Math 414-501 — Spring 2024

Assignment 1 - Due Friday, 1/26/2024.

• Read sections 0.1-0.5

• Problems.
  1. Chapter 0: 2 (Do Example 0.3, p. 4.), 3, 10, 11, 12, 13, 15, 28
  2. Use the inner product in Example 0.3, pg. 4. Find the projection of w onto v, where the vectors are displayed below. Verify that if p is the required projection, then q := w − p is orthogonal to v.

     

  3. Find the energy in the signal $f(t)=e^{-|t|}$, $t\in \mathbb R$.

  4. Find the inner product of $f(t)= \sin(t)$ and $g(t)=t$, $t\in [0,\pi]$.

  5. Bonus: Show that $L^2(\mathbb R)$ is a vector space.

Assignment 2 - Due Friday, 2/2/2024.

• Read sections 1.1-1.1.3, 1.2.1-1.2.4

• Problems.
  1. Chapter 0: 17

  2. Chapter 1: 1

  3. Suppose that $V$ is an inner product space and that $V_0$ is a finite dimenstional subspace of $V$. Show that if $v\in V$ and $v_0=\mathrm{Proj}_{V_0}(v)$, then $\|v-v_0\|^2 = \|v\|^2 - \|v_0\|^2$. (Hint: see Theorem 0.20.)

  4. Let $\{e_1,\ldots,e_n\}$ be an orthonormal basis for a complex inner product space space $V$. If in this basis $v=\sum_{j=1}^n a_je_j$ and $w=\sum_{j=1}^n b_je_j$, show that $\langle v,w\rangle = \sum_{j=1}^n {\overline b}_j a_j$. In matrix notation, $\langle v,w\rangle =b^*a$, where $a$ and $b$ are column vectors having $a_j$'s and $b_j$'s as entries.

  5. Let $B=\{v_1,\ldots,v_n\}$ be an orthogonal set such that $\|v_j\|>0$ but $\|v_j\|$ might not equal 1 for some or all $j$'s. Show that $v=\sum_{j=1}^n \frac{\langle v,v_j\rangle}{\|v_j\|^2}v_j$.

  6. You are given that in $L^2[-\pi,\pi]$ the projection of $f(x)=x^2$, onto the orthonomal set $\{\frac{1}{\sqrt{2\pi}},\frac{\cos(x)}{\sqrt{\pi}}\}$ is $p(x) = \frac{\pi^2}{3} - 4\cos(x)$. Let $q(x)=\frac{\pi^2}{4}-5\cos(x)$. Which is larger $\|f-p\|^2$ or $\|f-q\|^2$? Do you need to compute the two norms or can you give an answer without finding them? Explain.

Assignment 3 - Due Friday, 2/9/2024.

• Read sections 1.2.4-1.2.5, 1.3.1-1.3.3

• Problems.

  1. Chapter 1: 4, 7, 8 (Do only $N=5$ and $N=10$.) 10

  2. Let $\mathcal T_N=\text{Span} \{1,\cos(x), \sin(x), \cdots, \cos(Nx),\sin(Nx)\}$. Recall that in class (1/31/24) we showed that $S_N=\text{Proj}_{\mathcal T_N}(f)$. Use problems 3 and 4 from assignment 2 to show the following.
     1. $\|S_N\|_{L^2[-\pi,\pi]}^2=\pi\big( 2a_0^2 + \sum_{n=1}^N a_n^2+b_n^2\big) $.
     2. Use problem 3 (HW 2) to show that $\|f-S_N\|_{L^2[-\pi,\pi]}^2= \|f\|_{L^2[-\pi,\pi]}^2-\pi\big( 2a_0^2 + \sum_{n=1}^N a_n^2+b_n^2\big). $
     3. The partial sum $S_N$ is said to converge in the mean to $f$ if and only if $\lim_{N\to \infty}\|f-S_N\|_{L^2[-\pi,\pi]}=0$. Use the previous result to prove Parseval's Theorem: $S_N$ convereges in the mean to $f$ if and only if Parseval's equation, $\|f\|_{L^2[-\pi,\pi]}^2=\pi\big(2a_0^2 + \sum_{n=1}^\infty a_n^2+b_n^2\big)$, holds.

Assignment 4 - Due Friday, 2/16/2024.

• Read sections 1.3.4-1.3.5

• Problems.

  1. Chapter 1: 12 (skip the plots), 13 (See the problem below.), 20, 21 (Hint: you need to find an $x$ where the series and function give the answer.), 32

  2. Show that the Fourier series of the sum or difference of two functions is the sum or difference of the Fourier series of those functions.

  3. Find the complex Fourier series for $|x|$ on $-\pi \le x \le \pi$. Sketch (by hand is okay) three periods of the function to which the Fourier series converges.

  4. Find the complex Fourier series for $e^x$ on the interval $0\le x \le 2\pi$. Sketch (by hand is okay) three periods of the function to which the Fourier series converges.

Assignment 5 - Due Friday, 2/23/2024.

• Read sections 1.3.4-1.3.5, 2.1

• Problems.
  1. Chapter 1: 18, 23(a,b,c,d) (Hand drawn sketches are fine.), 33

  2. 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 the $k^{th}$ derivative of $f$, $f^{(k)}$, is continuously differentiable, use induction to derive a similar formula for the the Fourier coefficients of $f^{(k)}$.

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

  3. Suppose that $f$ is defined on $[0,\pi]$ has the cosine series $ f(x) = a_0 +\sum_{n=1}^\infty a_n \cos(nx)$. Show this version of Parseval's theorem for $f$ holds: \[ \|f\|_{L^2[0.\pi]}^2= \frac{\pi}{2}\big(2a_0^2+\sum_{n=1}^\infty a_n^2\big) \]

  4. Find the complex version of the Fourier series for $f(x) =\cosh(2x)$, $-\pi\le x \le \pi$. Use the complex form of Parseval's theorem in eqn. (1.41), pg. 81, to find the sum of the series \[ \sum_{n=-\infty}^\infty \frac{1}{(n^2+4)^2}. \]

Assignment 6 - Due Friday, 3/8/2024.

• Read sections 2.1 and 2.2.1.

• Problems.
  1. Chapter 2 exercises: 1, 2, 4.

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

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

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

Assignment 7 - Due Friday, 3/22/2024.

• Read sections 2.2 and 2.3.

• Problems.
  1. Chapter 2 exercises: 5, 6.

  2. In the previous assignment you showed that $\mathcal F[e^{-|t|}] = \sqrt{\frac{2}{\pi}}\frac{1}{1+\lambda^2}$. 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: How are Fourier transforms and inverse Fourier transforms related? Use the answer to this and #6)
  3. Let $\phi(t) := \begin{cases} 1 & 0 \le t < 1, \\ 0 & \text{otherwise},\end{cases}$ and $\psi(t) := \begin{cases} 1 & 0 \le t < 1/2, \\ -1 & 1/2 \le t <1, \\0 & \text{otherwise.}\end{cases}$ Find $\phi\ast \psi(t)$. (Compute the convolution directly from the definition. Do not use the convolution theorem.)

  4. Let $h(t) = \begin{cases} 1 & -\pi \le t < \pi , \\ 0 & \text{otherwise}. \end{cases} \ $ Recall that $\hat h(\lambda) = \sqrt{\frac{2}{\pi}} \frac{\sin(\pi \lambda)}{\lambda} $. Use Plancheral's theorem to find $\int_0^\infty \frac{\sin^2(x)}{x^2}dx$.

  5. Let $f(t)=\begin{cases}1& \text{if } 0\le t \le 1, \\ 0 & \text{otherwise.} \end{cases}$ Show that, if $L$ is the Butterworth filter, one has \[L[f] = \begin{cases}A e^{-\alpha t}\int_0^{\min(1,t)} e^{\alpha \tau}f(\tau)d\tau & \text{if }t > 0,\\ 0 & \text{if }t \le 0.\end{cases}\]

Assignment 8 - Due Monday, 4/1/2024.

• Read sections 3.1.1-3.1.4.

• Problems.
  1. Chapter 2 exercise 13.

  2. Chapter 3 exercise 2. (Hint: use problem 4 below.)

  3. 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 if $h=h_1\ast h_2$ is the impulse response for $L[f]=h\ast f$, then $L$ is causal. If $h_1=h_2=h$, where $h$ is 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} \ $.

  4. Suppose that $\mathbf x$ is an n-periodic sequence (i.e., $\mathbf x \in \mathcal S_n$). Show that $ \sum_{j=m}^{m+n-1}{\mathbf x}_j = \sum_{j=0}^{n-1}{\mathbf x}_j $. (This is the DFT analogue of Lemma 1.3, p. 44.)

Assignment 9 - Due Monday, 4/8/2024.

• Read sections 4.1-4.4.

• Problems.
  1. Chapter 4: (All sketches may be done by hand.) 1 (Do this by the method in Example 11.) 2 (Do this problem by using the formulas in Theorem 4.12 to find the coefficients, and then the component parts.) 6, 7.

  2. Let $f\in L^2(\mathbb R)$. Find the projection of $f$ onto the scaling space $V_0$ and onto the wavelet space $W_0$.

  3. Finish the proof of Theorem 4.6, p. 147.

  4. Start putting groups together for projects. Groups must have at least 2 members, and may have 4 with my permission.

Assignment 10, Due Monday, 4/22/2024.

• Read sections 5.1 and 5.2
• Problems.
  1. Chapter 5 exercises: 2(a,b), 8(b,c,d)
  2. Let the p_k's be the scaling coefficients in Example 5.8, p. 195. For these, the scaling and wavelet relations are
     φ(x) = p₀φ(2x) + p₁φ(2x−1) + p₂φ(2x−2) + p₃φ(2x−3),
     ψ(x) = p₃φ(2x+2) − p₂φ(2x+1) + p₁φ(2x) − p₀φ(2x−1).
     1. Show these p_k's satisfy the four properties in Theorem 5.9, p. 196.
     2. Find all four filters corresponding to these coefficients: low pass and high pass decomposition and reconstruction filters.
     3. Use Theorem 5.9 and the wavelet ψ(x) above to show that ∫ ψ(x)dx = 0.
     4. BONUS: Show that ∫ xψ(x)dx = 0.
  3. Send me an email about your group and your project proposal. The email should contain the names of the members of the group, a contact person for it and a brief summary of the project.