ on an interval $[-L,L]$, we have
$$
f(x)= \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos \frac{n\pi x}{L}+ b_n \sin \frac{n\pi x}{L} \right]
$$
where
$$
a_n = \left<f(x),\cos \frac{n\pi x}{L} \right> \, \mbox{ and } \, b_n = \left<f(x),\sin \frac{n\pi x}{L} \right>
$$
and $<f,g>=\frac{1}{L} \int_{-L}^L f(x) g(x) dx$.

Fourier transform

$$
\mathcal{F}(f)(y)= \hat{f}(y) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i xy} dx
$$
and
$$
\mathcal{F}^{-1}(\hat{f}(x)) = f(x) = \int_{-\infty}^{\infty} \hat{f}(y) e^{2\pi i xy} dy \, .
$$

Fourier spectrum

$|\hat{f}(y)|$.

Examples

$$

f(x)=\left\{

\begin{array}{c}

A \mbox{ for $-x_0<x<x_0$} \\

0 \mbox{ otherwise.}

\end{array}

\right.

$$

$$

f(x)=\left\{

\begin{array}{c}

e^{-ax} \mbox{ for $x>0$} \\

0 \mbox{ otherwise.}

\end{array}

\right.

$$

Some difficulties: what are $\mathcal{F}(1)$ and $\mathcal{F}(\cos 2\pi x)$?

Reading Materials

Lecture Notes p.80-82