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\begin{document}
 \centerline{\large Math.\ 401, Sec.\
501\hfill    Spring, 2006}
 \bigskip
\centerline{\bfseries\large Homework 12, due April 24}
\bigskip

\begin{enumerate}


\item {\it [Schaum's, p.\ 64, Ex.\ 3.22]\/}\quad
Let $\mathbf{r}$ be any three-dimensional vector, and 
$\{\mathbf i,\mathbf j,\mathbf k\}$ the standard unit vectors 
along the 
coordinate axes.  Prove the following finite-dimensional 
analogues of the theorems about orthonormal bases of functions: 
\begin{enumerate}
\item {\bf (Parseval's equation)} \quad $(\mathbf 
r\cdot\mathbf i)^2 + 
(\mathbf r\cdot\mathbf j)^2 +(\mathbf r\cdot\mathbf k)^2 = 
\mathbf r\cdot \mathbf r$.
 \item {\bf (Bessel's inequality)} \quad
$(\mathbf r\cdot\mathbf i)^2 + 
(\mathbf r\cdot\mathbf j)^2 \le \mathbf r\cdot  \mathbf r$.
\item {\bf (least squares approximation)} \quad
$ \mathbf p \equiv (\mathbf r\cdot \mathbf i)\mathbf i + 
(\mathbf r\cdot \mathbf j)\mathbf j $ is, of all vectors in the 
$x$--$y$ plane, the 
closest to $\mathbf r$; furthermore, $\mathbf p$ is perpendicular 
to 
$\mathbf r-\mathbf p$.
\end{enumerate}

\item {\it [Schaum's, p.\ 64, Ex.\ 3.23]\/}\quad 
Suppose that one term in any orthonormal basis (i.e., a complete 
sequence of orthonormal functions) is omitted.
(For example, leave $\,\sin(10x)$ out of the basis 
functions for the Fourier series on $(-\pi,\pi)$.)
Call the resulting (amputated) sequence $\{\psi_n(x)\}$.

\begin{enumerate}
\item Can we expand an arbitrary function $f(x)$  
as a series $\sum_n c_n \psi_n(x)\,$?
\item Is Parseval's identity always satisfied?
Is it ever satisfied?
\item Is Bessel's inequality always satisfied?
\end{enumerate}
 Justify your answers.



 \item {\it [Logan, p.\ 195, Ex.\ 3.1(b)]\/} \quad
Find the eigenvalues and eigenfunctions for the problem
$$y''+\lambda y=0, \quad 0<x<1,$$
$$y(0)=0, \qquad y(1) -y'(1) =0.$$


 
\item {\it [Schaum's, p.\ 64, Ex.\ 3.43]\/} \quad
Show that the solution of the boundary value problem
$${\partial u \over \partial t} = {\partial ^2u\over \partial 
x^2} 
\qquad (0<x<L, \ 0<t),$$
$${\partial u\over \partial x} (t,0) = h u(t,0) , \qquad
{\partial u\over \partial x}(t,L) = -h u(t,L), \qquad u(0,x) = 
f(x),$$
where $h$ and $L$ are constants, is
$$u(t,x) = \sum_{n=1}^\infty
e^{-\omega_n{}\!^2 t} \,
{\omega_n \cos (\omega_n x) + h \sin (\omega_n x) \over 
(\omega_n{}\!^2 + h^2)L + 2h}\, c_n$$
where
$$c_n = \int_0^L f(x) (\omega_n \cos (\omega_n x) + h \sin 
(\omega_n x)) \,dx$$
and the $\omega_n$ are solutions of
$$\tan (\omega L) = {2h\omega\over \omega^2 - h^2}\,.$$


                                                                                
\item  Using the $\delta$-function method, construct a Green
function to solve the problem
$$ {d^2y\over dx^2} +\omega^2 y = f(x),
\qquad
y'(0)=0, \quad
y'(2)=0.$$
(Assume $\omega>0$.)  For what values of $\omega$ does no 
solution exist?
                                                                                
\item  Using the $\delta$-function method, construct a Green
function to solve the problem
$${d y\over dt} -y = f(t), \qquad y(0)=0.$$ 
{\sl Warning:\/} In this case, $G$ is {\sl not\/} continuous at
the location of the delta function.



  \item  Consider the Green function for the Laplace problem
in the upper half plane,
 $$G(x-z,y) = \frac1{\pi}\, {y\over (x-z)^2 +y^2}\,.$$
                                                                                
\begin{enumerate}
 \item Verify that $G$ satisfies Laplace's equation as a
function of $x$ and $y$.
                                                                                
 \item  Show that $\displaystyle \int_{-\infty}^\infty G(x-z,
y)\, dz =1$
 for each fixed $x$ and $y$.
                                                                                
 \item Let $z=1$ and sketch $G(x-1, y)$ as a function of $x$
for three representative values of $y$.
                                                                                
 \item  Justify the claim that
 $$\lim_{y\to 0^+} G(x-z, y) = \delta(x-z).$$
 This means that for any well-behaved function $f$ (say $f$
continuous and bounded),
 $$\lim_{y\to 0^+} \int_{-\infty}^\infty G(x-z, y)\, f(z)\,dz=
f(x).$$
{\sl Hints:\/}
 Write the integral as
 $\int_{-\infty}^{x-\delta} + \int_{x-\delta}^{x+\delta}
+\int_{x+\delta}^\infty$,
 where $\delta$ is an arbitrarily small positive number.
 Show that the two outside integrals approach zero in the limit,
using the assumption that $f$ is bounded.
 In the middle integral, write
 $f(z) = f(x) + [f(z)-f(x)]$
 and show that the integral involving the bracketed term can be
assumed arbitrarily small,
 using the assumption that $f$ is continuous.
                                                                                
                                                                                
 \item Argue from (a) and (d) that
 $G$ is the correct Green function for the problem --- without
appeal to Fourier transforms or any other external information.
 (That is, show that
 $$u(x,y) \equiv \int_{-\infty}^\infty G(x-z,y)\, f(z)\,dz$$
 is the bounded solution of Laplace's equation in the upper half 
plane with
the boundary data $u(x,0)=f(x)$.)
\end{enumerate}




\end{enumerate}
\end{document}