Math 642600 Assignments — Spring 2015
Assignment 1  Due Wednesday, 1/28/2015.
 Do the following problems.
 Problem 2, page 204 (§ 5.1).
 Problem 7, page 204 (§ 5.1).
 Hanging chain problem. A chain having uniform linear
density ρ hangs between the points (0,0) and (1,0). The total mass
m, which is fixed, and the total energy E of the chain are
Assuming that the chain hangs in a shape that minimizes the
energy, find the shape of the hanging chain.
 Consider the functional J(y) =
∫_{a}^{b}F(x,y,y′)dx, where y ∈
C^{1} and y(a)=A, y(b)=B are fixed.
 Let y extremize J, subject to all of the conditions above. Verify
that if η ∈ C^{1} and η(a)=η(b)=0,
then
Δ J = ∫_{a}^{b} (g(x) +
F_{y′}(x,y,y′))η′(x)dx, where g(x) =
∫_{a}^{x}F_{y}(u,y,y′)du.
 Use (a) to show that g+F_{y′}(x,y,y′)=c, a
constant. From this, it follows that
F_{y′}(x,y,y′)∈ C^{1}. (Du
BoisReymond, 1879. The point is that one need not assume that
y′′ exists.)
 Smoothing spline. Let J[y] =
∫_{0}^{1} (y′(x))^{2}dx. Let the
admissible set for J be all piecewise C^{1} curves on [0,1]
that satisfy y(k/n) = y_{k }, for k = 0, 1, 2, ..., n, with
the discontinuities in y′ appearing only at the points
x_{k} = k/n. Use
the previous problem to show that the minimizer for J is a linear
spline passing through all of the points {(x_{k},y_{k}), k = 0,
..., n}. (Hint: y(x) is in C^{1}[x_{k},
x_{k+1}]).)
 Let x = (x,y) and let J[x] =
x^{2}y(x^{2} + y^{2})^{− 1}
if x ≠ 0, J[0] = 0. Show that the
Gâteaux derivative exists at at x = 0 for every
direction η = (h,k). Also, show that the Fréchet derivative
doesn't exist at x = 0.
Assignment 2  Due Wednesday, 2/11/2015.
 Do the following problems.
 Problem 2, page 205 (§ 5.2).
 Problem 6, page 205 (§ 5.2).
 Problem 7, page 206 (§ 5.2). (See Fig. 5.4, p. 205 for a
diagram.)
 Problem 8, page 206 (§ 5.2). Also, find the Hamiltonian of
the system, and write out Hamilton's equations for it. Finding
integrals (constants) of the motion should follow immediately from
these equations. (Note: There is an error in the expression for L. The
(dθ/dt)^{2} sin^{2}θ) should be replaced
by (dφ/dt)^{2} sin^{2}θ.)
 Consider the torus (surface of a "doughnut") obtained by rotating
a circle with center $(R,0,0)$ and radius $r < R $ about the $z$
axis. Find the metric tensor for it and use this tensor to obtain
the 2nd order differential equations for the geodesics.
 Find the Legendre transformation H(p) for F(x) =
x^{T} A x, where x is in
R^{n} and A is a symmetric, positive definite n×n
matrix.
Assignment 3  Due Wednesday, 2/18/2015.
 Do the following problems.
 Problem 1, page 207 (§ 5.4). (Hint: Use Minimax Principle.)
 Problem 4, page 208 (§ 5.4).
 Problem 6, page 208 (§ 5.4). Take Ω to be a 2D disk
centered at 0 and having radius r = a. How does the lowest
eigenvalue change with a?
 Use a quadratic polynomial to estimate the lowest eigenvalue for
the SturmLiouville problem with Lu = − u′′ =
λu, u(0) = 0, u′(1)+u(1) = 0.
 In class we showed that the Hamiltonian $H=\frac{1}{2m}
(p_r^2+r^{2}p_\theta^2)  Kr^{1}$, for a planet (mass $m$) in
orbit around the sun (mass=$M$), where the constant $K=GmM$. In
addition, we also found that $H = E$ and $p_\theta = \ell$ are
constants of the motion.
 Show that $p_r = \ell r^{2} \frac{dr}{d\theta} = 
\ell \frac{d}{d\theta}(r^{1})$.
 Let $u = r^{1} mK\ell^{2}$. Show that $2m E\ell^{2} + m^2K^2\ell^{4}
= \big(\frac{du}{d\theta}\big)^2 + u^2$. Solve this first order differential
equation for u in terms of $\theta$.
 Use your solution to the previous part to establish Kepler's laws
of planetary motion:
 The orbit of a planet is an ellipse, with the sun at one
focus.
 The radius vector from the sun to the planet sweeps out equal
areas in equal times.
 The square of the period of any planet is proportional to the
cube of the semimajor axis of its orbit. The proportionality constant
is the same for all planets.
Assignment 4  Due Friday, 2/27/2015.
 Read section 6.5.
 Do the following problems.
 Section 6.2: 12, 13 (Assume $\alpha \ge 2$.)
 Section 6.4: 2, 4, 9, 25
 Let $\xi \ge 0$. Find $\hat f(\xi) := \frac{1}{2\pi}\int_{\infty}^\infty
\frac{e^{i\xi x}}{1+x^4}dx$.
Assignment 5  Due Monday, 3/9/2015.
 Read sections 7.1 and 7.2.
 Do the following problems.
 Problem 26, pg. 278 (§ 6.4). In this problem, assume that $z
f(z) \le e^{kz}$ for all $z\in \mathbb C$ and, for all real $x$,
$x f(x) \le M$, for some constant $M>0$.
 Problem 2, page 279 (§ 6.5).
 Problem 5(a), page 279 (§ 6.5).
 Problem 9, page 280 (§ 6.5). (Do $J_{3/2}(z)$.)
 Problem 10, page 280 (§ 6.5).
Assignment 6  Due Monday, 4/6/2015.
 Read section 7.2.1.
 Do the following problems.
 Section 7.1: 2(a), 2(b). For each operator in 2(a), 2(b), find
the norm and adjoint $L^\ast$. You may use any of the problems below.
 Let $L$ be a bounded linear operator on $\mathcal H$; that is, $L
\in {\mathcal B}(\mathcal H)$. Show that $\rho(L)$ contains the
exterior of the disk $\{\lambda \in {\mathbb C} \colon \lambda
\le L\}$. (Consequently, $\sigma(L)$ is contained in the disk.)
 Let $K \in {\mathcal B}(\mathcal H)$ and let $L:D \to \mathcal H$ be a
densely defined, closed linear operator.
 Show that $L+K$ is defined on $D$ and is also closed.
 Show that $(L+K)^\ast$ is defined on the domain of $L^\ast$ and
that $(L+K)^\ast = L^\ast + K^\ast$.
 Suppose that $0 \in \rho(L)$. Show that $0 \in \rho(L^\ast)$ and
that $(L^{1})^\ast = (L^\ast)^{1}$.
 Let $L = L^\ast$ be defined on a domain $D$. In class, we showed
that $\sigma_r(L) = \emptyset$ and that $\sigma_d(L) \subset \mathbb
R$. Show that $\sigma_c(L) \subset \mathbb R$. Thus, $\sigma(L)$ is
a subset of the reals.
 Let $L = L^\ast$ be defined on a domain $D$ and suppose that $L$
satisfies $\langle Lf,f \rangle \ge 0 \ $, for all $f\in D$. Show that
$\sigma(L) \subseteq [0, \infty)$.
Assignment 7  Due Monday, 4/20/2015.
 Read section 7.2.1.
 Do the following problems.
 Let $A\in \mathbb C^{n\times n}$, $A=A^\ast$. Let $C_\lambda$ be a simple
closed curve that encircles the eigenvalues $\lambda_j <
\lambda_{j+1} < \cdots <\lambda_k$, along with $\lambda$, where
$\lambda_k < \lambda <\lambda_{k+1}$ or $\lambda > \lambda_n$.
 Show that $E_\lambda = \frac{1}{2\pi}\oint_{C_\lambda} R_z(A)dz
= \sum_{\ell = j}^k E_j$, where $E_j$ is the orthogonal projection
onto the eigenspace of $\lambda_j$.
 If $m \in \mathbb Z$, $m \ge 1$, $j=1$ and $\lambda > \lambda_n$,
show that $A^m = \frac{1}{2\pi}\oint_{C_\lambda} z^m
R_z(A)dz$. (Hint: we did the $m = 1$ case in class on 31 Mar. Use this
case and induction to obtain the result.)
 Adopt the notation and assumptions of the previous problem.
Define $E_{\lambda^\pm} = \lim_{\varepsilon \downarrow 0} E_{\lambda
\pm \varepsilon}\,$.
 Show that, for $1 \le k\le n$, $E_{\lambda_k^+} =
\sum_{j=1}^k E_j$ and $E_{\lambda_k^}= \sum_{j=1}^{k1} E_j$. (Note:
$\sum_{j=1}^{0} E_j = 0$.)
 Normalize $E_\lambda$ to be right continuous, so $E_{\lambda^+} =
E_\lambda$. Suppose that the $\lambda_j$'s all have multiplicity 1
and $u_j$, $\u_j\ = 1,$ is the eigenvector corresponding to
$\lambda_j$. Show that if $\lambda \ge \lambda_k$, $k=1,\ldots, n$,
then $E_\lambda v = \sum_{j=1}^k \langle v,u_j\rangle u_j$. Also,
use your knowledge of matrices to show that when $\lambda \ge
\lambda_n$, $E_\lambda = I $ and so $v = \sum_{j=1}^n \langle
v,u_j\rangle u_j$.
Remark: Let $\hat v_j := \langle
v,u_j\rangle$. Finding $\hat v_j$ is the analysis step
in processing $v$. The synthesis step is then
recovering $v$ via $v = \sum_{j=1}^n \hat v_j u_j$. As part of
processing $v$, the $\hat v_j$'s may be altered — to remove
noise, for example.
 If $E_\lambda$ is right continuous, show that
$
\Delta E_\lambda := E_\lambda  E_{\lambda^} =
\begin{cases}
E_k & \lambda = \lambda_k,\\
0 & \lambda \ne \lambda
\end{cases}.
$
Assignment 8  Due Wednesday, 4/29/2015.
 Read sections 7.2.3, 10.110.3.
 Do the following problems.
 Prove the following theorems:

If $f,g \in L^1(\mathbb R) \cap L^2(\mathbb R)$, then $f\ast g\in
L^1(\mathbb R) \cap L^\infty(\mathbb R)$.
 If $f\in L^1(\mathbb R)$, then its Fourier transform $F$ is in
$C_0(\mathbb R)$.
 If $f,F,g,G$ are all in $L^1(\mathbb R) \cap L^2(\mathbb
R)$, then, $\int_{\infty}^\infty f(t)G(t)dt = \int_{\infty}^\infty
F(t)g(t)dt$.
 Let $L=d^2/dx^2$, $D(L) =\{f\in L^2[0,\infty) \colon Lf \in
L^2[0,\infty) \text{and} \ f'(0)=0\}$. Use Stone's formula for the
family of projections to obtain the Fourier cosine transform.
 Consider the one dimensional heat equation, $u_t =
u_{xx}$, with $u(x,0) = f(x)$, where $\infty < x <
\infty$ and $0 < t < \infty$. By taking the Fourier transform in
$x$, show that the solution $u(x,t)$ is given by
\[
u(x,t) = \int_{\mathbb R}K(xy,t)f(y)dy,\ K(\xi,t) =
(4\pi t)^{1/2} e^{\xi^2/4t}.
\]
(The function $K(\xi,t)$ is the onedimensional heat kernel.)
 Problems 7.2.6(a,c).
Assignment 9  Not to be handed in.
 Suggested problems.
 Let $f$ be in Schwartz space, $\mathcal S$, and let $g$ be $C^\infty$
and satisfy $g^(m)(x) \le c_m(1+t^2)^{n_m}$
for all nonnegative integers m. Here $c_m$ and $n_m$
depend on $g$ and $m$. Show that $fg$ is in Schwartz space. Use this
result to explain how to
define the product $g(x)T(x)$, where $T$ is a tempered distribution.
 Let T be a tempered distribution. Find the Fourier transforms for
T′, xT(x), T(x−a), and e^{ibx}T(x).
 (Problem 1, p. 7, Feldman.)
Suppose that $f$ is locally Lebesgue integrable on $\mathbb R$ and
that there is a polynomial $P(x) = \sum_{k=N_1}^{N_2} a_kx^k$, where
$0\le N_1 < N_2 <\infty$, such that $f(x) \le P(x)$ a.e. in
$\mathbb R$.
 Show that there is a constant $C>0$ such that
$ f(x)(1+x^2) \le C(x^{N_1} +x^{N_2})$.
 Show that for all $\phi\in \mathcal S$ we have
$ \big\langle f, \phi\rangle \big \le \pi C\big( \\phi\_{N_1,0} + \
\phi \_{N_2+2,0} \big)$.
 Problem 7, page 464 (§10.3)
 Problem 9, page 465 (§10.3)
 Problem 11, page 465 (§10.3)
Updated 5/7/2015 (fjn)