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\centerline{Mathematics 174 -- Applied Mathematics 2}
\centerline{Problem Set 5}
\centerline{March 30, 2000}
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I.  Compute the Fourier Transforms of each of the
following functions using the indicated methods)
\item{A)}  (by direct computation)
$$f(x) = \cases{1 & if $|x| \le 1$\cr
                2 & if $1 < |x| < 2$\cr
                0 & otherwise\cr}$$
\item{B)}  (by direct computation)
$$f(x) = \cases{ \sin(x) & if $|x| \le {\pi\over 2}$\cr
                    0    & otherwise\cr}$$
\item{C)}  Show that if $xf(x)$ is in $L^1({\bf R})$, then
$${\cal F}[xf(x)](\omega) = -i {d\over d\omega}{\cal F}[f](\omega)$$
and use that to evaluate the Fourier Transform of $f(x) = x e^{-x^2}$.
\item{D)} (by using the general scaling and modulation properties)
$$f(x) = \cos(4x) e^{-3x^2}$$
({\it Answer:} ${\sqrt {3}\over 12\sqrt{\pi}} \left ({e^{-\left (\omega+4\right )^{2
}/12}}+{e^{-\left (\omega-4\right )^{2}/12}}\right)$)
\item{E)}  (by using the Fourier Inversion theorem and 
a known transform)  $f(x) = {1\over 1+ x^2}$
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II.  For real numbers $a$ and $c$, let 
$$g_{a,c} = {1\over \sqrt{2\pi a^2}} e^{-(x-c)^2/(2a^2)}$$
(a scaled shifted Gaussian).
\item{A)}  What are the meanings of the constants $a,c$ in 
terms of the ``bell-shaped'' curve $y = g_{a,c}(x)$?
\item{B)}  Show that if $a,c,b,d$ are any real numbers, then
$$g_{a,c}(x) * g_{b,d}(x) = g_{\delta,c+d}(x)$$
where $*$ = convolution, and $\delta = \sqrt{a^2 + b^2}$.  
(Hint:  Take the Fourier transform of the convolution, work with what
you get, then take the inverse transform.)
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III.  Let $f : {\bf R} \to {\bf C}$ be an $L^1$ function, and let
$F$ denote its Fourier transform.  Show that if $F(\omega) = 0$
for all $|\omega| > \omega_0$, then 
$$f(x) * {\sin(ax)\over\pi x} = f(x)$$
for all $a > \omega_0$.
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{\it Due:} In class, Thursday, April 6.
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