Math 641-600 Fall 2015
Assignments
Assignment 1 - Due Wednesday, September 9, 2015.
- Read sections 1.1-1.4
- 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) $\min_{u\in U}\|v-u\| = \|v-p\|$ or (ii) $v-p\in
U^\perp$, 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={φ1,.., φ1} be a basis for V
and let A be the matrix of L relative to E. Show that the
characteristic polynomial pA(λ) := det(A −
λ I) is independent of the choice of E, and so
pL(λ) := pA(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 $L(p)= \big((1-x^2)p'\big)' +
7p$. 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 self-adjoint 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
self-adjoint 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
uαvβ ≤ αu + βv.
- Let x,y ∈ Rn, and let p > 1 and define
q by q-1 = 1 - p-1. Prove Hölder's
inequality,
|∑j xjyj| ≤ ||x||p
||y||q.
Hint: use the inequality in part (a), but with appropriate choices of
the parameters. For example, u =
(|xj|/||x||p)p
- Let x,y ∈ Rn, 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
Rn. 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) : x2, -1 ≤ x ≤ 2. Find f
−1(Ej) for Ej = [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 non-zero 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 2.2.2-2.2.4
- 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 $\omega(f,\delta) := \sup_{\,|\,s-t\,|\,\le\,
\delta,\,s,t\in [0,1]}|f(s)-f(t)|$.
- 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 C2 on an interval
[a,b]. Let h = b − a. Show that if g(a) = g(b) = 0, then
||g||C[a,b] ≤ (h2/8)
||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 ∈
C2[0,1], then the equally spaced linear spline interpolant
fn satisfies
||f −
fn||C[0,1] ≤ (8n2) −
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 cn
einx, where ∑n |cn| <
∞. Show that ∑n cn einx
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 $\sum_{k=-\infty}^\infty
\big|c_k^{(1)}\big|^2$ 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
$k=0,\cdots ,n-1$.)
- Suppose that x is an n-periodic sequence (i.e., x
∈ Sn). Show that $ \sum_{j=m}^{m+n-1}{\mathbf
x}_j = \sum_{j=0}^{n-1}{\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 H0 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 ∈ H0 is also in
C(2)[0,1] and that u satisfies
⟨u,v⟩H0 = ∫01 f(x)
v(x) dx for all v ∈ H0.
Show that u satisfies the BVP.
- Consider S0 := {s ∈S1/n(1,0) :
s(0)=s(1)=0}. Show that S0 is spanned by φj(x) :=
N2(nx-j+1), j = 1 ... n-1. (Here, N2(x) is the
linear B-spline.)
- Show that the least-squares approximation s ∈
S0 to the solution u is given by s = ∑j
αjφj(x), where the
αj's satisfy Gα = β, with
βj = ⟨ y,φj
⟩H0 = ∫01 f(x)
φj(x) dx, j=1 ... n-1 and Gkj = ⟨
φj, φk ⟩H0.
- Show that Gkj = ⟨ φj,
φk ⟩H0 is given by
Gj,j = 2n, j = 1 ... n-1
Gj,j-1 = - n, j = 2 ... n-1
Gj,j+1 = - n, j = 1 ... n-2
Gj,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 $m=3$ B-spline, 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 $N_3(x) = \alpha \big( (x)_+^2
-3 (x-1)_+^2 +3 (x-2)_+^2 - (x-3)_+^2\big)$, 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 n-periodic sequences,
and let a, x, y be column vectors with entries a0, ...,
an-1, 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 Hilbert-Schmidt 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 $\|p-f\|=\min_{v\in
V}\|v-f\|$ 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 H0 := {v ∈
H1[0,1] : v(0) = v(1) = 0}, with the inner product < f,g
>H0 = ∫01f
′(x) g ′(x)dx. Show that there is a function gy
in H0 for which f(y) = < f,gy
>H0. (gy is called a
reproducing kernel and H0 is a reproducing
kernel Hilbert space.) Hint: Use
problem 4, HW3 to show that the Φy(f) :=
f(y) is a bounded linear functional on H0.
- 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 {fn} in a Hilbert space H is said to
be weakly convergent to f ∈ H if and only if lim n
→ ∞ < fn,g> = < f,g> for every
g∈H. When this happens, we write f = w-lim fn. For
example, if {φn} is any orthonormal sequence, then
φn 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 ||fn|| ≤ C for all n). Prove the following:
Let K be a compact linear operator on a Hilbert space
H. If fn converges weakly to f, then Kfn
converges to Kf that is, lim n → ∞ ||
Kfn - Kf || = 0.
Hint: Suppose this doesn't happen, then there will be a subsequence of
{fn}, say {fnk}, such that ||
Kfnk - 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 self-adjoint 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 $Ku(x) = \int_{-1}^1 (1-|x-y|)u(y)dy$ and the eigenvalue
problem $\lambda u = Ku$.
- Show that $K$ is a self-adjoint, Hilbert-Schmidt 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, self-adjoint operator and let M be the span
of the set of eigenvectors {φj} corresponding to all
eigenvalues λj ≠ 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
\[
\big\|(I - \lambda L)^{-1} - \sum_{k=0}^{n-1}\lambda^k L^k\big\| \le
\frac{|\lambda|^k \|L\|^k}{1 - |\lambda| \|L\|}.
\]
- 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
\[
G(x,y)=\left\{
\begin{array}{rl}
-(2y-1)x, & 0 \le x < y \le 1\\
-(2x-1)y, & 0 \le y< x \le 1.
\end{array} \right.
\]
- 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.