Math 658-600 Assignments — Fall 2012
Assignment 1 -Due Friday, 9/7/12.
- Read sections 1.3, 2.1-2.3.
- Problems
- Let f be in L1(R) and define the translation
operator Tuf(x) = f(x-u). The aim of this problem is
to show that limu → 0 ‖f -
Tuf‖1 = 0.
- Show that for all f in L1(R) the translation
operator satisfies ‖Tuf‖1 = ‖f‖1.
- Let g:R → R be continuous and have compact
support. Show that for ε > 0
there is a δ > 0 such that
‖g - Tug‖1 < ε, provided |u| <
δ.
- Recall that a function in L1(R) can be
approximated arbitrarily closely by a compactly supported,
continuous function. Use this to show that
limu → 0 ‖f - Tuf‖1 = 0.
- Give an example to show that this (the limit above) fails in
L∞(R)
- Verify the Fourier series for the functions in entries 1, 5, 6,
13, 18, 19 of Table 1, p. 26-28.
- Consider the trig polynomial T(x) = ∑|n|≤N
dn einx.
- Show that the Fourier coefficients of T are just the
dn's for |n|≤ N and 0 for |n|> N.
- Show that if f has the Fourier series ∑n
cn einx, then
(2π)-1
∫−ππ f(x)T(x)dx
=
∑|n|≤N
cndn.
- Show that (2π)-1 ∫−ππ
|T(x)|2dx = ∑|n|≤N
|dn |2, so Bessel's inequality becomes an
equation for trig polynomials.
- Show that if f has the Fourier series ∑n
cn einx, then
(2π)-1
∫−ππ|f(x) − T(x)|2dx
= (2π)-1
∫−ππ|f(x)|2dx −
∑|n|≤N |cn|2
+ ∑|n|≤N |dn
− cn|2
Assignment 2 Due Monday, 9/17/2012.
- Read sections 2.4-2.6, 7.1-2.
- Problems
- Section 2.2: 7
- Section 2.3: 5
- Section 2.6: 1
- Let p=1 or ∞. Suppose that f ∈ Lp and g
∈ L1 are both $2\pi$ periodic functions. Show that
the convolution
f∗g(θ) :=
(2π)-1 ∫02π f(θ
− φ)g(φ)dφ
is in Lp and that ‖f∗g‖p ≤
(2π)-1‖f‖p
‖g‖1.
Bonus: Show that the inequality above holds for all 1 ≤ p
≤ ∞. (You will need to use both the Fubini-Tonelli Theorem
and Hölder's inequality.)
-
The Cesàro means of the partial sums sn(θ) of
a Fourier series a function f are defined to be
σn(θ) =
(n+1)−1∑0≤k≤n
sk(θ)
- Show that σn(θ) = ∑|k|≤
n(1 − |k|∕(n+1)) ckeikθ.
- The Fejér kernel Fn(u) is defined to be the
Cesàro mean of the Dirichlet kernels Dk(u). Show
that
Fn(u) = [2π(n+1)
sin2(½u)]− 1
sin2(½(n+1)u).
- Show that σn(θ) is also given by the
convolution
σn(θ) = 2πFn
∗f(θ).
- Show that if f is a 2π periodic continuous function, then
σn converges uniformly to f.
Assignment 3 Due Monday, 9/24/2012.
- Read sections 7.3, 7.5, 7.6
- Problems
- Section 7.1: 2, 3(a,b) (just do f∗f in (a)), 5, 7
- Section 7.2: 4, 7, 12(a), 13(b), 14 (Note: Problem 14 has a
misprint in it. The Hermite functions are defined on the bottom of
p. 186 and their properties are discussed on p. 187.)
- Suppose that f ∈ C1(R), and that both f
and f′ are bounded on R, and that g ∈
L1(R). Show that f∗g ∈
C1(R) and that (f∗g)′ =
f′∗g.
Assignment 4 Due Wednesday, 10/3/2012.
- Read my
Notes on the Discrete Fourier Transform.
- Problems
- Section 7.1: 8
- Section 7.2: 13(a)
- Section 7.3: 6, 8, 9, 10 (There is a misprint in 10. f′ +
cf = 0 should be f′ + cxf=0.)
- Section 7.5: 2
- Suppose that φ: Rn → C and that
φ ∈ L1(Rn) ∩
C0(Rn). Furthermore, suppose that $\hat
\varphi(\xi) > 0$. Prove the following result: If $\{x_j\}_{j=1}^N$
is a finite set of distinct points in Rn, then the
matrix $A$ with entries $A_{j,k} = \varphi(x_j - x_k)$ is Hermitian
and positive semi-definite. (Note: $\hat \varphi$ is in
L1(Rn). Prove it for a 5 pt. bonus.)
- Let f(t) be band-limited, with its Fourier transform f^(ω)
(f "hat") having support in [−2π, 2π]. The
Nyquist rate for f is 2π/π = 2, and the
Nyquist frequency is ½·2 = 1.
-
Show that if f is sampled at the Nyquist frequency, instead of the
Nyquist rate, then the sampling theorem doesn't recover f. Instead, it
gives another band-limited function, falias. Show that on
[−π, π],
f^alias(ω) = f^(ω − 2π) + f^(ω)
+ f^(ω + 2π),
and that f^alias = 0 otherwise.
- Strictly speaking, the function eiω0t
doesn't have a Fourier transform in the usual sense. However, if we
allow tempered distributions, then it does have one,2π δ(ω
− ω0). Another example of a function with a
distributional Fourier transform is cos(ω0t). Its
Fourier transform is 2π(δ(ω − ω0) +
δ(ω + ω0)/2. Keeping this in mind and
referring to the previous part, consider the band-limited function
f(t) = sin(5πt/3+π/6),
whose angular frequencies ±5π/3 are in the band [−2π,
2π]. What is falias(t) in this case? Use your favorite
software to make a plot showing f(t), falias(t) on the
interval − 6 ≤ t ≤ 6, and also use circles or x's to
indicate samples of these two functions at integer values of t.
Assignment 5 Due Friday, 10/12/2012.
- Read the article Radon Inversion — Variations on a Theme, by Robert S. Strichartz, Amer. Math. Monthly, 89 (1982), 377-384.
- Problems
- Section 7.5: 3, 6(b)
- Prove the three formulas listed at the end of section 2 on p. 3
of the
Notes on the Discrete Fourier Transform.
- Prove: Let L be a linear filter with impulse response h, where h
is in L1(R) and is continuous. Then, if L is
causal, h(t) = 0 for all t < 0. (This completes the proof
of Proposition 3.1 in
my
Notes on Filters.)
- Let {x1, x2, ..., xN} be a set
of distinct points in R. Show that the set of functions
{exp(ix1t), exp(ix2t), ...,
exp(ixNt)}
is linearly independent. (Hint: Consider a Fourier transform pair $f,
\hat f$ such that both functions are in L1, and hence also
in C0(R). Then,
c1exp(ix1ξ) +
c2exp(ix2ξ) + ... +
cN exp(ixNξ) = 0, for all ξ,
is equivalent to
c1f(x1) +
c2f(x2) + ... +
cNf(xN) = 0 for all f in L1.
Choose the pair $f$ and $\hat f$ wisely.)
- (This problem is a follow-up to problem 5, HW 4.) Suppose that
φ: R → C and that φ ∈
L1(R) ∩
C0(R). Furthermore, suppose that $\hat
\varphi(\xi) > 0$ on a set of positive measure. Use the previous
problem to prove the following result: If $\{x_j\}_{j=1}^N$ is a
finite set of distinct points in R, then the
matrix $A$ with entries $A_{j,k} = \varphi(x_j - x_k)$
is strictly positive definite. You may use the fact
that $\hat \varphi$ is in L1(R).
- Let f: R → C be continuous and have support in
the interval [0, A], and let n be a positive integer. If
yj = f(jΔt), where Δt = A/n, then show that
\[ \hat f(kΔω) \approx \hat y_k \Delta t, \quad \Delta
\omega = \frac{2\pi}{A}. \]
- Let n be a positive integer. Consider the space of continuous
piecewise functions on [0,1] with corners at j/n, j = 0, $\ldots$,
n. These are linear splines and have a basis {T(x − j/n}, where
T is the tent function, \[ T(x):= \left\{ \begin{array}{cc} 1 - |x| &
|x|\le 1 \cr 0 & |x| > 1 \end{array} \right.. \]
- Let Δx = 1/n. Recall that if f is in C[0,1], then \[ s_f(x)
= \sum_{j=0}^n f(j\Delta x)T\big(n(x - j\Delta x)\big) \] interpolates f on the
interval [0, 1] at the points jΔx. Use this to show that if f
∈ C(2)[0,1], then \[ \|f-s_f\|_{C[0,1]} \le
\|f''\|_{C[0,1]} n^{-2}. \]
- Show that the trapezoidal rule for $\int_0^1 f(x)dx$ is just
$\int_0^1 s_f(x)dx$, and then use the previous part to show that, for
f ∈ C(2)[0,1], the error for the trapezoidal rule is
$\|f''\|_{C[0,1]} n^{-2}$.
- Let $\{\hat y_k\}$ be the DFT for a 2π periodic function f
that is twice continuously differentiable on R, and let
$\{c_k\}$ be the Fourier coefficients for f. Show that the error $|c_k
- \hat y_k/n|$ is at worst of order $(k/n)^2$. (We can do better!)
Assignment 6 Due Friday, 11/2/2012.
- Read sections 9.1-9.4 in the text.
- Problems
- Find the Radon transform of the function of the characteristic
function χD for the unit disk, D.
- Let $f\in \mathscr S$ (Schwartz space) and recall that we defined
$
\|f\|_k:=\max_{\,|\,\alpha|\le k,\,|\beta|\le k}\|f\|_{\alpha,\beta}.
$
- Show that $\|f\|_k$ is a seminorm for
$\mathscr S$.
- The following identity will be needed in the next part. Show that
if $x,y,z$ are positive and if $x \le y + z $, then
\[
\frac{x}{1+x}\le \frac{y}{1+y} + \frac{z}{1+z}.
\]
Hint: $f(t) = \frac{t}{1+t}$ is increasing, so $f(x)\le f(y+z)$.
- Let $\rho(f) := \sum_{k=0}^\infty \frac{\|f\|_k
}{1+\|f\|_k}2^{-1-k}$ and let $d(f,g):=\rho(f-g)$. Show that
$d(f,g)$ is a metric for $\mathscr S$. You will need to use the
inequality in (b).
- Let $f_j$ be a sequence of functions in $\mathscr S$ and let
$\rho$ be as in problem 2. Show that for each fixed $k$ we have
$\lim_{j\to \infty}\|f_j\|_k=0$ (not necessarily uniformly in $k$)
if and only if $\lim_{j\to \infty} \rho(f_j)=0$.
- Use induction on the number of variables to prove the
multivariate product rule below. (Multi-index notation is used
throughout. See the Wikipedia entry for
Multi-index notation.)
\[ \partial^\alpha (fg) = \sum_{0\le \beta \le
\alpha}\binom{\alpha}{\beta}\partial^\beta f \partial^{\alpha
-\beta}g.
\]
- Section 9.4: 3(a) (Use the previous problem.)
Assignment 7 Due Monday, 11/12/2012.
- Read the
Notes on
Scattered-Data Radial Function Interpolation.
- Problems
- Section 9.4: 4, 6, 14(b), 15
- The setting for this problem is $\mathbb{R}^2$. Consider the
$f(r) = 1/r$.
- Show that $F(\phi):=\langle f,\phi \rangle$ defines a tempered
distribution - i.e., it is in $\mathscr{S}'$.
- Consider the sequence of functions $f_k(r) =
\frac{e^{-r/k}}{r}$, all of which are in $L^1(\mathbb R^2)$. Let
$F_k(\phi) = \langle f_k,\phi \rangle$ be the tempered distributions
corresponding to $f_k$. Show that $F_k \to F$ in $\mathscr{S}'$.
- Show that \[ \hat{f}_k(\xi) = \frac{2\pi}{\sqrt{k^{-2}+|\xi|^2}}.
\] Hint: First show that $\hat{f}_k(\xi) =
\int_0^{2\pi}(k^{-1}+i|\xi| \cos(\theta))^{-1}d\theta$. Then evaluate
the integral using contour integration techniques. (If you haven't
had contour integration, then see me.)
- Show that $\hat{f}(\xi)=\displaystyle{\frac{2\pi}{|\xi|}}$.
- Let $\phi$ be in $\mathscr S$ and recall that, in the discussion
of the Radon transform, we used the Riesz transform $I_1[\phi](x)=
(\phi*f)(x)/(2\pi)$, $x \in \mathbb{R}^2$. Find the Fourier
transform of $I_1[\phi]$.
Assignment 8 Due Wednesday, 11/28/2012.
- Read the
Notes on Spherical Harmonics.
- Problems
- Let $\{c_n\}_{n\in \mathbb Z}$ satisfy $|c_n|\le (1+|n|)^m$ for
$n\not=0$ and $m$ some fixed positve integer. Show that the Fourier
series $\sum_{n\in \mathbb Z}c_ne^{inx}$ defines a $2\pi$-periodic
tempered distribution.
- Consider the function $\Phi(r) := \max(1-r,0)$. On $\mathbb R$,
this is just the linear "tent" function and is positive definite. By
finding its Fourier transform in $\mathbb R^3$, show that $\Phi(r)$
is not positive definite in $\mathbb R^3$.
- Suppose that $\Psi( r )$ is a positive definite radial function on $\mathbb R^n$. Show that it is positive definite on $\mathbb R^d$ for all $1\le d\le n$.
- Let $\kappa(x,y)$ be a strictly positive definite kernel on
${\mathbb R}^n$, say. Suppose that $\mathscr F^0$ is the set of all
$f=\sum_{\xi \in X}a_\xi \kappa(\cdot,\xi)$, where $X$ is a pairwise
distinct subset of ${\mathbb R}^n$. Show that for $f,g\in \mathscr
F^0$ the form below is bilinear in $f$ and $g$.
\[
\langle f,g\rangle = \sum_{\xi\in X,\,\eta\in Y} a_\xi b_\eta
\kappa(\xi,\eta).
\]
- Let $\mathscr F$ be a reproducing kernel Hilbert space (RKHS), with kernel $\kappa(\cdot,\cdot)$. Assume the functions in $\mathscr F$ are defined on $\mathbb R^n$. Let $X$ be a finite set of distinct points in $\mathbb R^n$ and define $V_X = {\rm span}\{\kappa(\cdot, \xi); \xi\in X\}$.
- Let $f$ be in $\mathscr F$. Recall that the interplant $I_X f$ is the orthogonal projection of $f$ onto $V_X$. Use this and the reproducing property of $\kappa$ to show that
\[
|f(x)-I_Xf(x)|\le \|f-I_Xf\|_{\mathscr F} \|\kappa(\cdot,x)-\sum_{\xi\in X}c_\xi \kappa(\cdot, \xi)\|_{\mathscr F},
\]
where the coefficients $\{c_\xi\}_{\xi\in X}$ are arbitrary.
- Show that $\|f-I_Xf\|_{\mathscr F} \le \|f\|_{\mathscr F}$.
- Define the power function by $\mathcal P_X(x) : = \min_{\,c_\xi,\, \xi\in X}\|\kappa(\cdot,x)-\sum_{\xi\in X}c_\xi \kappa(\cdot, \xi)\|_{\mathscr F}$. Use the previous parts to show that for all $x\in \mathbb R^n$ we have that
\[
|f(x)-I_Xf(x)|\le \|f\|_{\mathscr F}{\mathcal P}_X(x).
\]
- Let $A$ be an $n\times n$ self-adjoint, real matrix that is positive definite. Define the condition number of $A$ to be $c(A):=\lambda_{max}/\lambda_{min}$ — i.e., the ratio of maximum and minimum eigenvalues.
- Consider the equation $Ax = b$. Suppose that there is an error $\Delta b$ in $b$, resulting in an error $\Delta x$ in $x$. Show that
\[
c(A)^{-1}\frac{\| \Delta b \|_2}{ \|b\|_2} \le \frac{\|\Delta x\|_2}{\| x \|_2} \le c(A)\frac{\| \Delta b \|_2}{ \|b\|_2}.
\]
- Let $\varepsilon >0$ and consider the matrix $A = \begin{pmatrix} 1 & 1\\ 1 & 1+\varepsilon \end{pmatrix}$. Show that $c(A) \approx 4\varepsilon^{-1}$.
- Let $b=(1\ 1)^T$ and note that $x=A^{-1}b=(1 \ 0)^T$. Suppose that we know that the error in $b$ is roughly $\Delta b= 10^{-4}(1 \,\ -\!1)^T$. (This means that the entries in $b$ are accurate to about $10^{-4}$.) If $\varepsilon = 4\times 10^{-6}$, show that $\|\Delta x\|_2\approx 10^{2}$. With this in mind, how reliable are the entries in $x$?
- Suppose that $\Phi$ is a strictly positive definite radial function in $\mathbb R^n$ for all $n$ i.e., $\Phi( r )$ is on order 0 RBF.
- Show that $\Phi( r )>0$ for $0 < r $.
- Show that $\Phi$ is in $C^\infty((0,\infty))$. (Actually, it is analytic.)
Assignment 9 Not to be handed in.
- Problems
- Let $L_\pm$ be the raising and lowering operators in
the
Notes on Spherical Harmonics. Use them to show that, for $m\ge
0$, if $Y_{\ell,m}(\theta,\varphi) =
Q_{\ell,\,m}(\cos(\theta))\sin^m(\theta)e^{im\varphi}$, where
$Q_{\ell,\,m}(t)$ is a polynomial in $t$, then $Q_{\ell,\,m-1} =
C((1-t^2)Q'_{\ell,\, m}-2m t Q_{\ell,\,m})$ and, in addition, that $
Q_{\ell,\,m+1} = C Q_{\ell,\,m}'(t)$. Use these formulas along with
$Y_{\ell,\,\ell}(\theta,\varphi)=C\sin^\ell(\theta)e^{i\ell \varphi}$
to show that the $Y_{\ell,\, m}$ have the form in Theorem 3.1, up to
normalization constants. ("C" is a generic constant.)
- Use Theorem 3.1 in
the
Notes on Spherical Harmonics to expand the function
$f(\theta,\varphi) = 2 - \cos^2(\theta)+
\cos(\theta)\sin(\theta)^2\sin^2(\varphi)$ in spherical harmonics.
- Recall that a function $f$ in $\mathbb R^3$ is said to be
harmonic if and only if $\Delta_{\mathbb R^3}f = 0$, and also that a
harmonic polynomial homogeneous of degree $\ell$ is related to a
spherical harmonic $Y_\ell$ via $p(\mathbf x)=r^\ell Y_\ell(\mathbf
x/r)$. (Here $\Delta_{\mathbb S^2}Y_\ell + \ell(\ell+1)Y_\ell =0$.)
Find a basis for harmonic polynomials homogeneous of degree $\ell=3$
using Theorem 3.1 in
the
Notes on Spherical Harmonics.
- Fix $m\ge 1$. Show that the polynomials $\{P^{(m)}_\ell : \ell\ge m \}$ are
orthogonal with respect to the inner product $\langle f,g\rangle =
\int_{-1}^1 f(t)\overline{g(t)}(1-t^2)^m dt$. (Hint: use Theorem
3.1.)
- For the rotation matrix $R$ below, find these: the Euler angles, the angle of rotation, and the axis of rotation.
\[
R =\begin{pmatrix}
\frac{2}{3} & \frac{2}{3} &-\frac{1}{3}\\
-\frac{1}{3} & \frac{2}{3} & \frac{2}{3} \\
\frac{2}{3} & -\frac{1}{3} & \frac{2}{3}
\end{pmatrix}.
\]
- Let $A$ be an n×n matrix such that $A^3=-A$. Show that $e^{\theta A} = I + \sin(\theta) A + (1-\cos(\theta))A^2$. Using the axis of rotation $\mathbf \omega$ and angle of rotation $\theta$ for $R$ found in the previous problem, find $A$ such that $R=e^{\theta A}$. (Hint: recall that $A\mathbf x = \mathbf \omega \times \mathbf x$ and that $R\mathbf x = \mathbf x + \sin(\theta)\mathbf \omega \times \mathbf x+ (1-\cos(\theta))\mathbf \omega\times (\mathbf \omega \times \mathbf x)$.)
- Let $\mathbf 1 = \begin{pmatrix} 1&0\\0&1\end{pmatrix}$, $\mathbf
i = \begin{pmatrix} i&0\\0&-i\end{pmatrix},\ \mathbf j =
\begin{pmatrix} 0&1\\ -1&0\end{pmatrix},\ \mathbf k = \begin{pmatrix}
0&i\\i&0\end{pmatrix}.\ $ For $q_0\in \mathbb R$ and $\mathbf q\in
\mathbb R^3$, a quaternion $Q = q_0\mathbf 1 + \mathbf q$, where
$\mathbf q = q_1\mathbf i+q_2\mathbf j + q_3\mathbf k$ is the "pure"
part of $Q$. The set of quaternions is denoted by $\mathbb H$. Also,
$SU(2)$ comprises all $U\in \mathbb H$ such that $u_0^2 + |\mathbf
u|^2 = 1$.
- Show that $PQ = (p_0q_0 - \mathbf p\cdot \mathbf q)\mathbf 1 +
q_0\mathbf p + p_0\mathbf q+\mathbf p \times \mathbf q$.
- Let $U=u_0\mathbf 1 + \mathbf u$ be in $SU(2)$ and let $X=\mathbf x$ be a pure quaternion representing a point in $\mathbb R^3$. In addition, let $\overline{U} = u_0\mathbf 1 - \mathbf u$. Show that
\[
UX\overline{U} = \mathbf x + 2u_0\mathbf u \times \mathbf x+ 2\mathbf u \times (\mathbf u\times x).
\]
- If $u_0=\cos(\psi)$ and $\mathbf u = \sin(\psi)\mathbf \omega$, where $\mathbf \omega \in S^2$, then show that
\[
UX\overline{U} = \mathbf x + \sin(2\psi)\mathbf \omega \times \mathbf x+ (1-\cos(2\psi))\mathbf \omega \times (\mathbf \omega\times \mathbf x).
\]
- Find all quaternions $U$ that correspond to the rotation matrix $R$ in problem 5. For each of them, what is $\psi$?
Updated 12/11/2012 (fjn)