Math 414-501 — Spring 2023

Assignment 1 - Due Wednesday, 1/25/2023.

• Read sections 0.1-0.5
• Problems.
  1. Chapter 0, exercises: 2, 3, 4, 5

Assignment 2 - Due Wednesday, 2/1/2023.

• Read sections 0.5.1-0.5.3, 1.1, 1.2.1
• Problems.
  1. Chapter 0, exercises: 7, 9, 11

  2. Consider the real inner product $\langle f,g\rangle := \int_0^1 f(x) g(x)dx$. If $f(x)=x^2$ and $g(x)=1+x$, show that Schwarz's inequality holds, and find the angle between $f$ and $g$.

  3. Show that the functions $\phi(x)$, $\psi(x)$, $\psi(2x)$ and $\psi(2x -1)$ defined in exercise 0.15 are orthogonal.

Assignment 3 - Due Wednesday, 2/8/2023.

• Read sections 0.5.1-0.5.3, 1.1, 1.2.1
• Problems.
  1. Chapter 0, exercises: 12 (Only do $1,x,x^2$.) 14 (Only do n=1 and 2, and skip $g(x)=x^3$.), 15

  2. Let V be a vector space with a complex inner product < ·,· >. Suppose that the set B = {u₁, u₂, ..., u_n} is an orthonormal basis for V. Show that if
     v = a₁u₁ + a₂u₂ + ... + a_nu_n and w = b₁u₁ + b₂u₂ + ... + b_nu_n,
     then <v, w > = ∑_j a_j b_j = b^T a. (Hint: Put the expression for v in the inner product and then use additivity and homogeneity. Finally, identify the coefficients that multiply the a's in the resulting sum.)

  3. Show that a projection is unique; i.e., if $v_0$ and $v_1$ are both projections of $v\in V$ onto the subspace $V_0$, then $v_0=v_1$.

Assignment 4 - Due Wednesday, 2/15/2023.

• Read sections 1.1, 1.2.1-1.2.3
• Problems.
  1. Chapter 0, exercises: 23, 27

  2. Chapter 1, exercises: 1, 2, 4, 7

  3. Let $V$ be an inner product space. Suppose that $u$ and $v$ are orthogonal. Prove the Pythogorean Theorem: $\|u+v\|^2=\|u\|^2+\|v\|^2$. Next, let $V_0$ be a finite dimensional subspace of $V$. Use the Pythagorean Theorem to show that if $u=f$ and $v=-\text{Proj}_{V_0}(f)$ then $\|f-\text{ Proj}_{V_0}(f)\|^2=\|f\|^2-\|\text{Proj}_{V_0}(f)\|^2$. (Hint: $f-\text{ Proj}_{V_0}(f)$ is orthogonal to $V_0$.)

Assignment 5 - Due Wednesday, 3/1/2023.

• Read sections 1.2.5 and my notes on pointwise convergence of Fourier series.
• Problems.
  1. Chapter 1, exercises: 9, 10, 20
     1. In part 10(c), the period is $\pi$ and the function is defined on $[0,\pi]$. The general formula for a Fourier series with period $T$ defined on the interval $[0,T]$ is \[ f(x) \sim a_0 + \sum_{n=1}^\infty a_n \cos\big(\frac{2n\pi x}{T}\big) + b_n \sin\big(\frac{2n\pi x}{T}\big), \] where $a_0 = \frac{1}{T}\int_0^T f(t)dt$, $a_n=\frac{2}{T}\int_0^T f(t) \cos\big(\frac{2n\pi t}{T}\big)dt$, $b_n=\frac{2}{T}\int_0^T f(t) \sin\big(\frac{2n\pi t}{T}\big)dt$. Replace $T$ by $\pi$ to get the series that you want.

     2. In exercise 20, an explanation using a sketch will suffice.

  2. Exercise 11 in Chapter 1 has errors in it replace; it by this: Let $f(x)=e^{-x/3}$ for $-\pi \le x \le \pi$ and $g(x)=e^{-x/3}$ for $0\le x \le 2\pi$.
     1. Find the complex Fourier series for both $f$ and $g$.
     2. Sketch by hand 3 periods of the function to which the series for $f$ converges.
     3. Sketch by hand 3 periods of the function to which the series for $g$ converges. (Use $\alpha_n=\frac{1}{2\pi}\int_0^{2\pi}g(x)e^{-inx}dx$.)
     4. These functions are different. Does this contradict Lemma 1.3? Why or why not?

  3. Using the series that for $f$ in the problem above and the formulas in section 1.2.5, find the cosine/sine series for $f$.

Assignment 6 - Due Wednesday, 3/8/2023.

• Read sections 1.3.4 and 1.3.5.
• Problems.
  1. Chapter 1, exercises: 18, 22, 23(a,b,c,d) (Hand drawn sketches are fine.), 32(c,d,e,f), 33 (In 32(f), numerically evaluate the integral using a computer.)

  2. Prove that $P(u)$ satisfies the first 3 properties stated in my notes on pointwise convergence of Fourier series.

  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 $b_n'=-na_n$. (This was essentially done in Theorem 1.30 in the text. The result is also true if $f'$ is only piecewise continuous.) If f is k times continuously differentiable, use induction to derive a similar formula for the the Fourier coefficients of $f^{(k)}$, the $k^{th}$ derivative of $f$.

     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 \[ f(x) = \sum_{n=1}^\infty\frac{(-1)^n}{n^3} \sin(nx) \] (Hint: Run the result in part (a) "backwards": If $n\ne 0$, $a_n=-b_n'/n$ and $b_n=a_n'/n$.)

  4. Use Parseval's equation and the series above to find the sum $\sum_{n=1}^\infty \frac{1}{n^6}$.

Assignment 7 - Due Wednesday, 3/22/2023.

• Read sections 2.1, 2.2.1
• Problems.
  1. Chapter 2, exercises: 1, 2, 4

  2. Show that if $f(x)= a_0 +\sum_{n=1}a_n\cos(nx)$ is the cosine series for $f$ defined on $0\le x\le \pi$, then \[ \int_0^\pi |f(x)|^2dx = \pi |a_0|^2+ \frac{\pi}{2} \sum_{n=1}^\infty |a_n|^2$. \]

  3. Show that if $f(x)= \sum_{n=1}^\infty b_n\sin(nx)$ is the sine series for $f$ defined on $0\le x\le \pi$, then \[ \int_0^\pi |f(x)|^2dx = \frac{\pi}{2} \sum_{n=1}|b_n|^2. \]

  4. The complex form of the Fourier series for $f(x)=e^x$, $-\pi \le x \le \pi$ is given by $f(x) = \sum_{n=-\infty}^\infty \frac{(-1)^n(e^\pi - e^{-\pi})}{2\pi (1-in)}e^{inx}$. Find the sum of the series below. \[ \sum_{n=-\infty}^\infty \frac{1}{1+n^2} \]

Assignment 8 - Due Monday, 3/27/2023.

• Read sections 2.2.1, 2.2.2, 2.2.4
• Problems.
  1. Show that the Fourier transform of $f(t)=e^{-|t|}$ is \[ \widehat f(\lambda)= \sqrt{\frac{2}{\pi}}\frac{1}{1+\lambda^2}.\] Hint: Break up the FT into the integrals below and do each integral separately. \[\widehat f(\lambda)=\frac{1}{\sqrt{2\pi}}\int_0^\infty e^{-t}e^{-i\lambda t}dt+ \frac{1}{\sqrt{2\pi}}\int_{-\infty}^0 e^{t}e^{-i\lambda t}dt. \]

  2. Use the properties listed in Theorem 2.6 to find the Fourier transforms of the following functions:
     1. $te^{-|t|}$ (Use #2.)
     2. $e^{-2|t-3|}$ (#6 and #7)
     3. ${\rm sign}(t)e^{-|t|}$ (Hint: differentiate $e^{-|t|}$; use #4.)
     4. $\begin{cases}2\pi -t & \pi \le t\le 2\pi \\ t & 0\le t \le \pi \\ 0 & \text{otherwise} \end{cases}$. (Hint: How is this related to the tent function in Example 2.5? Use property 6.)
     5. $(1+(t-2)^2)^{-1}$  (Hint: How is this function related to $\widehat f(\lambda)$, where $f(t)=e^{-|t|}$? Once you've gotten done that, use #6.)

  3. Find the Fourier transforms of these functions.
     1. $g(t) = \left\{\begin{array}{cl} 1 & \text{if }-1 \le t \le 2 \\ 0 & \text{otherwise}. \end{array} \right.$

     2. $h(t) = \left\{\begin{array}{cl} 1 & \text{if }-3 \le t \le 0 \\ -1 & \text{if }\ 0 < t \le 3 \\ 0 & \text{otherwise}. \end{array} \right.$

Assignment 9 - Due Wednesday, 4/5/2023.

• Read sections 2.2.4, 2.3, 2.4
• Problems.
  1. Chapter 2, exercises: 5, 6, 10

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

Assignment 10 - Due Monday, 4/17/2023.

• Read sections 4.2 and 4.3.
• Problems.
  1. Chapter 4: 2, 6, 7, Bonus: 11 (10 points)

  2. 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 $L[f]=L_2\big[L_1[f]\big]$, 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} \ $.

  3. Let $f(x)=x$, $0\le x \le 1$, 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 $V_3$.

  4. Start putting together groups for projects. Groups with 2-4 students are okay. Once you have a group, choose someone to represent it, and have the person email me with a list of group members, their email addresses, and a short abstract for the project. For suggestions, click on the link projects

Assignment 11 - Due Wednesday, 4/26/2023.

• Read sections 5.1.1-5.1.4, 5.2.1-5.2.3.
• Problems.
  1. Chapter 5: 2, 4, 8(c,d,e,f)

  2. Consider the $p_k$'s in Example 5.8, p. 195. Show that the wavelet $\psi(x)$ in equation (5.5) is orthogonal to the $\phi(x-k)$'s.

  3. For the $p_k$'s in Example 5.8, write out the high pass decomposition and reconstruction filters, and do the same for the low pass filters.