Math 641600 — Fall 2014
Assignments
Assignment 1  Due Wednesday, September 10.
 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 (a) $\min_{u\in U}\vu\ = \vp\$ or (b) $vp\in
U^\perp$, then it satisfies both (a) and (b). Moreover, $p$ is
unique. (If $v\in U$, then $p=v$.)
 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 2 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 17.
 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).
 (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 24.
 Read sections 2.1 and 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(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 Wednesday, October 1.
 Read sections 2.1.1 and 2.2.
 Do the following problems.
 Section 2.1: 10, 11
 Section 2.2: 1 (Do $w=1$ and $w=\frac{1}{\sqrt{1x^2}}$.)
 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 $s=0$.
 Let $\{f_n\in C^1[0,1]\}$. Note that $\{f_n\}$ also belongs to
$H^1[0,1]$. Show that if $f_n$ is Cauchy in $H^1[0,1]$, then, in the $C[0,1]$
norm, $\{f_n\}$ actually converges to a continuous function $f$.
 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 Wednesday, October 8.
 Read sections 2.2.22.2.4
 Do the following problems.
 Section 2.2: 8(a,b,d), 9
 Let $0\le \delta \le 1$. 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 δ. Let S_{δ} = { ε > 0  f(t)
− f(s) < ε for all s,t ∈ [0,1], s − t
≤ δ}. In other words, for given δ, S_{δ}
is in the set of all ε such that f(t) − f(s) <
ε holds for all s − t ≤ δ. Show that
ω(f;δ) = inf S_{δ}
 Show that ω(f;δ) is non decreasing as a
function of δ. (Or, more to the point, as δ ↓ 0,
ω(f;δ) gets smaller.)
 Show that lim_{ δ↓0} ω(f;δ) = 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 31.
Assignment 7  Due Wednesday, November 5.
 Read sections 3.1 and 3.2.
 Do the following problems.
 Section 2.2: 25(a,b), 26(b), 27(a)
 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 the set of all continuous
functions vanishing at x = 0 and x = 1, and having L^{2}
derivatives. Also, let H_{0} have the inner product:
⟨u,v⟩_{H0} = ∫_{0}^{1} u
′(x) v ′(x) dx.
Use integration by parts to convert
the BVP into its ``weak'' form:
⟨u,v⟩_{H0} =
∫_{0}^{1} f(x) v(x) dx for all v ∈ H_{0}.
 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 Wednesday, November 12.
 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.)
 Section 3.3: 2 (Assume the appropriate
operators are closed and that λ is real.)
 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 Wednesday, November 19.
 Read sections 3.5 and 3.6.
 Do the following problems.
 Section 3.4: 2(b)
 Consider space H = {v ∈ H^{1}[0,1] : v(0) = v(1) =
0}. Let q(x) be continuous and strictly positive on [0,1]. On H,
define the (real) inner product
< f,g >_{H} = ∫_{0}^{1}(f
′(x) g ′(x) +q(x)f(x)g(x))dx,
A weak solution u to the boundary value problem (BVP)
u''+q(x)u = f(x), u(0) = u(1) = 0, f ∈
L^{2}[0,1].
is a function u ∈ H such that for all
v ∈ H we have < u, v >_{H} =
∫_{0}^{1}v(x)f(x)dx.
 Show that, in the inner product above, H is a Hilbert
space. (Hint: use the fact that H^{1} is a Hilbert space.)
 Show that the BVP has a unique weak solution in H.
 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. 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
strongly 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.

Consider the Hilbert space $\ell^{\,2}$. Let $S=\{\{a_j\}_{j=1}^\infty
\in \ell^{\,2}\colon \sum_{j=1}^\infty (1+j^2)\,a_j^2\le 1 \}$. Show
that $S$ is a compact 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 Wednesday, December 3.
 Read sections 3.5, 3.6, and 4.1.
 Do the following problems.
 Section 3.4: 2(c,d), 3
 Section 3.5: 1(b), 2(a)
 Section 3.6: 1(a), 5
 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.
 Let $F:C[0,1]\to C[0,1]$ be defined by $F[u](t) :=
\int_0^1(2+st+u(s)^2)^{1}ds$, $0\le t\le 1$. Let $\ \cdot
\:=\\cdot \_{C[0,1]}$. Let $B_r:=\{u\in C[0,1]\,\, \u\\le
r\}$.
 Show that $F: B_1\to B_{1/2}\subset B_1$.
 Show that $F$ is Lipschitz continuous on $B_1$, with Lipschitz
constant $0<\alpha<1$  i.e., $\F[u]F[v]\\le \alpha \uv\$.
 Show that $F$ has a fixed point in $B_1$.
Extra Problems  These are not to be handed
in.
 Section 4.1: 1(b), 4, 6
 Section 4.2: 1, 4, 8
 Section 4.3: 3
 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 $Kf(x) :=
\int_0^1G(x,y)f(y)dy$.
 Verify that $u=x^2$ satisfies the boundary conditions required
for it to be in the domain of $L$ and that $\langle Lu,u \rangle <
0$. Also verify if we let $u=x(x1)^2$, then $u$ is in the domain for $L$
and that $\langle Lu,u \rangle > 0$. Thus $L$ does not satisfy the
conditions of Keener's Theorem 4.7.
 Even though the Theorem 4.7 is not satisfied, the orthonormal set
of eigenfunctions for $L$ form a complete set in $L^2[0,1]$,
Explain.
Updated 12/8/2014.