Math 414-501 Spring 2022
Assignments
Assignment 1 - Due Friday, 1/28/2022.
- Read sections 1.2.1-1.2.5.
- Do the following problems.
- Chapter 1: 1, 3, 4, 7, 10(a,b), 25
- Find the Fourier series for $f(x)$, where $f(x) = 1$ if $0 < x \le
\pi$, and $f(x)=0$ if $-\pi < x \le 0 $.
Assignment 2 - Due Friday, 2/4/2022.
- Read sections 1.2.4-1.2.5.
- Do the following problems.
- Chapter 1: 9, 10(c), 21
- Find the Fourier series for $e^{x/3}$ valid on $-\pi < x \le \pi$
and the Fourier series for the same function, this time on the interval $0
< x \le 2\pi$. These two series are different. Does this contradict
Lemma 1.3? Explain your answer. (Hint: What is the $2\pi$ periodic
extension for $e^{x/3}$ on $(-\pi,\pi]$, and for it on $(0,2\pi]$.)
- Find the Fourier sine and cosine series for $f(x) = 1$, $0 \le x
\le \pi$.
Assignment 3 - Due Friday, 2/11/2022.
- Read sections
3.1-1.3.4, notes
on point-wise convergence of Fourier series.
- Do the following problems.
- Chapter 1: 8, 17, 18, 19
- Find the complex form of the Fourier series for $e^{i\beta x}$ on
the interval $-\pi < x \le \pi$, where $\beta$ is real and
is not an interger. Use this to show that $\csc(\beta \pi) =
\sum_{n=-\infty}^\infty \frac{(-1)^n}{\pi(\beta -n)}$. (Hint: what does
the Fourier series converge to when $x=0$?)
- Show that the Fourier series for the sum of two functions is the
sum of the Fourier series of the functions.
- Suppose that is $f'$, the derivative of a function $f$, has the
Fourier series $f'(x)=\sum_{n=1}^\infty
a'_n\cos(nx)+b'_n\sin(nx)$. Show that $f(x)$ has the Fourier series
$f(x)=a_0+ \sum_{n=1}^\infty (-b'_n/n) \cos(nx)+(a'_n/n)\sin(nx)$. As
usual, $a_0=\frac{1}{2\pi}\int_{-\pi}^\pi f(x)dx$. (Hint: integrate by
parts in the formulas for the $a'_n$'s and $b'_n$'s.)
Assignment 4 - Due Friday, 2/25/2022.
- Read sections 0.2-0.5 and 1.3.5.
- Do the following problems.
- Chapter 0: 1, 5, 13 (You may use the book's answer to 12 given on
pg. 288.)
- Chapter 1: 23(a,b,c,d)
Assignment 5 - Due Monday, 3/7/2022.
- Read sections 0.2-0.5 and 1.3.5.
- Do the following problems.
- Chapter 0: 15 (The o.n set to be
used for the projection is $\{\phi(x), \psi(x), 2^{1/2}\psi(2x),
2^{1/2}\psi(2x-1)\}$.), 17
- Chapter 1: 33
- Show that if $f\in L^2[0,2\pi]$, then Parseval's formula holds
for both the real and complex cases, with the interals being over
$[0,2\pi]$ instead of $[-\pi,\pi]$. (Hint: If on $[0,2\pi]$, $f$ has
the Fourier series $f(x)=a_0+\sum_{n=1}^\infty
a_n\cos(nx)+b_n\sin(nx)$, then on $[-\pi,\pi]$,
$f(x+\pi)=a_o+\sum_{n=1}^\infty (-1)^n a_n\cos(nx) + (-1)^n
b_n\sin(nx)$.)
- You are given that, on $[0,2\pi]$, $e^{-x}= \sum_{n=-\infty}^\infty
\frac{1-e^{-2\pi}}{2\pi(1+in)}e^{inx}$. Use this and the result from
above to find $\sum_{n=-\infty}^\infty \frac{1}{n^2+1}$.
- Let V be a vector space with a complex inner product $\langle
\cdot,\cdot\rangle$. Let $u$ and $v$ be orthogonal. Prove the
Pythagorean Theorem: $\|u+v\|^2=\|u\|^2+\|v\|^2$.
- Let V be a vector space with a complex inner product $\langle
\cdot,\cdot\rangle$. Suppose that the set $S =
\{e_1,e_2,\cdots,e_n\}$ is an orthonormal basis for $V$. Show that
if $v=\sum_{j=1}^n a_j e_j$ and $w=\sum_{k=1}^n b_k e_k$, then
$\langle v,w\rangle= \sum_{j=1}^n a_j\bar b_j=\bar b^Ta$, where
$a=(a_1 \cdots a_n)^T$ and $b=(b_1\cdots b_n)^T$.
Assignment 6 - Due Friday, 3/11/2022.
- Read sections 2.1 and 2.2.1.
- Do the following problems.
- Chapter 2: 1, 2
- Find the Fourier transform of each of these functions.
- $f(t) = e^{-|t|}$.
- $g(t)=\begin{cases}1 & -1 \le t \le 2, \\ 0 & \text{otherwise}.
\end{cases}$
- $h(t) = \begin{cases} 0, & t<0 \\
Ae^{-\alpha t},& t\ge 0. \end{cases} $, where $A>0$ and $\alpha>0$.
- $F(t) = \begin{cases} 1 & 0 \le t \le 1,\\ -1 & -1 \le t < 0 \\ 0
& \text{otherwise.} \end{cases} $
Assignment 7 - Due Friday, 4/1/2022.
- Read sections 2.3 and 2.4
- Problems.
- Chapter 2: 5, 6.
- Let $\phi(t) := \begin{cases} 1 & 0 \le t < 1, \\ 0 &
\text{otherwise}\end{cases}$ and $\psi(t) := \begin{cases} 1 & 0
\le t < 1/2, \\ -1 & 1/2 \le t <1, \\0 &
\text{otherwise.}\end{cases} \ $
- Use the definition of the convolution to find $\phi\ast \psi(t)$.
- Without using the convoluion theorem, find the Fourier transform
${\mathcal F}(\phi\ast \psi)$.
- Use the convolution theorem to find ${\mathcal F}(\phi\ast
\psi)$. Compare this with the result above.
- Let $f(t) = \begin{cases} \pi + t & -\pi \le t \le 0 , \\
\pi-t & 0\le t \le \pi \\
0 & \text{otherwise}. \end{cases} \ $ Recall that $\hat f(\lambda) =
\sqrt{\frac{8}{\pi}} \frac{\sin^2(\pi \lambda/2)}{\lambda^2} $. Use
Plancheral's theorem to find $I=\int_0^\infty
\frac{\sin^4(t)}{t^4}dt$.
- Let $h(t):=\begin{cases} 1/d, & 0\le t\le d, \\ 0,
&\text{otherwise}\ \end{cases}$ be the impulse response for a filter
$L$. (The "running average filter.")
- Show that $L[f] = \frac{1}{d} \int_{ t - d}^t f(\tau)d\tau$.
- Show that if $f(t)=0$ for $t<0$, then
$L[f](t) = d^{-1}\begin{cases} 0, & t<0, \\ \int_0^t
f(\tau)d\tau, & 0\le t \le d \\ \int_{t-d}^t f(\tau)d\tau, & d\le
t .\end{cases}$
- From problem 6, chapter 2, the Fourier transform of the Gaussian
$f_s(t) = \sqrt{s} e^{-s t^2}$ is $\hat f_s(\lambda) =
\frac{1}{\sqrt{2}} e^{-\lambda^2/(4s)}$. Consider the two Gaussians
$f_3(t) = \sqrt{3}e^{-3 t^2}$ and $f_6(t) = \sqrt{6}e^{-6t^2}$. Use
the convolution theorem to show that $f_3 \ast f_6(t) = \sqrt{\pi}
f_{2}(t)=\sqrt{2\pi} e^{-2t^2}$.
- Let f(t) be a signal that is 0 when t < 0 or t > 1. Show
that, for the Butterworth filter, one has
\[
L[f] = Ae^{-\alpha t} \int_0^{\min(1,t)} e^{\alpha \tau} f(\tau)d\tau,
\ \text{if }t\ge 0, \ \text{and } L[f] = 0 \ \text{if } t<0.
\]
Assignment 8 - Due Friday, 4/8/2022.
- Read sections 4.1 and 4.2. Begin forming groups and looking at
possible projects.
- Problems.
- Chapter 2: 13.
- Chapter 4: 2.
- Let $\hat f(\lambda)= \begin{cases} \pi-|\lambda| & |\lambda|\le
\pi\\ 0 & \text{otherwise}\end{cases}$. (This is a tent function in
$\lambda$.)
- Find $f(t)=\mathcal F^{-1}[\hat f(\lambda)](t)$.
- Note that this function is band limited. What are $\Omega$, the
natural frequency and Nyquist frequency? Express $f(t)$ in terms of
the series in the sampling theorem. (See pg. 120, eqn. 2.22).
- This is a version of problem 12, chapter 2. Let
\[ h(t):=\begin{cases} 1/d, & 0\le t\le d \\ 0,
&\text{otherwise,}\end{cases} \ f(t) =\begin{cases}
e^{-t}(\sin(3t) +\sin(40t)), & 0\le t\\ 0,
&\text{otherwise,}\end{cases} \ \text{and } L[f]=f\ast h.
\]
- Find $\hat h(\lambda)$ and $\hat f(\lambda)$. Show that $
|\widehat L[f](\lambda)|^2= \big(\frac{2 \sin(\lambda d/2)}{\lambda
d}\big)^2|\hat f(\lambda)|^2$. Using this, choose a value of $d$ that
will suppress the frequency $\lambda = 40$, which we regard as
noise.
- Use the formula from Asignment 7, problem 4(b) and your choice of
$d$ from above to filter $f(t)$. Plot $f(t)$
and $L[f](t)$ on the same axes. In your estimation, did this
choice of $L$ do a good job of removing the noise?
Assignment 9 - Due Wednesday, 4/20/2022.
- Read sections 4.3, 5.1
- Problems.
- Chapter 4: 5, 6, 7
- Chapter 5: 2
- Let $f(x)=x$, $0\le x \le 1$, and 0 otherwise.
- Find $f_3$, the orthogonal projections of $f$ onto the Haar
approximation space $V_3$, in terms of the $\{\phi(2^3x-k)\}$ basis for
$V_3$.
- Use the filter decomposition method given in section 4.3.3 to
find the coefficients for $f_2$ and $w_2$ in terms of the bases
$\{\phi(2^2x-k)\}$ and $\{\psi(2^2x-k)\}$, for $V_2$ and
$W_2$, respectively.
- Show that for the Haar MRA $a_k^j$ is the average of $f$ over the
interval $[2^{-j}k,2^{-j}(k+1)]$, and that $b_k^j$ is 1/2 of the
difference of the average of $f$ over $[2^{-j-1}k,2^{-j}(k+1/2)]$ and
the average of $[2^{-j}(k+1/2),2^{-j}(k+1)]$. (The average of a function
$f$ over an interval $[a,b]$ is $\text{avg}(f)=\frac{1}{b-a}\int_a^b
f(x)dx$.)
Assignment 10 - Due Tuesday, 5/3/2022.
- Read section 5.1
- Problems.
- Chapter 5: 8(c,d), 11
Updated 4/12/2022