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{ \large \textbf{\textsc{Math 5588 -- Homework 5 (Due Thursday February 23)}}}
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\begin{enumerate}
\item Properties of the convolution.
\begin{enumerate}
\item Show that $(f*g)' = f'*g = f*g'$.
\item Show that $f*(g*h) = (f*g)*h$.
\end{enumerate}
\item Compute the convolution of a Gaussian kernel $u(x) = e^{-x^2/2}$ with itself. [Hint: Use the Fourier transform]
\item Find the Fourier transform of the box function
\[u(x) = \begin{cases}
1,&\text{if } |x| < 1\\
0,&\text{if } |x| > 1.
\end{cases}\]
\item Use the Fourier transform to solve the damped heat equation $u_t -u_{xx} + u = 0$ with initial condition $u(x,0)=f(x)$.
\item Write out Plancheral's identity for the box function from problem 3. What is
\[\int_0^\infty \frac{\sin^2x}{x^2}\,dx \text{?}\]
\item \textbf{Nyquist-Shannon sampling theorem}: Let $f$ be an integrable function whose Fourier transform $\hat{f}$ vanishes outside of $(-\Omega,\Omega)$\footnote{This means $\hat{f}(k) =0$ for $|k|\geq \Omega$.}.  
\begin{enumerate}
\item Show that when $\ell \geq \Omega$
\begin{equation}\label{eq:Fsampled}
\frac{1}{\sqrt{2\pi}}\hat{f}(k) = \frac{1}{2\ell} \sum_{n=-\infty}^\infty f\left(\tfrac{n\pi}{\ell}\right) e^{-in\pi k/\ell} \ \ \text{ for } -\ell \leq k \leq \ell.
\end{equation}
[Hint: Write down the complex form of the Fouier series for $\hat{f}$ on the interval $(-\ell,\ell)$. Simplify the coefficients with the inverse Fourier transform formula. You may assume that the Fourier series converges.]
\item Show that when $\ell \geq \Omega$
\begin{equation}\label{eq:fsampled}
f(x) = \sum_{n=-\infty}^\infty f\left(\tfrac{n\pi}{\ell}\right) \text{sinc}\left(\ell x - n \pi\right) \ \ \text{ for all } x \in \R,
\end{equation}
where $\text{sinc}(x):=\sin(x)/x$ for $x\neq 0$, and $\text{sinc}(0):=1$. [Hint: Express $f$ via the inverse Fourier transform formula and substitute the expression \eqref{eq:Fsampled} for $\hat{f}$. You may exchange the infinite summation and integral without justification.]
\item Part (b) shows that $f$ can be exactly reconstructed from evenly spaced samples $f\left(\tfrac{n\pi}{\ell}\right)$ for $n \in \Z$ provided $\ell \geq \Omega$. Find an expression for the \emph{sampling frequency} $f_s$ in terms of $\ell$ (use units of \# samples per unit length). The \emph{bandwidth} of $f$, denoted $f_b$, is the highest frequency (in units of cycles per unit length) present in the Fourier representation 
\[f(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty \hat{f}(k) e^{ikx} \, dk.\]
Since $\hat{f}$ vanishes outside of $(-\Omega,\Omega)$, $f_b$ is the frequency of the complex wave
\[e^{\pm i\Omega x} = \cos(\Omega x) \pm i\sin(\Omega x).\]
Find an expression for $f_b$, and show that $\ell \geq \Omega$ is equivalent to $f_s \geq 2f_b$. [This is the famous Nyquist-Shannon sampling theorem: A signal can be exactly reconstructed from evenly spaced samples provided the sampling frequency is at least twice the bandwidth of the signal. The reconstruction formula is given in \eqref{eq:fsampled}, and is often called \emph{sinc interpolation}.]
\end{enumerate}
\end{enumerate}

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