Math 414-501 — Spring 2022

Assignment 1 - Due Friday, 1/28/2022.

• Read sections 1.2.1-1.2.5.
• Do the following problems.
  1. Chapter 1: 1, 3, 4, 7, 10(a,b), 25
  2. 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.
  1. Chapter 1: 9, 10(c), 21
  2. 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 $(0,2\pi]$.)
  3. 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.
  1. Chapter 1: 8, 17, 18, 19
  2. 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 $x=0$?)
  3. Show that the Fourier series for the sum of two functions is the sum of the Fourier series of the functions.
  4. 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.
  1. Chapter 0: 1, 5, 13 (You may use the book's answer to 12 given on pg. 288.)
  2. 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.
  1. 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
  2. Chapter 1: 33
  3. 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 $[-\pi,\pi]$, $f(x+\pi)=a_o+\sum_{n=1}^\infty (-1)^n a_n\cos(nx) + (-1)^n b_n\sin(nx)$.)
  4. 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 $\sum_{n=-\infty}^\infty \frac{1}{n^2+1}$.
  5. 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: $\|u+v\|^2=\|u\|^2+\|v\|^2$.
  6. 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.
  1. Chapter 2: 1, 2

  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} $

Assignment 7 - Due Friday, 4/1/2022.

• Read sections 2.3 and 2.4
• Problems.
  1. Chapter 2: 5, 6.

  2. 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} \ $
     1. Use the definition of the convolution to find $\phi\ast \psi(t)$.
     2. Without using the convoluion theorem, find the Fourier transform ${\mathcal F}(\phi\ast \psi)$.
     3. Use the convolution theorem to find ${\mathcal F}(\phi\ast \psi)$. Compare this with the result above.

  3. 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 $I=\int_0^\infty \frac{\sin^4(t)}{t^4}dt$.

  4. 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.")

     1. Show that $L[f] = \frac{1}{d} \int_{ t - d}^t f(\tau)d\tau$.
     2. Show that if $f(t)=0$ for $t<0$, then $L[f](t) = d^{-1}\begin{cases} 0, & t<0, \\ \int_0^t f(\tau)d\tau, & 0\le t \le d \\ \int_{t-d}^t f(\tau)d\tau, & d\le t .\end{cases}$

  5. 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 $f_3 \ast f_6(t) = \sqrt{\pi} f_{2}(t)=\sqrt{2\pi} e^{-2t^2}$.

  6. 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.
  1. Chapter 2: 13.
  2. Chapter 4: 2.

  3. Let $\hat f(\lambda)= \begin{cases} \pi-|\lambda| & |\lambda|\le \pi\\ 0 & \text{otherwise}\end{cases}$. (This is a tent function in $\lambda$.)
     1. Find $f(t)=\mathcal F^{-1}[\hat f(\lambda)](t)$.

     2. 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).

  4. This is a version of problem 12, chapter 2. Let \[ h(t):=\begin{cases} 1/d, & 0\le t\le d \\ 0, &\text{otherwise,}\end{cases} \ f(t) =\begin{cases} e^{-t}(\sin(3t) +\sin(40t)), & 0\le t\\ 0, &\text{otherwise,}\end{cases} \ \text{and } L[f]=f\ast h. \]
     1. 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.

     2. 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.
  1. Chapter 4: 5, 6, 7
  2. Chapter 5: 2
  3. Let $f(x)=x$, $0\le x \le 1$, and 0 otherwise.
     1. 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 $V_3$.
     2. 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.
  4. 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.
  1. Chapter 5: 8(c,d), 11