Math 641-600 Fall 2020
Assignments
Assignment 1 - Due Friday, 8/28/2020.
- Read sections 1.1-1.4 and my
notes Adjoints
and Self-Adjoint Operators, Finite Dimensional Case.
- Do the following problems.
- Section 1.1: 4, 5, 6, 9(a) (In 9(a), do the first 3, but without
software.)
- In problem 1.1.4, suppose that the continuous functions are
replaced by the space of polynomials of degree $n-1$, $\mathcal
P_{n-1}=\{a_0+a_1 x+ \cdots+a_n x^{n-1}\}$. Show that the formula for
$\langle F,G\rangle$, with $F,G \in \mathcal P_{n-1}$ is an inner
product for $\mathcal P_{n-1}$. (Hint: How many roots can a degree
$n-1$ polynomial have?)
- Prove the Corollary in my notes
on Inner
Products and Norms.
- 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 is a vector $p \in U$ that
satisfies either $\min_{u\in U}\|v-u\| = \|v-p\|$ or $v-p\in
U^\perp$, then it satisifes both. In addition, show that if such a
vector exists, then 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 $P$ has the following properties:
- $P$ maps $V$ onto $U$.
- $P$ is a linear map.
- $P$ satisfies $P^2 = P$ and $P^\ast=P$. ($P$ is called an
orthogonal projection. The vector $p$ is the orthogonal projection
of $v$ onto $U$.)
- $U^\perp= \{w\in V\colon Pw=0\}$ and $V=U\oplus
U^\perp$, where $\oplus$ indicates the direct sum of the two
spaces. (This and the next exercise are easy, but important.)
- $I -P$ is the projection of $V$ onto $U^\perp$, and that the
Pythagorean theorem, $\|v\|^2=\|Pv\|^2+\|(I-P)v\|^2$, holds.
- Let $U$ and $V$ be as in the previous exercise. Suppose that $U$
is finite dimensional and that $B=\{v_1,v_2,\ldots,v_n\}$ is an
ordered basis for $U$. In addition, let $G$ be the $n\times n$
matrix with entries $G_{jk}= \langle v_k,v_j\rangle$.
- Show that $G$ is positive definite and thus invertible.
- In part 3(a), you showed that $p\in U$ is the minimizer of
$\min_{u\in U}\|v-u\|$ if and only if $v-p$ is in $U^\perp$. Use
this to show the following: Let $v\in V$ and $d_k := \langle
v,v_k\rangle$. Show that $p$ exists for every $v$ and, relative to
the basis $B$, is given by $p=\sum_j x_j v_j\in U$, where the
$x_j$'s satisfy the normal equations, $d_k = \sum_{j=1}^n
G_{kj}x_j$. (In matrix form, $\mathbf x = G^{-1}\mathbf d$.) Remark:
Since the normal equations don't depend on the choice of $B$, $p$
itself is independent of the choice of basis.
- Show that if B is orthonormal, then $p=Pv=\sum_j \langle
v,v_j\rangle v_j$, so $G=I$ in this case.
Assignment 2 - Due Friday, September 4, 2020.
- Read the notes
on
Banach spaces and Hilbert Spaces, and sections 2.1 and 2.2 in
Keener.
- Do the following problems.
- Section 1.2: 1(a,b), 3(a,b)
- Section 1.3: 2(a,b)
- Section 1.4: 4
- Show $\lambda_2<0$ for
$
A=\begin{pmatrix} 0 & 1 & 3\\
1& 0 &2\\
3 & 2 & 0
\end{pmatrix}.
$
- Let $V$ be an $n$ dimensional vector space and suppose $L:V\to V$ is
linear.
- Let $B=\{v_1,v_2,\ldots,v_n\}$ and $C=\{w_1,w_2,\ldots,w_n\}$ be
bases for $V$, and let $A_L$ and $\tilde A_L$ be the matrices of $L$
relative to $B$ and $C$, repectively. Show that
$p_{A_L}(\lambda)=p_{\widetilde A_L}(\lambda)$, where the $p_{A_L}$ and
$p_{\widetilde A_L}$ are the characteristic polynomials for $A_L$ and
$\tilde A_L$. Thus, the characteristic polynomials are independent of
the choice of basis, and so $p_{L} = p_{A_L}$ is well defined.
- Use the previous part to show that $\text{Trace}(L) :=
\text{Trace}(A_L)$ is independent of the basis chosen, and is also
well defined.
- Let $L:P_2\to P_2$ be given by $L(p)= \big((1-x^2)p'\big)' +
7p$. Find $p_L(\lambda)$ and $\text{Trace}(L)$.
- Let $\mathcal P_n$ be the polynomials of degree $n$ or less. If
$L(u) = 2xu'-u''$, show that $L:\mathcal P_n \to \mathcal P_n$. For
$n=2$, find the eigenvalues and eigenfunctions (these are polynomials)
of $L$.
- 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.)
Assignment 3 - Due Friday, September 11, 2020.
- Read Keener's section 2.2 and my notes
on Lebesgue
integration.
- Do the following problems.
- Section 1.3: 5
- This problem concerns several important inequalities.
- Show that if $\alpha, \beta$ are positive and $\alpha + \beta
=1$, then for all $u,v \ge 0$ we have $ u^\alpha v^\beta \le \alpha u
+ \beta v$.
- Let $x,y \in \mathbb C^n$ (or $\mathbb R^n$), $1 < p <\infty$, and define
$q$ by $q^{-1}= 1 - p^{-1}$. Prove Hölder's
inequality,
$\sum_{j=1}^n |x_j| |y_j| \le \|x\|_p \|y|\|_q, $
where $\|x\|_p=(\sum_{j=1}^n|x_j|^p)^{1/p}$ and
$\|x\|_\infty=\sup_{1\le j\le n}|x_j|$. (Hint: use the inequality in
part (a), but with appropriate choices of the parameters. For example,
$u = (|x_j|/\|x\|_p)^p$.
- Let $x,y \in \mathbb C^n$, and let $1 \le p \le\infty$. Prove
Minkowski's inequality,
$\|x+y\|_p \le \|x\|_p + \|y\|_p$.
(Hint: you will need to use Hölder's
inequality, along with a trick.)
- Consider the vector space of sequences
$\{x=\{x_j\}_{j=1}^\infty\}$, where the $x_j$'s can be real or
complex. For $1\le p<\infty$, let
$\ell^p=\{x=\{x_j\}_{j=1}^\infty\colon \|x\|_p=(\sum_{j=1}^\infty
|x_j|^p)^{1/p}<\infty\}$ and, for $p=\infty$, $\ell^\infty =
\{x=\{x_n\}_{n=1}^\infty\colon \|x\|_\infty=\sup_j|x_j|<\infty
\}$. Use the inequalities from the previous problem to show that, for
$1\le p\le \infty$, $\|x\|_p$ defines a norm on $\ell^p$.
- Show that $\ell^2$ is an inner product space, with $\langle
x,y\rangle = \sum_{j-1}^\infty x_j \bar y_j$ being the inner product, and
that with this inner product it is a Hilbert space. Bonus: show that
it is separable.
- Let $C^1[0,1]$ be the set of all continuously differentiable
real-valued functions on $[0,1]$. Show that $C^1[0,1]$ is a Banach
space if $\|f\|_{C^1} := \max_{x\in [0,1]}|f(x)| + \max_{x\in
[0,1]}|f'(x)|$.
- 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 $\|f\|_{H^1[0,1]}^2 := \int_0^1\big( |f(x)|^2 +
|f'(x)|^2\big)dx$.
Assignment 4 - Due Friday, September 18, 2020.
- Read the notes on
Lebesgue
integration and
on Orthonormal
sets and expansions.
- Do the following problems.
- Section 2.1: 10, 11
- A measurable function whose range consists of a finite number of
values is a simple function
see Lebesgue
integration, p. 5. Use the definition of the Lebesgue integral in
in terms of Lebesgue sums, from eqn. 2, to show that, in terms of this
definition, the integral of a simple function ends up being the one in
eqn. 3 on p. 6.
- Consider the function $f:[0,1] \to \{0,1\}$, where $f(x)=0$, if
$x=1/n$, $n=1,2, 3, \ldots$ and $f(x)=1$, otherwise. Show that the
Lebesgue integral of $f$ is 1. Does the Riemann integral for this
function exist? Prove your answer.
- Consider the function $f(x)=x^{-1/2}$. Use the formula in the
middle of p. 6 of the notes to show that, in the Lebesgue sense,
$\int_0^1 f(x)dx=2$. (This is a standard improper Riemann integral
whose value is also 2.)
- Let F(s) = ∫ 0∞ e − s
t f(t)dt be the Laplace transform of f ∈
L1([0,∞)). Use the Lebesgue dominated convergence
theorem to show that F is continuous from the right at s = 0. That is,
show that
lim s↓0 F(s) = F(0) = ∫
0∞f(t)dt.
- Let fn(x) = n3/2 x e-n x, where
x ∈ [0,1] and n = 1, 2, 3, ....
- Verify that the pointwise limit of fn(x) is f(x) = 0.
- Show that ||fn||C[0,1] → ∞ as n
→ ∞, so that fn does not converge uniformly to
0.
- Find a constant C such that for all n and x fixed
fn(x) ≤ C x−1/2, x ∈ (0,1].
- Use the Lebesgue dominated convergence theorem to show that
lim n→∞ ∫ 01
fn(x)dx = 0.
Assignment 5 - Due Wednesday, September 30, 2020.
- Read sections 2.2.2-2.2.4 and the notes on
Approximation of Continuous Functions.
- Do the following problems.
- Section 2.2: 1 (Use $w=1$.), 8(a,b,c) (FYI: the formula for
$T_n(x)$ has an $n!$ missing in the numerator.), 9, 10
- 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 $f=\sum_{j=1}^\infty \langle f, u_j\rangle u_j$.
- This problem is aimed at showing that the Chebyshev polynomials
form a complete set in $L^2_w$, which has the weighted inner product
\[ \langle f,g\rangle_w := \int_{-1}^1
\frac{f(x)\overline{g(x)}dx}{\sqrt{1 - x^2}}. \]
- Show that the continuous functions are dense in $L^2_w$. Hint: if
$f\in L^2_w$, then $ \frac{f(x)}{(1 - x^2)^{1/4}}$ is in $L^2[-1,1]$.
- Show that if $f\in L^\infty[-1,1]$, then $\|f\|_w \le
\sqrt{\pi}\|f\|_\infty$.
- Follow the proof given in the notes
on
Orthonormal Sets and Expansions showing that the Legendre
polynomials form a complete set in $L^2[-1,1]$ to show that the
Chebyshev polynomials form a complete orthogonal set in $L^2_w$.
- Let $\delta>0$. We define the modulus of continuity for $f\in
C[0,1]$ by $\omega(f,\delta) := \sup_{\,|\,s-t\,|\,\le\,
\delta,\,s,t\in [0,1]}|f(s)-f(t)|$.
- 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 $\lim_{\delta \downarrow 0} \omega(f,\delta) = 0$.
- Let $g$ be in $C^2[a,b]$. Let $h = b-a$.
- Show that if g(a) = g(b) = 0, then $ \|g\|_{C[a,b]} \le (h^2/8)
\|g''\|_{C[a,b]}$. Give an example that shows that $1/8$ is the best
possible constant.
- Use the previous part to show that if f ∈
C2[0,1], then the equally spaced linear spline interpolant
$s_f$ satisfies $\|f - s_f\|_{C[0,1]} \le (8n^2)^{-1}\|f''\|_{C[0,1]}$.
- Show that $ x^k =
\sum_{j=k}^n\frac{\binom{j}{k}}{\binom{n}{k}}\beta_{j,n}(x)$,
$k=0,\ldots, n$.
Assignment 6 - Due Monday, October 19, 2020.
- Read sections 2.2.2-2.2.4, the notes
on Fourier
series, and the notes on
the
discrete Fourier transform.
- Do the following problems.
- Section 2.2: 14, 16
- Prove this: Let $g$ be a $2\pi$ periodic function (a.e.) that is
integrable on each bounded interval in $\mathbb R$. Then,
$\int_{-\pi+c}^{\pi+c} g(u)du$ is independent of $c$. In particular,
this imples that $\int_{-\pi+c}^{\pi+c} g(u)du=\int_{-\pi}^\pi
g(u)du=\int_0^{2\pi} g(u)du$.
- Compute the Fourier series for the following $2\pi$ functions. In
each case sketch three periods of the function to which the series
converges.
- $f(x) = x$, $0\le x \le 2\pi$. (sine/cosine form)
- $f(x) = |x|$, $-\pi \le x \le \pi$. (sine/cosine form)
- $f(x) = e^{2x}$, $-\pi \le x \le \pi$. (complex form)
- $f(x) = e^{2x}$, $0 \le x \le 2\pi$. (complex form) Does the FS
you get contradict the result in problem 2 above? Explain your answer.
- Use the FS from 6(d) above 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 $L^2[-\pi,\pi]$.
- Show that the FS for a linear spline $s(x)$ that satisfies
$s(-\pi)=s(\pi)$ is uniformly convergent to $s(x)$ on $[-\pi,\pi]$.
- Show that such splines are dense in $L^2[-\pi,\pi]$.
- Show that $\{e^{inx}\}_{n=-\infty}^\infty$ is complete in
$L^2[-\pi,\pi]$.
- Let $\mathcal S_n$ be the set of $n$-periodic,
complex-valued sequences.
- Suppose that $\mathbf x \in \mathcal S_n$. 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 problem 2 above.)
- Prove the Convolution Theorem for the
DFT. (See
Notes on the Discrete Fourier Transform, pg. 3. This is the DFT
analogue of problem 1 above.)
Assignment 7 - Due Wednesday, October 28, 2020.
- Read section 2.2.7 and the notes
on
Splines and Finite Element Spaces.
- Do the following problems.
- Section 2.2: 25(a,b), 26(b)
- A $n\times n$ matrix $A$ is called a circulant if all of its
diagonals are constant. Equivalently, $A$ is a circulant if the
columns of $A$ are ciclic permutations of its first column. For
example, the matrix below is a circulant.
\[
\begin{pmatrix}
3 &5 &4 &1 \\
1 &3 &5 &4 \\
4 &1 &3 &5\\
5 &4 &1 &3
\end{pmatrix}
\]
- Suppose that $\mathbf a \in \mathbb C^n$ is the first column of a
circulant matrix $A$. Let $\alpha \in \mathcal S_n$, where, for
$j=0,\ldots n-1$, $\alpha_j=\mathbf a_j$. In addition, let $x,y$ be
column vectors in $\mathbb C^n$, with indexes starting at $j=0$
instead of $j=1$. Then let $\xi,\eta \in \mathcal S_n$, such that
$\xi_j=x_j$, $\eta_j=y_j$, for $j=0,\ldots, n-1$. Show that $Ax=y$ is
equivalent to $\eta = \alpha \ast \xi$.
- Use the DFT and the convolution theorem to show that the
eigenvalues of $A$ are the coefficents of $\widehat {\mathbf a}$.
- Find the corresponding eigenvectors of $A$.
- Use this method, along with your favorite software, to find the
eigenvalues and eigenvectors of the matrix given as an exmple
above. (For this matrix, $\mathbf a =(3\ 1\ 4\ 5)^T$.)
- Let $S^{1/n}(1,0)$ be the space of piecewise linear splines, with
knots at $x_j=j/n$, and let $N_2(x)$ be the linear B-spline ("tent
function", see Keener, p. 81 or my notes on splines.)
- Let $\phi_j(x):= N_2(nx +1 -j)$. Show that
$\{\phi_j(x)\}_{j=0}^n$ is a basis for $S^{1/n}(1,0)$.
- Let $S_0^{1/n}(1,0):=\{s\in S^{1/n}(1,0):s(0)=s(1)=0\}$. Show that
$S_0^{1/n}(1,0)$ is a subspace of $S^{1/n}(1,0)$ and that
$\{\phi_j(x)\}_{j=1}^{n-1}$ is a basis for it.
- Consider the space of cubic Hermite splines
$S_0^{1/n}(3,1)\subset S^{1/n}(3,1)$ that satisfy $s(0)=s(1)=0$. Show
that $\langle u,v\rangle = \int_0^1 u''v''dx$ defines an inner product
on $S_0^{1/n}(3,1)$.
Assignment 8 - Due Wednesday, November 11, 2020.
- Read sections 3.1-3.3, the notes
on
the projection theorem, the Riesz representation theorem, etc, on
a
resolvent
example, and on
compact operators.
- Do the following problems.
- Section 3.1: 2
- Section 3.2: 3(c) (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.
- Let $V$ be a closed subspace of a Hilbert space $\mathcal
H$. Show that $\mathcal H =V\oplus V^\perp$, and that
$(V^\perp)^\perp= V$.
- Let $k(x,y)$ be defined by
\[
k(x,y) = \left\{
\begin{array}{cl}
y, & 0 \le y \le x\le 1, \\
x, & x \le y \le 1.
\end{array}
\right.
\]
-
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
$\|L\|_{C[0,1]\to C[0,1]}\le 1$. Bonus (5 pts.): Show that
$\|L\|_{C[0,1]\to C[0,1]}=1/2$.
- 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.
- 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 Hilbert space $\mathcal H=\ell^2$ and let
$S=\{x=(x_{1}\ x_{2}\ x_3\ \ldots)\in \ell^2:
\sum_{n=1}^\infty (n^2+1)|x_n|^2 <1\}$. Show that $S$ is a
precompact subset of $\ell^2$.
- Show that every compact operator on a Hilbert space is bounded.
Assignment 9 - Due Monday, November 30, 2020.
- Read sections 4.1, 4.2, 4.3.1, 4.3.2, 4.5.1 and my notes on and
my notes on
example problems for distributions.
- Do the following problems.
- Section 3.5: 2(b)
- Section 4.1: 4, 7
- Section 4.2: 1, 3
- Section 4.3: 3
- Let $Kf(x)=\int_0^1 k(x,y)f(y)dy$, where $k(x,y)$ is defined by $
k(x,y) = \left\{
\begin{array}{cl}
y, & 0 \le y \le x\le 1, \\
x, & x \le y \le 1.
\end{array}
\right.$
- Show that $0$ is not an eigenvalue of $K$.
- Show that $Kf(0)=0$ and $(Kf)'(1)=0$.
- Find the eigenvalues and eigenvectors of $K$. Explain why the
(normalized) eigenvectors of $K$ are a complete orthonormal basis for
$L^2[0,1]$. (Hint: Show that $K$ is a Green's function for $Lu=-u''$,
$u(0)=0$, $u'(1)=0$.)
- Let $Lu=-u''$, $u(0)=0$, $u'(1)=2u(1)$.
- Show that the Green's function
for this problem is
\[
G(x,y)=\left\{
\begin{array}{rl}
-(2y-1)x, & 0 \le x < y \le 1\\
-(2x-1)y, & 0 \le y< x \le 1.
\end{array} \right.
\]
- Let $Kf(x) := \int_0^1 G(x,y)f(y)dy$. Show that $K$ is a self-adjoint
Hilbert-Schmidt operator, and that $0$ is not an eigenvalue of $K$.
- Use (b) and the spectral theory of compact operators to show the
orthonormal set of eigenfunctions for $L$ form a complete set in
$L^2[0,1]$.
- Do the following problems.
Updated 11/19/2020.