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%\journal{J. Comput. Phys.}
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\begin{document}
\title{Problem set 1}
\author{Due date: Feb 5}
\maketitle

\begin{enumerate}
\item If $\{ h_{n} \}$ is a sequence in a Hilbert space $\cH$ such that $\sum_{n} \| h_{n} \| < \infty$, then show
that $h_{n}$ converges.
\item Suppose that $E$ is a linear subspace of a Hilbert space $\cH$, then show that the closure of $E$ is also a linear 
subspace
\item Suppose that $E$ is a subspace of a Hilbert space $\cH$, then show that 
$(E^{\perp})^{\perp}$ is the closure of the span of elements in $E$, i.e.
$$
(E^{\perp})^{\perp} = \overline{\left\{ \sum_{j=1}^{N} c_{j} f_{j} \, , \quad f_{j} \in E  \right\}}
$$ 
\item Suppose that $\cH = \ell^{2} (\bN)$. 
\begin{enumerate}
\item Show that if $\{ a_{n} \} \in \cH$, then the power series $\sum_{n=1}^{\infty} a_{n} z^{n}$ has radius
of convergence at least $1$
\item For $\lambda < 1$, show that $L(\{ a_{n} \}):= \sum_{n=1}^{\infty} a_{n} \lambda^{n}$ is a bounded linear functional
\item Find the element $h_{0} \in \cH$ such that $L(h) = (h,h_{0})$ and find $\|L \|$
\end{enumerate}
\item Let $\cH_{1} = \bL^{2} ([-\pi,\pi])$ be the Hilbert space of functions $F(e^{i \theta})$ on the unit circle with the inner product
$$
(F,G)= \frac{1}{2\pi} \int_{-\pi}^{\pi} F(e^{i \theta}) \overline{G(\eit)} d \theta \, .
$$
Let $\cH_{2}$ be the space $\bL^{2} (\R)$. 
Using the mapping 
$$
x \to \frac{i-x}{i+x} $$
of $\R$
 to the unit circle, show that:\\
a) The correspondence $U:\cH_{1} \to \cH_{2}$ given by 
$$
U[F] = \frac{1}{\pi^{1/2}(i+x)}F(\frac{i-x}{i+x}) \, 
$$
is a unitary mapping.\\
b) As a result show that 
$$
\left\{ \frac{1}{\pi^{1/2}(i+x)}\left( \frac{i-x}{i+x} \right)^{n} \right\}_{n=-\infty}^{\infty} \, 
$$
is an orthonormal basis of $\bL^{2}(\R)$. 
\item Prove that the operator $T: \bL^{2}[0,\infty] \to \bL^{2}[0,\infty]$ 
$$
T[f](x) = \frac{1}{\pi}\int_{0}^{\infty} \frac{f(y)}{x+y} \, dy
$$
is bounded operator with norm $\| T \| \leq 1$.
\item Suppose that the multiplication operator $A: \ell^{2}(\bN) \to \ell^{2}(\bN)$ is defined via
$Ae_{n} = \alpha_{n} e_{n}$ where
$\{ e_{i} \}_{i=1}^{\infty}$ are the standard coordinate vectors and $\alpha_{n} \in \R$. 
Then show that $A$ is bounded if and only if $\sup_{n} |\alpha_{n}| \leq M$.
\item Suppose that $\cK: \bL^{2}([0,1]) \to \bL^{2}([0,1])$ is defined by
$$
\cK[f] = \int_{0}^{1} k(x,y) f(y) \, dy \, ,
$$ 
where $k(x,y) \in \bL^{2}([0,1]\times[0,1])$.
Show that $\cK$ is a bounded linear operator.
\item Give two examples of linear subspaces of $\bL^{2}(\R)$ which are not closed and find their closure.
\item Suppose that $P_{1}$ and $P_{2}$ are orthogonal projections onto subspaces $S_{1}$ and $S_{2}$. 
Show that $P_{2}P_{1}$ is an orthogonal projection if and only if $P_{1}$ and $P_{2}$ commute, i.e.
$P_{1}P_{2} = P_{2}P_{1}$ and in this case
$P_{2}P_{1}$ projects onto $S_{2} \cap S_{1}$. 
Give an example of two projection operators which do not commute.
\item Let $\cH = \bL^{2}(\R)$. Let $\cF: \cH \to \cH$ be the Fourier transform
$$
\cF[f](x) = \int_{-\infty}^{\infty} e^{i 2\pi  x y} f(y) \, dx \, .
$$
Then it is well known that $\cF$ is a unitary map with the inverse
$$
\cF^{-1}[f](x) = \int_{-\infty}^{\infty} e^{-i 2\pi x y } f(y) \, dx \, .
$$
Let $f \ast g$ denote the convolution operator
$$
f \ast g(x) = \int_{-\infty}^{\infty} f(x-y) g(y) \, dy
$$
Further, it is also known that
$$
\cF[fg](x) = \cF[f] \ast \cF[g] \, ,
$$
and
$$
\cF[f \ast g] = \cF[f] \cdot \cF[g] \, .
$$
\begin{enumerate}
\item Let $\chi_{A}(x)$ denote the indicator function of the set $A$, i.e.
$\chi_{A}(x) =1$ if $x\in A$ and $0$ otherwise. Suppose $k_{0}>0$.
Show that 
$$ 
\cF[\chi_{[-k_{0},k_{0}]}] =  \frac{\sin(2\pi k_{0}x)}{\pi x}
$$
\item Let $K(x) = \cF[\chi_{[-k_{0},k_{0}]}](x)$. Show that
$$
\int_{-\infty}^{\infty} K(x-z) K(z-y) \, dz = K(x-y) \, .
$$
\item Let $\cK: \bL^{2}(\R) \to \bL^{2}(\R)$ denote the operator defined by
$$
\cK[f](x) = \int_{-\infty}^{\infty} K(x-y) f(y) \, dy \, .
$$
Show that $\cK$ is a bounded operator.
\item Use part (b) to show that $\cK$ is a projection operator in the following sense, $\cK[\cK[f]] = \cK[f]$
\item Let $\cH_{0} \subset \cH$ denote the subspace defined by:
$$
f \in \cH_{0} \quad \text{if} \quad \cF[f](x) = 0 \quad \forall |x|>k_{0} \, .
$$
Show that $\cH_{0}$ is a closed linear subspace. $\cH_{0}$ is the subspace of band-limited functions with
band-limit $k_{0}$.
\item Show that $\cK$ is the projection operator onto $\cH_{0}$.
\end{enumerate}
\end{enumerate}
\end{document}