Math 641-600 Fall 2021
Assignments
Assignment 1 - Due Wednesday, 9/8/2021.
- For the problems, refer to sections 1.1 and my notes
on
Inner Products. Read section 2.1 and the notes on
Banach spaces and Hilbert Spaces
- Do the following problems.
- Section 1.1: 4, 5, 8, 10 (In 10, do $\{1,x,x^2\}$, but without
software.)
- In problem 1.1.4, suppose that the continuous functions are
replaced by the space of polynomials of degree $n-1$ or less,
$\mathcal P_{n-1}=\{a_0+a_1 x+ \cdots+a_n x^{n-1}\}$. 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?)
- Let $V$ be a finite dimensional, complex inner product
space. Suppose that $B=\{u_1,u_2, \ldots,u_n\}$ is an ordered
orthonomal basis for $V$. If $v=\sum_{j=1}^n a_j u_j$ and
$w=\sum_{j=1}^n b_j u_j$, show that $\langle v,w\rangle = \sum_{j=1}^n
a_j\bar b_j=[w]_B^*[v]_B$, where $[v]_B$ and $[w]_B$ are the
coordinate vectors for $v$ and $w$, respectively. As usual, $[w]_B^*$
is the transpose conjugate of $[w]_B$.
- 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 i.e., for all $\mathbf
x\in \mathbb C^n$, $\mathbf x\ne \mathbf 0$, we have $\mathbf
x^*G\mathbf x >0$. Why is $G$ invertible? (Hint: $B$ is linearly
independent.)
- In the problem above, 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 Wednesday, 9/15/2021.
- Read Keener's section 2.2.1 and my notes
on Lebesgue
integration and
on Orthonormal
sets and expansions.
- Do the following problems.
- Section 2.1: 10, 11
- 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.)
- 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 $\ell^2$, with $\langle x,y\rangle = \sum_{j-1}^\infty
x_j \bar y_j$ being the inner product, is a Hilbert space.
- Show that, in terms of the partial sums in eqn. 2 of my notes, the
integral of a simple function ends up being the one in eqn. 3.
- This problem illustrates how to find the (improper) integral of
an unbounded function. Consider the function $f(x)=x^{-1/2}$. Use the
formula in the eqn. 5 of the notes to show that, in the Lebesgue
sense, $\int_0^1 f(x)dx=2$. Do this just using $x^{-1/2}$ and its properties.
- 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 3 - Due Friday, 9/24/2021.
- 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 $\lim_{\delta \downarrow 0} \omega(f,\delta) = 0$.
Assignment 4 - Due Wednesday, 9/29/2021.
- Read sections 2.2.2-2.2.4 and the notes on
Approximation
of Continuous Functions.
- Do the following problems.
- Section 2.2: 12
- In proving the Weierstrass Approximation, we did the case
$x>j/n$. Do the case $x < j/n$.
-
- 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]} \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 $\|f - s_f\|_{C[0,1]} \le (8n^2)^{-1}\|f''\|_{C[0,1]}$.
- Let $f(x)$ be continuous on $[0,1]$ and let $s_f(x)$ be the
linear spline for $f$ with equally spaced points $j/n$, where $j=0,
1,2,\ldots,n$.
- Show that $\int_0^1s_f(x)dx$ is equal to the trapezoidal
(quadrature) rule for approximating $\int_0^1f(x)dx$.
- Let $E=\big|\int_0^1f(x)dx - \int_0^1 s_f(x)dx\big|$ be the
quadrature error. If $f\in C^2[0,1]$, use the previous problem to show
that $E\le (8n^2)^{-1}\|f''\|_{C[0,1]}$.
- Show that, in terms of the Bernstein polynomials $\beta_{j,n}$,
\[
x^k = \sum_{j=k}^n\frac{\binom{j}{k}}{\binom{n}{k}}\beta_{j,n}(x),
\]
where $k=0,1, 2, \ldots,x^n$.
Assignment 5 - Due Wednesday, 10/6/2021.
- 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: 2(a,c) (Just find the series.), 14
- 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,
$\int_{-\pi+c}^{\pi+c} g(u)du=\int_{-\pi}^\pi g(u)du$.
- Compute the Fourier series for the following functions. For (a)
and (b), use the symmetry of the function to help find the series.
- $f(x) = x$, $-\pi < x < \pi$. (sine/cosine form)
- $f(x) = |x|$, $-\pi \le x \le \pi$. (sine/cosine form)
- $f(x) = e^{2x}$, $-\pi < x < \pi$. (complex form).
- Compute the complex form of the Fourier series for $f(x) =
e^{2x}$, $0 < x < 2\pi$. Why is this different from 3(c) above?
Use this Fourier series and Parseval's theorem to sum the series
$\sum_{k=-\infty}^\infty (4+k^2)^{-1}$.
Assignment 6 - Due Wednesday, 10/20/2021.
- Read sections 2.2.4 and 2.2.7, the notes on
the
discrete Fourier transform and the notes
on
Splines and Finite Element Spaces.
- Do the following problems.
- 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 the formula in problem 2, assignment
5.))
- Prove the Convolution Theorem for the
DFT. (See
Notes on the Discrete Fourier Transform, pg. 3.)
- Let $\mathbf a\in \mathcal S_n$ and consider an n×n matrix
$\mathbf A$ whose first column is $\mathbf a=[a_0 \ a_1 \ \cdots \
a_{n-1}]^T$, and whose remaining columns are entries of $\mathbf a$
cyclically permuted. For example, the second colunm is $[a_{n-1} \ a_0
\ \cdots \ a_{n-2}]^T$, third, $[a_{n-2} \ a_{n-1} \ a_0 \cdots \
a_{n-3}]^T$ and so on. $\mathbf A$ is said to be a cyclic or circulant
matrix.
- Consider the the matrix equation $\mathbf A \mathbf x= \mathbf
y$, where $\mathbf x$ and $\mathbf y$ are columns vectors. These
vectors may be extended to vectors in $\mathcal S_n$ by repeating
the entries $0,\ldots,n-1$ to form a sequence. For $\mathbf x$
this is given by
\[
\mathbf x = \{\cdots x_{0} \ x_{1} \ \cdots \
x_{n-1} \ x_0 \ x_1 \ \cdots \
x_{n-1} \ x_{0} \ x_{1} \ \cdots \
x_{n-1}\ \cdots\}
\]
Show that the equation $\mathbf y = \mathbf A \mathbf x$ is equivalent
to the convolution equation $\mathbf y = \mathbf a\star \mathbf x$.
- Use the DFT and the convolution theorem to show that the
eigenvalues of a cyclic matrix $\bf A$ are the entries in $\widehat
{\mathbf a}$, the DFT of $\mathbf a$.
- Use your favorite software to find $\widehat {\mathbf a}$, and
hence, the eigenvalues of the matrix below. Also, find the
corresponding eigenvectors.
\[
\begin{pmatrix}
3 &5 &4 &1 \\
1 &3 &5 &4 \\
4 &1 &3 &5\\
5 &4 &1 &3
\end{pmatrix}
\]
- Let $f(t)=10\cos(2t)$ and consider the ODE $u''+2u'+2u=f(t)$.
- Find the general solution $u$ to the equation, and use it to
obtain its "steady state" periodic solution, $u_p$.
- Let $n=2^L$ and $h=\frac{2\pi}{n}$. For $L=3,5,8,\text{and}\ 10$,
sample $f$ at $jh$, $j=0\ldots n-1$; let $f_j:=f(jh)$. Use your
favorite program to find the FFT of $\{f_0,f_1,\ldots,f_{n-1}\}$ and,
using the method outlined in the notes on
the
discrete Fourier transform, find $\hat u_k$. Finally, apply your
program's inverse FFT to the $\hat u_k$'s to obtain the approximation
$u_j$ to $u_p(jh)$. For each of these $L$, plot the approximate solution and
the exact solution. (The $u_j$'s may have a small complex part due to
roundoff error; just plot the real parts of the $u_j$'s you found by
the procedure above.) Be sure to label your plots.
- 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.
Assignment 7 - Due Friday, October 29, 2021.
- Read sections 3.1-3.3, the notes on Bounded
Operators & Closed Subspaces.
on
the projection theorem, the Riesz representation theorem, etc, and
the notes on
an
example of the Fredholm alternative and finding a resolvent
.
- Do the following problems.
- Section 2.2: 26(a,c), 27(a)
- 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 $S_0^{1/n}(3,1)$.
- Let V be a Banach space. Show that a linear operator
L:V → V is bounded if and only if L is continuous.
- Consider the Sobolev space $H^1[0,1]$, with the inner product
$\langle f, g\rangle_{H^1} := \int_0^1 \big(f(x)\overline {g(x)} +
f('x)\overline {g'(x)}\big)dx$. For $f\in H^1$, let $Df=f'$. Show that
$D:H^1[0,1]\to L^2[0,1]$ is bounded, and that $\|D\|_{H^1 \to L^2}=1$.
- 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
$\|L\|_{C[0,1]\to C[0,1]}\le 1$.
- 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:
- $\|L\| = \sup \{\|Lu\| : u \in {\mathcal H},\ \|u\| = 1\}$
- $\|L\| = \sup \{|\langle Lu,v\rangle| : u,v \in {\mathcal H},\
\|u\|=\|v\|=1\}$
Assignment 8 - Due Friday, November 5, 2021.
- Read sections 3.3-3.5, and my notes on Compact
Operators, and on the
Closed Range Theorem.
- Do the following problems.
- Section 3.3: 1 (Assume the appropriate
operators are closed and that λ is real.)
- Section 3.4: 2(b)
- Consider all (absolutely) continuous functions $f$ whose
derivatives are in $L^2[0,1]$ -- i.e., $f(x)=\int_0^x f'(t)dt$ --,
and satisfy $f(0)=0$ and $f(1)=0$. Let $H_0$ be the set of all such
$f$. Show that $\langle f,g\rangle_{H_0} :=\int_0^1f'(x)g'(x)dx$
defines a real inner product on $H_0$. You are given that $H_0$ is a
Hilbert space.
- Show that if $f\in H_0$, then $\|f\|_{C[0,1]} \le
\|f\|_{H_0}$. Use this to show that $H_0$
is a reproducing kernel Hilbert space
- Show that for all $f\in H_0$ there exists a function $G_y$ in $H_0$
for which $f(y) = \langle f, G_y\rangle_{H_0}$. ($G(x,y):= G_y(x)$ is
called a reproducing kernel and associated with $H_0$.)
- Show that $G_x(y) = \big\langle G_x,G_y\big\rangle_{H_0}$;
equivalently,
\[
G(y,x)=\big\langle G(\cdot,x),G(\cdot,y)\big\rangle_{H_0}.
\]
Use this to show that $G(x,y)=G(y,x)$.
- Let $X:=\{ 0 < x_1 < x_2 < \cdots < x_n < 1 \}$. Show that the
$n\times n$ matrix $A_{j,k}:=G(x_j,x_k)$ is self adjoint and positive
definite.
- Let $X:=\{0 < x_1 < x_2 < \cdots < x_n < 1 \}$. Suppose that
$f\in H_0$ and $y_j:=f(x_j)$. Show that there are unique coefficients
$c_k$ for which $s(x) := \sum_{k=1}^n c_k G(x,x_k)$ interpolates $f$
at the points in $X$; i.e., $s(x_j)=y_j$, $j=1,\ldots n$.
- 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.
- Consider the finite rank (degenerate) kernel
k(x,y) =
φ1(x)ψ1(y) +
φ2(x)ψ2(y),
where φ1 = 6x-3, φ2 = 3x2,
ψ1 = 1, ψ2 = 8x − 6.
Let Ku= ∫01 k(x,y)u(y)dy. Assume that L =
I-λ K has closed range,
-
For what values of λ does the integral equation
u(x) - λ∫01 k(x,y)u(y)dy =f(x)
have a solution for all f ∈ L2[0,1]?
- For these values, find the solution u = (I −
λK)−1f i.e., find the resolvent.
- For the values of λ for which the equation
does not have a solution for all f, find a condition on f
that guarantees a solution exists. Will the solution be unique?
Assignment 9 - Due Friday, November 12, 2021.
- Read sections 3.3-3.5, and my notes on the
Closed Range Theorem.and my notes on
Spectral Theory for Compact Operators.
- Do the following problems.
- Section 3.4: 2(c,d), 6 (The condition in 6 should be $\lambda
\mu_i\ne 1$.)
- Section 3.5: 1(b), 2(b)
- (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 $\lambda u = Ku$.
- Let $L\in \mathcal B(\mathcal H)$. Suppose that for all $f\in
N(L)^\perp$ there is a constant $c>0$ such that $\|Lf\|\ge c\|f\|$, where
$c$ is independent of $f$. Show that $R(L)$ is closed.
- In the following, $\mathcal H$ is a Hilbert space, with inner
product $\langle Lu,v\rangle$, $\mathcal
B(\mathcal H )$ is the set of bounded linear operators on $\mathcal
H$. Let $L$ be in $\mathcal B(\mathcal H )$ and let $ N:= \sup
{|\langle Lu, u \rangle | : u\in \mathcal H, \|u\| = 1}$.
- Verify the identity
$\langle L(u+\alpha v), u+\alpha v\rangle - \langle L(u-\alpha v),
u-\alpha v\rangle = 2\bar \alpha \langle Lu, v\rangle + 2\alpha
\langle Lv, u\rangle$, where $\alpha \in \mathbb C$.
- Show that $N \le \|L\|$.
- Let $L=L^\ast$ be a self-adjoint operator on $\mathcal H$, which
may be a real or complex Hilbert space. Use (a) and (b) to show that
$N= \|L\|$. (Hint: In the complex case, choose $\alpha$ so that $
\alpha \langle Lu,v\rangle = |\langle Lu,v\rangle|$. For the real
case, use $\alpha=\pm 1$, as needed.)
- Suppose that $\mathcal H$ is a complex Hilbert space. If
$ L\in \mathcal B(\mathcal H)$. Use (a) and (b) to show that $\|L\| \le 2N$.
- For the real Hilbert space, $\mathcal H = \mathbb R^2$,
let $L = \begin{pmatrix} 0& 1\\ -1 & 0 \end{pmatrix}.\ $ Show that
$\|L\| = 1$, but $N=0$.
Assignment 10 - Due Friday, November 19, 2021.
- Read sections 3.3-3.5 my notes on
Spectral Theory for Compact Operators.
- Do the following problems.
- Section 3.6: 1(a,b)
-
In
problem 3, assignment 9, you found the eigenvalues and
eigenfunctions of the compact, self-adjoint operator $Ku(x) =
\int_{-1}^1 (1-|x-y|)u(y)dy$. Show that the eigenfunctions of $K$
form a complete orthogonal set.
- Let $M$ be a closed subspace of a Hilbert space $\mathcal H$ and suppose
that $K\in \mathcal C(\mathcal H))$. If $M$ is an invariant subspace
for $K$, show that $K|_M$ is also compact.
- Consider the matrix $A=\begin{pmatrix}1 & -3 & 7 & 1 \\ -5 & 2 &
1 &1 \\ 0 & 0 &9 & -1 \\ 0& 0 & -3 & 4 \end{pmatrix}$. Show that
$M:=\text{span}\{ [1 \ 0 \ 0 \ 0]^T, [0 \ 1 \ 0 \ 0]^T\}$ is invariant
under $A$, but $M^\perp$ is not.
- Let $k(x)$ be a 2$\pi$-periodic function in $L^2[-\pi,\pi]$ and
consider the operator $Ku(x):=\int_{-\pi}^\pi k(x-y)u(y)dy$. Use the
Fourier series $k(x)=\sum_{n=-\infty}^\infty k_n e^{i n x}$ to find
the eigenvalues and eigenfunctions of $K$.
Assignment 11 - Due Wedneday, December 1, 2021.
- Read sections 4.1, 4.2 and my notes on
example problems for distributions.
- Do the following problems.
- Section 4.1: 4, 7
- Section 4.2: 1, 3, 4
- 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 $0<\alpha \le 1/2$.
- Show that $F$ has a fixed point in $B_1$.
- Let $Lu=-u''$, $u(0)=0$, $u'(1)=2u(1)$.
- Show that $L$ is self adjoint.
- Find the Green's function, $g(x,y)$, for $L$.
- Let $Gf(x) := \int_0^1g(x,y)f(y)dy$. Show that $G$ is a
self-adjoint Hilbert-Schmidt operator, and that $0$ is not an
eigenvalue of $G$.
- Use (b) and the spectral theory of compact operators to show the
orthonormal set of eigenfunctions for $L$ form a complete set in
$L^2[0,1]$.
Assignment 12 - These are not to be handed
in.
- Section 4.3: 5, 6
- Suppose that $Lu= u''+\lambda u$, with Dom$(L)=\{u\in
L^2[0,\infty):Lu\in L^2[0,\infty)\ \text{and}\ u(0)=0\}$, where
$\lambda\in \mathbb C \setminus [0,\infty)$. In addition, choose
$\text{Im}\sqrt{\lambda}>0$. Show that the Green's function for $L$ is given by
\[ g(x,y, \lambda)=\begin{cases}
\frac{1}{\sqrt{\lambda}}\sin(\sqrt{\lambda} xe^{i\sqrt{\lambda} y} &
0\le x\le y<\infty\\ \frac{1}{\sqrt{\lambda}}\sin(\sqrt{\lambda}
y)e^{i\sqrt{\lambda}x} & 0\le y\le x<\infty \end{cases}\]
Hint: follow the procedure used in Keener, pg. 150, to solve a similar
problem.
- Consider a Sturm-Liuville operator
$Lu=-\frac{d}{dx}p(x)\frac{du}{dx}+q(x)u$, where $p\in C^1[0,1]$ and
satisfies $p(x)>0$. In addition, we suppose that $q\in C[0,1]$ and is
real valued. If Dom$(L)=\{u\in L^2[0,1]:Lu\in L^2[0,1] \ \text{and
}u(0)=0, \ u'(1)+u(1)=0\}$. Show that a Green's function exists if and
only if there is no homogeneos solution $u$ to $Lu=0$ that is also in
Dom$(L)$.
- Let $Lu=-\frac{d}{dx}x\frac{du}{dx}$, with Dom$(L)=\{u\in
L^2[0,1]: Lu\in L^2[0,1] \ \text{and}\ \lim_{x\downarrow 0}xu'(x)=0, \
u(1)=0\}$. Find the Green's function for $L$.
- Use the Courant-Fischer Theorem to show that the eigevalues of
the Sturm-Liousville operator $L$ defined in problem 3 above, subject
to $u(0)=0$ and the three boundary conditions $\{u'(1)=0\}$ (Neumann),
$\{u'(1)+\sigma u(1)=0,\ \sigma>0\}$ (Mixed) and $\{u(1)=0\}$
(Dirichlet), are ordered so that $\lambda_n^N \le \lambda^\sigma_n\le
\lambda^D_n$.
Updated 12/1/2021.