Math 641600 — Fall 2015
Assignments
Assignment 1  Due Wednesday, September 9, 2015.
 Read sections 1.11.4
 Do the following problems.
 Section 1.1: 3(c), 5, 7(a), 8
 Section 1.4: 3
 Let $U$ be a subspace of an inner product space $V$, with the
inner product and norm being $\langle\cdot,\cdot \rangle$ and
$\\cdot\$. Also, let $v$ be in $V$. (Do not assume that
$U$ is finite dimensional or use arguments requiring a basis.)
 Fix $v\in V$. Show that if there exists $p\in U$ such that $p$
satisfies either (i) $\min_{u\in U}\vu\ = \vp\$ or (ii) $vp\in
U^\perp$, then it satisfies both (i) and (ii). Moreover, if $p$
exists, it is unique.
 Suppose $p$ exists for every $v\in V$. Since $p$ is uniquely
determined by $v$, we may define a map $P: V \to U$ via
$Pv:=p$. Show that $P$ is a
linear map and that $P$ satisfies $P^2 = P$. ($P$ is called
an orthogonal projection. The vector $p$ is the orthogonal
projection of $v$ onto $U$.)
 If the projection $P$ exists, show that for all $w,z\in V$,
$\langle Pw,z\rangle = \langle Pw,Pz\rangle= \langle
w,Pz\rangle$. Use this to show that $U^\perp= \{w\in
V\colon Pw=0\}$.
 Suppose that the projection $P$ exists. Show that $V=U\oplus
U^\perp$, where $\oplus$ indicates the direct sum of the two spaces.
 Let $U$ and $V$ be as in the previous exercise. Suppose that $U$
is finite dimensional and that $B=\{u_1,u_2,\ldots,u_n\}$ is an
ordered basis for $U$. In addition, let $G$ be the $n\times n$
matrix with entries $G_{jk}= \langle u_k,u_j\rangle$.
 Let $v\in V$ and $d_k := \langle v,u_k\rangle$. Show that $p$
exists for every $v$ and is given by $p=\sum_j x_j u_j\in U$, where
the $x_j$'s satisfy the normal equations, $d_k = \sum_j
G_{kj}x_j$. (Hint: use exercise 3 above.)
 Explain why the projection $P$ exists. Show that if B is
orthonormal, then $Pv=\sum_j \langle v,u_j\rangle u_j$.
Assignment 2  Due Wednesday, September 16, 2015.
 Read sections 2.1 and 2.2
 Do the following problems.
 Section 1.2: 9, 10(a,b)
 Section 1.3: 2(b)
 Section 1.4: 4
 Let V be an n dimensional vector space and suppose L:V→V is
linear.
 Let E={φ_{1},.., φ_{1}} be a basis for V
and let A be the matrix of L relative to E. Show that the
characteristic polynomial p_{A}(λ) := det(A −
λ I) is independent of the choice of E, and so
p_{L}(λ) := p_{A}(λ) is well defined, in
the sense that it is independent of the choice of basis for V.
 Use the previous part to show that Trace(L) := Trace(A) and
det(L) := det(A) are also well defined,
 Let $L:P_2\to P_2$ be given by $L(p)= \big((1x^2)p'\big)' +
7p$. Find Trace(L) and det(L).
 Let A and B be n×n matrices. Suppose that the range of B,
Range(B), is an invariant subspace for A. Show that there is an
n×n matrix X such that AB = BX.
 (This is a generalization of Keener's problem 1.3.5.) Let $A$ be
a selfadjoint matrix with eigenvalues $\lambda_1\ge
\lambda_2,\ldots,\ge \lambda_n$. Show that for $ 2\le k < n$ we have
\[ \max_U \sum_{j=1}^k \langle Au_j,u_j \rangle =\sum_{j=1}^k
\lambda_j, \]
where $U=\{u_1,\ldots,u_k\}$ is any o.n. set. (Hint: Put $A$ in
diagonal form and use a judicious choice of $B$.)
 Let U be a unitary, n×n matrix. Show that the following hold.
 < Ux, Uy > = < x, y >
 The eigenvalues of U all lie on the unit circle, λ=1.
 Show that U is diagonalizable. (Hint: follow the proof for the
selfadjoint case.)
 Let $k(x,y) = x+ 3x^2y + xy^2$ and $\langle
f,g\rangle=\int_{1}^1 f(x)g(x)(1+x^2)dx$. Consider the operator
$Lu=\int_{1}^1 k(x,y) u(y)dy$. In the notes, we have shown that
$L:P_2\to P_2$.
 Relative to the inner product above, find $L^\ast$ and
$\text{Null}(L^\ast)$.
 Find a condition on $q\in P_2$ for which $Lp=q$ always has a
solution. Is this different from what was in the notes?
Assignment 3  Due Wednesday, September 23, 2015.
 Read sections 2.1, 2.2.1 and 2.2.2
 Do the following problems.
 Section 2.1: 3, 5, 6
 Show that $\ell^2$, under the inner product $\langle
x,y\rangle = \sum_{j=1}^\infty x_j \overline{y_j}$, is a Hilbert
space.
 This problem concerns several important inequalities.
 Show that if α, β are positive and α + β
=1, then for all u,v ≥ 0 we have
u^{α}v^{β} ≤ αu + βv.
 Let x,y ∈ R^{n}, and let p > 1 and define
q by q^{1} = 1  p^{1}. Prove Hölder's
inequality,
∑_{j} x_{j}y_{j} ≤ x_{p}
y_{q}.
Hint: use the inequality in part (a), but with appropriate choices of
the parameters. For example, u =
(x_{j}/x_{p})^{p}
 Let x,y ∈ R^{n}, and let p > 1. Prove
Minkowski's inequality,
x+y_{p} ≤ x_{p} + y_{p}.
Use this to show that x_{p} defines a norm on
R^{n}. Hint: you will need to use Hölder's
inequality, along with a trick.
 Let $f\in C^1[0,1]$. Show that
$\f\_{C[0,1]}\le C\f\_{H^1[0,1]}$, where $C$ is a constant
independent of $f$ and $\f\_{H^1[0,1]}^2 := \f\_{L^2\,[0,1]}^2 +
\f'\_{L^2\,[0,1]}^2$. Hint: you will need the mean value theorem for
integrals.
 Let f(x) : x^{2}, 1 ≤ x ≤ 2. Find f^{
−1}(Ej) for E_{j} = [j/2, (j+1)/2), j
= 0, …, 7. Using these, find the numerical value of
the Lebesgue sum corresponding to y*_{j} = (2j+1)/4
Assignment 4  Due Friday, October 2, 2015.
 Read sections 2.1.1 and 2.2.
 Do the following problems.
 Section 2.1: 10, 11
 Section 2.2: 1 (Do $w=1$.), 8. (Problem 8(e) has an $n!$ missing in
the numerator.)
 Let $F(s) := \int_0^\infty e^{st}f(t)dt $ be the Laplace
transform of $f \in L^1([0,\infty))$. Use the Lebesgue dominated
convergence theorem to show that $F(s)$ is continuous from the right
at $s=0$.
 You are given that $C^1[0,1]$ is dense in $H^1[0,1]$, and that
$H^1[0,1]$ is complete. Use
problem 4, HW3 to show that every $f$ in $H^1[0,1]$ is continuous.
 Let $U:=\{u_j\}_{j=1}^\infty$ be an orthonormal set in a Hilbert
space $\mathcal H$. Show that the two statements are
equivalent. (You may use what we have proved for o.n. sets in
general; for example, Bessel's inequality, minimization properties,
etc.)
 $U$ is maximal in the sense that there is no nonzero vector in
$\mathcal H$ that is orthogonal to $U$. (Equivalently, $U$ is not a
proper subset of any other o.n. set in $\mathcal H$.)
 Every vector in $\mathcal H$ may be uniquely represented as the
series $f=\sum_{j=1}^\infty \langle f, u_j\rangle u_j$.
Assignment 5  Due Friday, October 9.
 Read sections 2.2.22.2.4
 Do the following problems.
 Section 2.2: 9, 10
 Let $\delta>0$. We define the modulus of continuity for $f\in
C[0,1]$ by $\omega(f,\delta) := \sup_{\,\,st\,\,\le\,
\delta,\,s,t\in [0,1]}f(s)f(t)$.
 Explain why $\omega(f,\delta)$ exists for every $f\in C[0,1]$.
 Fix $\delta>0$. Let $S_\delta = \{ \epsilon >0 \colon f(t)  f(s)
< \epsilon \forall\ s,t \in [0,1], \ s  t \le \delta\}$. In other
words, for given $\delta$, $S_\delta$ is in the set of all
$\epsilon$ such that $f(t)  f(s) < \epsilon$ holds for all $s 
t\le \delta$. Show that $\omega(f, \delta) = \inf S_\delta$
 Show that $\omega(f,\delta)$ is non decreasing as a
function of $\delta$. (Or, more to the point, as $\delta \downarrow 0$,
$\omega(f,\delta)$ gets smaller.)
 Show that $\lim_{\delta \downarrow 0} \omega;(f,\delta) = 0$.
 Calculus problem: Let g be C^{2} on an interval
[a,b]. Let h = b − a. Show that if g(a) = g(b) = 0, then
g_{C[a,b]} ≤ (h^{2}/8)
g′′_{C[a,b]}.
Give an example that shows
that $1/8$ is the best possible constant.
 Use the previous problem to show that if f ∈
C^{2}[0,1], then the equally spaced linear spline interpolant
f_{n} satisfies
f −
f_{n}_{C[0,1]} ≤ (8n^{2})^{ −
1} f′′_{C[0,1]}
Assignment 6  Due Friday, October 30.
 Read section 2.2.7
 Do the following problems.
 Section 2.2: 14
 Compute the complex form of the Fourier series for $f(x) =
e^{2x}$, $0 \le x \le 2\pi$. Use this Fourier series and Parseval's
theorem to sum the series $\sum_{k=\infty}^\infty (4+k^2)^{1}$.
 The following problem is aimed at showing that
$\{e^{inx}\}_{n=\infty}^\infty$ is complete in $L^2[0,2\pi]$.
 Consider the series ∑_{n} c_{n}
e^{inx}, where ∑_{n} c_{n} <
∞. Show that ∑_{n} c_{n} e^{inx}
converges uniformly to a continuous function f(x) and that the series
is the Fourier series for f. (It's possible for a trigonometric
series to converge pointwise to a function, but not be the FS for
that function.)
 Use the previous problem to show that if $f$ is a continuous,
$2\pi$periodic function having a piecewise continuous derivative,
then the FS for $f$ converges uniformly to $f$. (Hint: Note that since
$f'\in L^2[0,2\pi]$, the series $\sum_{k=\infty}^\infty
\bigc_k^{(1)}\big^2$ is convergent. Also, for $k\ne 0$, one has that
$c_k = (ik)^{1}c_k^{(1)}$.)
 Apply this result to show that the FS for a linear spline $s(x)$,
which satisfies $s(0)=s(2\pi)$, is uniformly convergent to
$s(x)$. Show that such splines are dense in $L^2[0,2\pi]$.
 Show that $\{e^{inx}\}_{n=\infty}^\infty$ is complete in
$L^2[0,\pi]$.
 Show that $\\hat y\_{{\mathcal S}^n} = \sqrt{n}
\y\_{{\mathcal S}^n}$. (Note: The inner product for ${\mathcal
S}^n$ is the same as the ${\mathbb C}^n$ inner product on entries
$k=0,\cdots ,n1$.)
 Suppose that x is an nperiodic sequence (i.e., x
∈ S_{n}). Show that $ \sum_{j=m}^{m+n1}{\mathbf
x}_j = \sum_{j=0}^{n1}{\mathbf x}_j $. (This is the DFT analogue of
the lemma in my notes on
Pointwise convergence of Fourier series
.)
 Use the previous problem to prove the Convolution Theorem for the
DFT. (See
Notes on the Discrete Fourier Transform, pg. 3.)
Assignment 7  Due Friday, November 6.
 Read sections 3.1 and 3.2.
 Do the following problems.
 Section 2.2: 25(a,b), 26(b), 27(a)
 Let $H_0$ be the set of all $f\in C^{(0)}[0,1]$ such that
$f(0)=f(1)=0$ and that $f'$ is piecewise continuous. Show that
$\langle f,g\rangle_{H_0} :=\int_0^1f'(x)g'(x)dx$ defines a real inner
product on $H_0$.
 We want to use the Galerkin method to numerically solve the
boundary value problem (BVP): −u" = f(x), u(0) = u(1) = 0,
f ∈ C[0,1]
 Weak form of the problem. Let H_{0} be as in the
previous problem. Suppose that $v\in H_0$. Multiply both sides of
the eqaution above and use integration by parts
to show that $ \langle u,v\rangle_{H_0} = \langle f,v\rangle_{L^2[0,1]}$.
This is called the ``weak'' form of the BVP.
 Conversely, suppose that u ∈ H_{0} is also in
C^{(2)}[0,1] and that u satisfies
⟨u,v⟩_{H0} = ∫_{0}^{1} f(x)
v(x) dx for all v ∈ H_{0}.
Show that u satisfies the BVP.
 Consider S_{0} := {s ∈S^{1/n}(1,0) :
s(0)=s(1)=0}. Show that S_{0} is spanned by φ_{j}(x) :=
N_{2}(nxj+1), j = 1 ... n1. (Here, N_{2}(x) is the
linear Bspline.)
 Show that the leastsquares approximation s ∈
S_{0} to the solution u is given by s = ∑_{j}
α_{j}φ_{j}(x), where the
α_{j}'s satisfy Gα = β, with
β_{j} = ⟨ y,φ_{j}
⟩_{H0} = ∫_{0}^{1} f(x)
φ_{j}(x) dx, j=1 ... n1 and G_{kj} = ⟨
φ_{j}, φ_{k} ⟩_{H0}.
 Show that G_{kj} = ⟨ φ_{j},
φ_{k} ⟩_{H0} is given by
G_{j,j} = 2n, j = 1 ... n1
G_{j,j1} =  n, j = 2 ... n1
G_{j,j+1} =  n, j = 1 ... n2
G_{j,k} = 0, all other possible k.
Assignment 8  Due Friday, November 13.
 Read sections 3.3 and 3.4.
 Do the following problems.
 Section 3.2: 3(d) (Assume the appropriate
operators are closed and that λ is real.)
 Let $N_3(x)$ be the $m=3$ Bspline, which is a piecewise
quadratic in $S^{\mathbb Z}(2,1)$. It is chosen so that $N_3(x)=0$
for $x\le 0$ and $x\ge 3$. Show that $N_3(x) = \alpha \big( (x)_+^2
3 (x1)_+^2 +3 (x2)_+^2  (x3)_+^2\big)$, where $\alpha>0$ is a
normalization constant; it is chosen to be $\frac12$.
 Let V be a Banach space. Show that a linear operator L:V → V
is bounded if and only if L is continuous.
 (DFT problem.) Let α, ξ, η be nperiodic sequences,
and let a, x, y be column vectors with entries a_{0}, ...,
a_{n1}, etc. Show that the convolution η =
α∗ξ is equivalent to the matrix equation y = Ax, where
A is an n×n matrix whose first column is a, and whose remaining
columns are a with the entries cyclically permuted. Such matrices are
called cyclic. Use the DFT and the convolution theorem to find the
eigenvalues of A. An example of a cyclic matrix is given below.
\[
\begin{pmatrix}
3 &5 &4 &1 \\
1 &3 &5 &4 \\
4 &1 &3 &5\\
5 &4 &1 &3
\end{pmatrix}
\]
 Let L be a bounded linear operator on Hilbert space $\mathcal
H$. Show that these two formulas for $\L\$ are equivalent:
 $\L\ = \sup \{\Lu\ : u \in {\mathcal H},\ \u\ = 1\}$
 $\L\ = \sup \{\langle Lu,v\rangle : u,v \in {\mathcal H},\
\u\=\v\=1\}$

Consider the boundary value problem $u''(x)=f(x)$, where $0\le x \le
1$, $\, f\in C[0,1]$, $\, u(0)=0$ and $u'(1)=0$.

Verify that the solution is given by $u(x) = \int_0^1 k(x,y)f(y)dy$, where
\[
k(x,y) = \left\{
\begin{array}{cl}
y, & 0 \le y \le x, \\
x, & x \le y \le 1.
\end{array}
\right.
\]

Let $L$ be the integral operator $L\,f = \int_0^1
k(x,y)f(y)dy$. Show that $L:C[0,1]\to C[0,1]$ is bounded and that the
norm $\L\_{C[0,1]\to C[0,1]}\le 1$. Actually, $\L\_{C[0,1]\to
C[0,1]}=1/2$. Can you show this?
 Show that $k(x,y)$ is a HilbertSchmidt kernel and that
$\L\_{L^2\to L^2} \le \sqrt{\frac{1}{6}}$.
 Finish the proof of the Projection Theorem: If for every $f\in
\mathcal H$ there is a $p\in V$ such that $\pf\=\min_{v\in
V}\vf\$ then $V$ is closed.
Assignment 9  Due Friday, November 20.
 Read sections 3.5 and 3.6.
 Do the following problems.
 Section 3.4: 2(b)
 Consider the real Hilbert space H_{0} := {v ∈
H^{1}[0,1] : v(0) = v(1) = 0}, with the inner product < f,g
>_{H0} = ∫_{0}^{1}f
′(x) g ′(x)dx. Show that there is a function g_{y}
in H_{0} for which f(y) = < f,g_{y}
>_{H0}. (g_{y} is called a
reproducing kernel and H_{0} is a reproducing
kernel Hilbert space.) Hint: Use
problem 4, HW3 to show that the Φ_{y}(f) :=
f(y) is a bounded linear functional on H_{0}.
 Let $L$ be a bounded operator on a Hilbert space $H$. Show that
the closure of the the range of $L$ satisfies $\overline{R(L)} =
N(L^\ast)^\perp$. (Hint: Follow the proof of the Fredholm
alternative, which is just the special case where $R(L)$ is closed.)
 A sequence {f_{n}} in a Hilbert space H is said to
be weakly convergent to f ∈ H if and only if lim_{ n
→ ∞} < f_{n},g> = < f,g> for every
g∈H. When this happens, we write f = wlim f_{n}. For
example, if {φ_{n}} is any orthonormal sequence, then
φ_{n} converges weakly to 0. You are given that every weakly
convergent sequence is a bounded sequence (i.e. there is a constant C
such that f_{n} ≤ C for all n). Prove the following:
Let K be a compact linear operator on a Hilbert space
H. If f_{n} converges weakly to f, then Kf_{n}
converges to Kf — that is, lim_{ n → ∞} 
Kf_{n}  Kf  = 0.
Hint: Suppose this doesn't happen, then there will be a subsequence of
{f_{n}}, say {f_{nk}}, such that 
Kf_{nk}  Kf  ≥ ε for all k. Use this
and the compactness of K to arrive at a contradiction. We remark that
the converse is also true. If a bounded linear operator $K$ maps
weakly convergent sequences into convergent sequences, then $K$ is
compact.

Consider the Hilbert space $\ell^{\,2}$. Let $S=\{\{a_j\}_{j=1}^\infty
\in \ell^{\,2}\colon \sum_{j=1}^\infty (1+j^2)\,a_j^2 < 1 \}$. Show
that $S$ is a precompact subset of $\ell^{\,2}$.
 Let L be a bounded selfadjoint linear operator on a
Hilbert space $\mathcal H$. Show that these two formulas for $\L\$
are equivalent:
 $\L\ = \sup \{\Lu\ : u \in {\mathcal H},\ \u\ = 1\}$
 $\L\ = \sup \{\langle Lu,u\rangle : u\in {\mathcal H},\
\u\=1\}$
Assignment 10  Due Monday, November 30.
 Read sections 3.5, 3.6, and 4.1.
 Do the following problems.
 Section 3.4: 2(c), 6
 Section 3.5: 1(b), 2(a)
 (This is a variant of problem 3.4.3 in Keener.) Consider the
operator $Ku(x) = \int_{1}^1 (1xy)u(y)dy$ and the eigenvalue
problem $\lambda u = Ku$.
 Show that $K$ is a selfadjoint, HilbertSchmidt operator.
 Let $f\in C[1,1]$. If $v= Kf$, show that $v"=2f$,
$v(1)+v(1)=0$, and $v'(1)+v'(1)$.
 Use the previous part to convert the eigenvalue problem $\lambda
u = Ku$ into this eigenvalue problem:
\[
\left\{
\begin{align}
u"+&\frac{2}{\lambda} u =0,\\
u(1)+&u(1) =0 \\
u'(1)+ &u'(1)=0.
\end{align}
\right.
\]
 Solve the eigenvalue above to get the eigenvalues and
eigenvectors of $\lambda u = Ku$. Explain why the eigenvectors form
a complete set for $L^2[1,1]$.
 Let K be a compact, selfadjoint operator and let M be the span
of the set of eigenvectors {φ_{j}} corresponding to all
eigenvalues λ_{j} ≠ 0. (Note: both M and
M^{⊥} may be infinite dimensional.)
 Show that M and M^{⊥} are both invariant under K.
 Show that K restricted to M^{⊥} is compact.
 Show that either M^{⊥} = {0} or that it is the
eigenspace for λ = 0.
 Show that one may choose a complete, orthonormal set from among
the eigenvectors of K. (Use Proposition 2.4 in
Spectral Theory for Compact Operators.)
Extra Problems  These are not to be handed
in.
 Let $L$ be in $\mathcal B (\mathcal H)$.
 Show that $\L^k\ \le \L\^k$, $k=2,3,\ldots$.
 (We did this in class.) Let $\lambda \L\<1$. Show that
\[
\big\(I  \lambda L)^{1}  \sum_{k=0}^{n1}\lambda^k L^k\big\ \le
\frac{\lambda^k \L\^k}{1  \lambda \L\}.
\]
 Let $L$ be as
in
problem 6, HW8. Estimate how many terms it would require to
approximate $(I  \lambda L)^{1}$ to within $10^{8}$, if
$\lambda\le 0.1$.
 Use Newton's method (see text, problem 3.6.3) to approximate the
cube root of 2. Show that the method converges.
 Section 4.1: 6
 Section 4.2: 1, 4, 8
 Let $Lu=u''$, $u(0)=0$, $u'(1)=2u(1)$.
 Show that the Green's function
for this problem is
\[
G(x,y)=\left\{
\begin{array}{rl}
(2y1)x, & 0 \le x < y \le 1\\
(2x1)y, & 0 \le y< x \le 1.
\end{array} \right.
\]
 Verify that $0$ is not an eigenvalue for $Kf(x) :=
\int_0^1G(x,y)f(y)dy$.
 Show the orthonormal set of eigenfunctions for $L$ form a complete set
in $L^2[0,1]$. (Hint: use tthe results from
problem 4, HW10.
Updated 12/8/2015.