\documentclass{amsart}


\usepackage{amsmath}
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\input{macros}
\newcommand{\sfe}{{\mathsf{e}}}
\newcommand{\cF}{{\mathcal{F}}}
\newcommand{\sig}{\sigma}

\begin{document}
\noindent
\Large Math 555\hfill Winter 2015

\noindent
{\bf Homework Set 8\hfill Due Friday, March 13}
\bigskip

\begin{enumerate}
\item
Compute the Fourier series for the following functions, and discuss the convergence
of these series.
\begin{enumerate}
\item $x(t)=x(t+2\pi)$, where on the interval $[-\pi, \pi)$
\[
x(t):=
\begin{cases}
0&\quad -\pi\le t< 0,\\
\sin t&\quad 0\le t<\pi.
\end{cases}
\]
\item $x(t)=x(t+2\pi)$, where on the interval $[-\pi, \pi)$, 
$x(t)=t^2$.
\item
Show that $t^3-\pi ^2 t=12\sum_{k=1}^\infty \frac{(-1)^k}{k^3}\sin (kt)$ on $[-\pi,\pi)$.
(Hint: differentiate)
\end{enumerate}
\item
Let $\l\in \C$. Show that if there is a non-zero solution to 
$v''=-\l v$ that is $2\pi$ periodic, then $\l =n^2$ for some
$n\in\{1,2,\dots\}$.
\item Apply Parseval's relation to the function $f(x)=x$ on
$(-\pi,\ \pi)$ to evaluate $\sum_{n=1}^\infty\frac{1}{n^2}$.
\item 
How would you define the Fourier series for the periodic function
$x(t)=x(t+10)$, where in the interval $[0,10)$
%\[
%x(t):=
%\begin{cases}
%1&\quad 0\le t\le 1,\\
%0&\quad1<t<10\quad ?
%\end{cases}
%\]
\[
x(t):=
\begin{cases}
1&\quad 0\le t\le 1,\\
0&\quad 1<t<2,\\
-1&\quad 2\le t\le 3,\\
0&\quad3<t<10\qquad ?
\end{cases}
\]
Using your definition, compute the Fourier series.
(Hint: Note that $x$ is a linear combination of a function and the shift of itself.)
\item
\begin{enumerate}
\item
Show that 
$$
\set{\sqrt{\frac{2}{\pi}}\, \sin(nx)}{n=1,2,\dots}
$$
is a complete orthonormal system in $L^2(0,\pi)$.
\item
Show that if $f\in C^1[0,\pi]$ satisfies $f(0)=f(\pi)=0$,
then the expansion of $f$ interms of the orthonormal basis in a)
converges uniformly to $f$ on $[0,\pi]$ (called the Fourier sine series).
\end{enumerate}
\item
Let $\map{f}{\R}{\C}$ be integrable ($f\in L^1(-\infty,\infty)$).
The \emph{Fourier transform} (FT) of $f$ is given by
\[
\cF(f)(\xi):=\hat f(\xi):=\int_{-\infty}^\infty f(x)\sfe^{-2\pi i x\xi} dx\quad\forall\, \xi\in\R\ .
\]
\begin{enumerate}
\item
Show that $\cF$ maps $L^2(-\infty,\infty)$ to $L^2(-\infty,\infty)$.
\item
Show that $\cF$ is a unitary linear transformation on $L^2(-\infty,\infty)$ 
($\norm{\hat f(\xi)}=\norm{f}$).
\item
Show that $\cF^\star$, the adjoint of $\cF$, is given by
\[
\cF^\star(\hat f)(x):=f(x):=\int_{-\infty}^\infty \hat f(\xi)\sfe^{2\pi i x\xi} d\xi\quad\forall\, \in\R\ .
\]
\item
Let $\map{\sig_{x_0}}{L^2(-\infty,\infty)}{L^2(-\infty,\infty)}$ be the mapping
$\sig_{x_0}(f)(x):=f(x+x_0)$, and $\map{\mu_{\xi_0}}{L^2(-\infty,\infty)}{L^2(-\infty,\infty)}$ 
the mapping
$\mu_{\xi_0}(f)(x):=\sfe^{2\pi i x\xi_0}f(x)$.
Establish the following properties of $\cF$.
\begin{enumerate}
\item 
$\cF\circ \sig_{x_0}=\mu_{x_0}\circ\cF$.
%$\cF(f(\cdot -x_0))(\xi)=\e^{-2\pi i x_0\xi}\cF(f)(\xi)=\e^{-2\pi i x_0\xi}\hat f(\xi)$.
\item 
$\cF\circ \mu_{\xi_0}=\sig_{-\xi_0}\circ\cF$.
\item $\cF(\bar{f})(\xi)=\overline{\hat f(-\xi)}$
\item
Given $f,\hat f,g, \hat g\in L^1(-\infty,\infty)\cap L^2(-\infty,\infty)$ 
and define the convolution of $f$ and $g$ to be 
\[
(f\star g)(x):=\int_{-\infty}^\infty f(y)\sig_{-y}(g)(x) dy=\int_{-\infty}^\infty f(y)g(x-y) dy\ .
\]
Show that $\cF(f\star g)=\cF(f)\cF(g)$.
\end{enumerate}
\item
What happens to the results given above if we now define $\cF$ on $L^1(\Rn)$ by
\[
\cF(f)(\xi):=\hat f(\xi):=\int_{\Rn} f(x)\sfe^{-2\pi i \ip{x}{\xi}} d\lambda_n(x)\quad\forall\, \xi\in\Rn\ .
\]
\end{enumerate}
\item {\bf (Bonus Problem: 10 points)}
This problem analyzes the Fourier series solution to the 
vibrating string problem:
\begin{alignat*}{3}
&\mbox{(DE)}&\qquad 
u_{tt}&=u_{xx}&\qquad &0\le x\le \pi,\ 0\le t\\
&\mbox{(IC)}&\qquad 
u(x,0)&=f(x),\quad u_t(x,0)=0&\qquad &0\le x\le \pi\\
&\mbox{(BC)}&\qquad 
u(0,t)&=u(\pi,t)=0&\qquad &0\le t\ .
\end{alignat*}
\begin{enumerate}
\item
Show directly that if $f\in C^2[0,\pi]$, then the series for $u$ obtained
by superposing fundamental modes converges uniformly on 
$[0,\pi]\times \R$ to a continuous function $u(x,t)$ that satisfies
$u(x,0)=f(x)$.
\item
Show that the function $u$ from part (a) agrees with d'Alembert's 
solution to this IBVP, and hence show that 
$u\in C^2\left([0,\pi]\times \R\right)$, and that $u$ satisfies
the wave equation.

\noindent
Note: Observe that there are difficulties in trying to justify 
term-by-term differentiation of the series for $u$ to check that
$u$ satisfies the wave equation. Find conditions of $f$ (e.g. smoothness,
values of $f,\ f'$, etc. at $0,\ \pi$) which would justify 
this approach.)
\end{enumerate}
\end{enumerate}
\end{document}
