Math 414-501 — Spring 2018

Assignment 1 - Due Wednesday, 1/31/2018.

• Read sections 1.2.1-1.2.5.
• Do the following problems.
  1. Chapter 1: 1, 3, 4, 7, 8, 9, 10, 14, 25

Assignment 2 - Due Wednesday, 2/7/18.

• Read sections 1.3.1-1.3.5.
• Do the following problems.
  1. (This an easy problem!) Let $f$ and $g$ have the Fourier series \[ f(x) = a_0 +\sum_{n=1}^\infty a_n \cos(nx)+b_n\sin(nx) \text{ and } g(x) = c_0 +\sum_{n=1}^\infty c_n \cos(nx)+d_n\sin(nx). \] Show that the Fourier series for $Af(x)+ Bg(x)$, where $A$ and$ B$ are constants is \[ Af(x)+Bg(x) = Aa_0 +Bc_0 + \sum_{n=1}^\infty (Aa_n+Bc_n) \cos(nx)+(Ab_n+Bd_n)\sin(nx). \]
  2. Chapter 1: 11 (Typo: $f(x)$ is defined on $-\pi \le x \le \pi$.), 12 (skip the plots), 13, 17, 18, 19, 21.

  3. Suppose that $f$ and $f'$ are continuous 2π-periodic functions, and that the Fourier series for $f$ and $f'$ are $ f(x) = a_0 +\sum_{n=1}^\infty a_n \cos(nx)+b_n\sin(nx)$ and $f'(x) = a'_0 +\sum_{n=1}^\infty a'_n \cos(nx)+b'_n\sin(nx)$, respectively.
     1. Use integration by parts to show that the coefficients of the two series are related this way for n ≥ 1: $a'_n = nb_n$ and $b_n'=-na_n$. If f is k times continuously differentiable, use induction to derive a similar formula for the the Fourier coefficients of the $k^{th}$ derivative of $f$, $f^{(k)}$.

     2. Let $f(x) = \frac{1}{12}(x^3 - \pi^2x)$, $-\pi \le x \le \pi$. In the text (cf. Example 1.9), we derived the Fourier series for $g(x) = x$ on $-\pi \le x <\pi$. Use the series for $g$ and the results from the previous problem to show that the Fourier series for $f(x) = \frac{1}{12}(x^3 - \pi^2x)$ is given by \[ f(x) = \sum_{n=1}^\infty\frac{(-1)^n}{n^3} \sin(nx) \]

Assignment 3 - Due Wednesday, 2/14/18.

• Read sections 0.1, 0.2, 0.3.1, 0.5.1, 0.5.2, 1.3.4-1.3.5, 2.1
• Do the following problems.
  1. Chapter 1: 22, 23(b,c,e), 24(b,c,e)
  2. Suppose that $\alpha$ is real and that it is not an integer. Find the complex Fourier series for $f(x)=e^{i\alpha x}$, $-\pi\le x \le \pi$. Use this series to show that $ \csc(\pi \alpha) = \sum_{n=-\infty}^\infty \frac{(-1)^n}{\pi(\alpha -n)}$. (Hint: choose $x=0$.)
  3. In the previous assignment, you showed that $f(x) = \frac{1}{12}(x^3 - \pi^2x) =\sum_{n=1}^\infty\frac{(-1)^n}{n^3} \sin(nx)\,$ on the interval $-\pi\le x \le \pi$.
     1. Use the series method to show that the Fourier series converges uniformly to $f$.
     2. Use Parceval's theorem to find the value of the series $\sum_{n=1}^\infty \frac{1}{n^6}$.
  4. Consider the inner product $\langle f,g\rangle =\int_{-1}^1 f(x)g(x)$. Show that the polynomials $p_0=1/\sqrt{2}, p_1=\sqrt{3/2}x, p_2=\sqrt{5/8} (3x^2-1)$ are othonormal in this inner product. Let $V:=\text{span}\{p_0,p_1,p_2\}$. Find the projection of $e^x$ onto $V$.

Assignment 4 - Due Wednesday, 2/28/2018.

• Read sections 2.1 and 2.2.1-2.2.2, 2.2.4
• Problems.
  1. Chapter 2: 1, 4.

  2. Find the Fourier transform of each of these functions.
     1. $f(t) = e^{-|t|}$.

     2. $g(t)=\begin{cases}1 & -1 \le t \le 2, \\ 0 & \text{otherwise}. \end{cases}$

     3. $h(t) = \begin{cases} 0, & t<0 \\ Ae^{-\alpha t},& t\ge 0. \end{cases} $, where $A>0$ and $\alpha>0$.

     4. $F(t) = \begin{cases} 1 & 0 \le t \le 1,\\ -1 & -1 \le t < 0 \\ 0 & \text{otherwise.} \end{cases} $
  3. Use Example 2.5, pg. 99, to find the the value of the integral $I = \int_0^\infty \frac{\sin^2(t)}{t^2}dt$. (Hint: Use the inversion formula, equation (2.3), in Theorem 2.1, pg. 94.)

Assignment 5 - Due Wednesday, 3/7/2018.

• Read sections 2.3 and 2.4.
• Problems. You may use the table below.

  "Quartet"

  Function	Fourier Transform
  $f(t)$	$\hat f(\lambda)$
  $\hat f(t)$	$f(-\lambda)$

  1. Chapter 2: 6.

  2. In the previous assignment, you found that the Fourier transform of $\mathcal F[e^{-|t|}](\lambda) = \sqrt{\frac{2}{\pi}}(1+\lambda^2)^{-1}$. Use this transform and the properties listed in Theorem 2.6 to find the Fourier transforms of the following functions:

     1. $t e^{-|t|}$ (Use #2.)

     2. $e^{-2|t-3|}$ (#6 and #7)

     3. $\text{sgn}(t)e^{-|t|}$ (Hint: differentiate $e^{-|t|}$; use #4.) Here, $\text{sgn}(t) = \begin{cases} 1 & t > 0,\\ 0 & t = 0, \\ -1 & t < 0.\end{cases} $

     4. $\frac{1}{1+(t-2)^2}$  (Hint: Use the table and #6.)

  3. Follow the proof on pg. 104 of properties 6 and 7 in Theorem 2.6 to show that $\mathcal F^{-1}[\hat f(b\lambda -a)](t)= e^{ita/b}f(t/b)/b$. Use this to show that if $f(t)=\begin{cases} e^{-t} & t\ge 0\\ 0 & t<0\end{cases}$, then $\mathcal F[f(t)\sin(at)] = \frac{a}{\sqrt{2\pi}}((1+i\lambda)^2+a^2)^{-1}$.

  4. Let $T(t)$ be the tent function defined in example 2.5. Let $T_1(t) =\begin{cases} 1+t & -1 \le t \le 0, \\ 1-t & 0 < t \le 1, \\ 0 & \text{otherwise}. \end{cases}$ Show that $T_1(t) = \frac{1}{\pi} T(\pi t)$ and find $\widehat T_1(\lambda)$. Consider the function $f(t) = \begin{cases} t+1 & -1\le t < 0, \\ 1 & 0\le t \le 1, \\ 2-t & 1 < t \le 2, \\ 0 & \text{otherwise}. \end{cases}$ Find $\hat f(\lambda)$. (Hint: Begin by showing that $f(t)=T_1(t) + T_1(t-1)$.)

Assignment 6 - Due Friday, 3/23/2018.

• Read sections 3.1.1-3.1.4.
• Problems.
  1. Chapter 2: 5, 13

  2. Let $h(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 h(\lambda) = \sqrt{\frac{8}{\pi}} \frac{\sin^2(\pi \lambda/2)}{\lambda^2} $. Use Plancheral's theorem to find $I=\int_0^\infty \frac{\sin^4(t)}{t^4}dt$.

  3. Let f(t) be a signal that is 0 when t < 0 or t > 1. Show that, for the Butterworth filter, which has impulse response $h(t)=\begin{cases} Ae^{-\alpha t}& t\ge 0\\ 0 &t<0\end{cases}$, one has

     \[ L[f] =\begin{cases} A e^{-\alpha t} \int_0^t e^{\alpha \tau} f(\tau)d\tau, & 0\le t\le 1 \\ Ae^{-\alpha t} \int_0^1 e^{\alpha \tau} f(\tau)d\tau & t\ge 1\\ 0 &t<0. \end{cases} \]

  4. This is a version of problem 12, chapter 2. Let $h(t):=\begin{cases} 1/d, & 0\le t\le d \\ 0, &\text{otherwise}\end{cases}$. Show that $L[f] = \frac{1}{d} \int_{ t - d}^t f(\tau)d\tau$.

  5. Let $f(t) =\begin{cases} \sin^2 (t)& 0\le t\le 4\pi \\ 0 & t<0 \ \text{or }t>4\pi. \end{cases}$ For $d=2\pi$, use the previous problem to find $L[f](t)$, $t\ge 0$. (Be aware that the cases $0 \le t < 2\pi$ and $t \ge 2\pi$ have to be treated separately.)

  6. Let $h_1$ and $h_2$ be impulse response functions for causal filters $L_1[f] = h_1\ast f$ and $L_2[f]=h_2\ast f$. Show that $h_3=h_2\ast h_1$ is the impulse response for $L[f]=L_2L_1[f]$, and that $L$ is causal. Also, let $h=h_1=h_2$ be the impulse response function for the Butterworth filter, show that $h*h(t) = \begin{cases} A^2 te^{-\alpha t},& t\ge 0\\ 0,& t \le 0 \end{cases} \ $.

Assignment 7 - Due Wednesday, 4/4/2018.

• Read sections 4.2,4.3.
• Problems.
  1. Chapter 2: 14
  2. Chapter 4: 2, 6, 7, 11 (10 pt. bonus.)
• Begin forming groups for projects. Each group should have 2 or 3 members — solo projects will not be accepted. The group leader should send me an email with a list of the group members and a short abstract for the project

Assignment 8 - Due Friday, 4/13/2018.

• Read sections 4.4, 5.1.
• Problems.
  1. Let $f(x)=A+Bx$. For the Haar scaling function $\phi(x)$ and wavelet $\psi(x)$, show that $a_k^j= A+2^{-j}(k+1/2)B$ and that $b_k^j = -2^{-j-2}B$.
  2. Do problem 1 in chapter 4 using the filter approach given in sestion 4.3.
  3. Chapter 5: 2, 4(b), 5, 8(c,d,e,f)
  4. If you haven't formed a group for a project, do so by Friday. Once you've done that, send me an email with a list of the group members and a short description of the project

Assignment 9 - Due Monday, 4/23/2018.

• Read sections 5.2 (Note: we covered this section, especially the decomposition and reconstruction diagrams, in class on 4/16/18. The only new thing is initialization.), 5.3.3 and 5.3.4.

• If you haven't formed a group for a project, do so by Friday. Once you've done that, send me an email with a list of the group members and a short description of the project.

• Problems.
  1. Chapter 5: 8(d,e,f), 11

  2. Do any two of the problems below.
     1. Write two programs in your favorite language to do the following. Consider the $p_k$'s from the Example 5.25, p. 227 (Daubechies' wavelet; db2 in matlab). The first program inputs a signal $a^j$ and outputs $a^{j-1}$ and $b^{j-1}$. It is the decomposition step in a wavelet analysis. The second program inputs $a^{j-1}$ and $b^{j-1}$ and outputs $a^j$. This is the reconstruction step. In matlab, both programs involve the conv (convolution) function, which will be used to filter inputs. In addition, the first program requires a down-sampling part; the second, up sampling.

     2. Let $f(t)=\begin{cases} t, & t \le \frac{3}{17}\\ 1.02t +\frac{6.06}{17}, & \frac{3}{17} < t \end{cases}$. Sample $f$ at $k 2^{-10}$, $k=0, \ldots, 1024 $. (The time interval is $2^{-10}$, so $a^{10}_k$ corresponds to the time $k2^{-10}$.) Either matlab's wavemenu or the decomposition program from the previous problem, applied to $a^{10}_k=f(k2^{-10})$ to obtain $b^{-9}$. (The time interval is $2^{-9}$, $b^9_k$ corresponds to the time $k2^{-9}= 2k2^{-10}$.) Keeping in mind the times involved, plot both $a^{10}$ and $b^9$ versus $t$, $0\le t\le 1$. Can you detect the change in slope from the wavelet coefficients?

     3. Chapter 5: 17

     4. Chapter 6: 2