\magnification=\magstep1
\def\sinc{{\rm sinc}}
\def\FT{{\cal F}}
\centerline{Mathematics 174 -- Applied Mathematics 2}
\centerline{Discussion 5 -- Properties of the Fourier Transform}
\centerline{March 28, 2000}
\bigskip
\noindent
{\it Background}
\bigskip
We have introduced the Fourier Transform of a complex-valued $f(x)$ on ${\bf 
R}$:
$$\FT[f](\omega) = {1\over 2\pi}\int_{-\infty}^\infty f(x) 
e^{i\omega x}\ dx.$$
If $f$ is piecewise continuous and in $L^1({\bf R})$ (that is,
if $\int_{-\infty}^\infty |f(x)|\ dx < \infty$), then the Fourier
transform of $f$ is another well-defined function on ${\bf R}$.

For instance, we have seen that if 
$$f(x) = \cases{1 & if $x \in [-a,a]$\cr
                0 & otherwise\cr}$$
then 
$$\FT[f](\omega) = {\sin(a\omega)\over \pi\omega}.$$
The function ${\sin(t)\over \pi t}$ is often called $\sinc(t)$,
so the above formula for $\FT[f](\omega)$ could also be written as
$$\FT[f](\omega) = a \sinc(a\omega).$$
Also, if $f(x) = e^{-x^2}$ is the ``standard'' Gaussian,
then
$${\cal F}[f](\omega) = {1\over \sqrt{4\pi}} e^{-\omega^2/4}.$$

In addition, we have seen the general Shift Property of the Fourier Transform:
If $h(x) = f(x+a)$ is a horizontal shift of $f$, then 
$$\FT[h](\omega) = e^{-ia\omega} \FT[f](\omega).$$
\bigskip
\noindent
{\it Comment:}  In
communications and electrical engineering, the process of 
multiplying a function $g(t)$ by an exponential $e^{\pm iat}$ is called 
{\it modulation}.  It corresponds to the idea of {\it amplitude modulation}
(AM) for radio transmissions, where the signal $g(t)$ would be multiplied
onto a ``carrier wave'' $\cos(at)$ (the real part of $e^{iat} = \cos(at) + i \sin(at)$)
before it is transmitted as $g(t) \cos(at)$.
\bigskip
\noindent
{\it Discussion Questions}
\bigskip
\item{A)}  Consider the function 
$f(x) = e^{-|x|}$.  Show that $f$ is piecewise continuous and in 
$L^1({\bf R})$, and compute its Fourier Transform.
\bigskip
\item{B)}  Determine the following additional properties of the Fourier
Transform and prove your statements.
\itemitem{1)}  What is the 
relationship between $\FT[cf + g](\omega)$, $\FT[f](\omega)$, and $\FT[g](\omega)$?
\itemitem{2)}  What is the relationship between
the Fourier Transform of $f(x)$ and the Fourier transform of 
a {\it modulation} of $f(x)$, $h(x) = e^{iax}f(x)$?  
\itemitem{3)}  What is the relationship between
the Fourier Transforms of $f(x)$ and a {\it horizontally 
scaled version} of $f(x)$, $h(x) = f(cx)$?  Be careful about the sign of
$c$!
\bigskip
\item{C)}  If $f$ is a real-valued function, then the real and imaginary
parts of its Fourier Transform are 
$${1\over 2\pi}\int_{-\infty}^\infty f(x)\cos(\omega x)\ dx,  \qquad
{1\over 2\pi}\int_{-\infty}^\infty f(x)\sin(\omega x)\ dx$$
respectively.  Give a sufficient condition for the Fourier Transform of $f$
to be real (imaginary part zero).  Similarly, give a sufficient condition for 
the Fourier Transform of $f$ to be purely imaginary (real part zero).
\bigskip
\noindent
{\it Assignment}
\bigskip
Group write-ups due next class, Thursday, March 30.

\bye