Math 414-501 Assignments — Spring 2021

Assignment 1 — Due Wednesday, 1/27/2021.

• Read sections 1.2.1-1.2.4.
• Do the following problems.
  1. For each function $f$ below sketch three periods of its $2\pi$ periodic extension and compute its Fourier series. Sketches done by hand are okay.
     1. $f(x)=\pi -|x|$, $-\pi\le x\le \pi$
     2. $f(x)=\begin{cases}1&0\le x\le \pi\\ 0 & -\pi < x <0\end{cases}$.
     3. $f(x)=x^2$, $-\pi\le x\le \pi$.
  2. For the series in 1(c) above, use Matlab or some other program to plot, on the same set of axes, $x^2$ and the partial sums of the Fourier series for $N=4,7,10$ on the interval $-\pi\le x\le \pi$. The m-file part_sum_c.m can help with the plots.
  3. Chapter 1: 3, 7, 14.

Assignment 2 — Due Wednesday, 2/3/2021.

• Read sections 1.2.5, 1.3.1-1.3.4, notes on point-wise convergence of Fourier series
• Problems:
  1. Chapter 1: 4, 8 (skip the plots), 19, 21, 25
  2. Let $f(x) = e^{-x/3}, - \pi \le x \le \pi$ and let $g(x) = e^{-x/3}, 0\le x \le 2\pi$.
     1. Find the complex form of the Fourier series for $f$.
     2. Find the complex form of the Fourier series for $g$.
     3. Notice that the coefficients in the FS for $f$ and for $g$ are different. By Lemma 1.3, shouldn't these be the same? Why are they different? (Hint: sketch three periods of the periodic extensions of $f$ and $g$.)

Assignment 3 — Due Wednesday, 2/10/2021.

• Read sections 1.2.5, 1.3.4-1.3.5, notes on point-wise convergence of Fourier series
• Problems:
  1. Chapter 1: 23(a,b,c,d) (Hand drawn sketches are fine.), 24(a,b,c,d), 29, 32(a,c,d,f).
  2. Consider the partial sum $S_N(x)=a_0+\sum_{n=1}^\infty a_n\cos(nx)+b_n\sin(nx)$. Do the following.
     1. Let $n\ge 1$. Show that the $n^{th}$ term of the sum has the form $a_n\cos(nx)+b_n\sin(nx)=\frac{1}{\pi}\int_{-\pi}^\pi f(t)\cos(n(x-t))dt$.
     2. Show that $S_N(x)=\int_{-\pi}^\pi f(t) \big(\frac1{2\pi}+\frac1{\pi}\sum_{n=1}^N \cos(n(x-t))\big)dt$.
     3. Show that $\frac1{2\pi}+\frac1{\pi}\sum_{n=1}^N \cos(nu) = \frac1{2\pi}\sum_{n=-N}^N e^{inu}= P_N(u)$, which is the Dirichlet kernel.
     4. Use the product to sum identity $\sin(A)\cos(B)=\frac12(\sin(A+B)-\sin(B-A))$ to show that if $u\ne 0$ then \[ \begin{align} \sin(\frac{u}{2})\bigg(\underbrace{\frac1{2\pi}+\frac1{\pi}\sum_{n=1}^N \cos(nu)}_{P_N(u)}\bigg)&= \frac{1}{2\pi}\big(\sin(\frac{u}{2}) + \big(\sin(\frac{3u}{2})-\sin(\frac{u}{2})\big) +\big(\sin(\frac{5u}{2}) - \sin(\frac{3u}{2})\big) +\cdots +\big(\sin(\frac{(N+1)u}{2})- \sin(\frac{Nu}{2}\big))\\ &= \frac{1}{2\pi}\sin(\frac{(2N+1)u}{2}) =\frac{1}{2\pi}\sin((N+1/2)u)) \end{align} \] Dividing by $\sin(u/2)$ then yields $P_N(u)=\frac{\sin((N+\frac12)u)}{2\pi \sin(\frac{u}{2})}$. (Of course, $P_N(0)=\frac{N+1/2}{\pi}$.)
  3. Carry out the proof of the Riemann-Lebesgue Lemma for the case $\int_a^b f(x)\sin(\lambda x)dx$. For the specific function $f(x)=x$ and limits $a=1$, $b=2$, show that the lemma holds, by directly computing the quantities involved.

Assignment 4 — Due Wednesday, 2/17/2021.

• Read sections 1.3.4-1.3.5, 0.2, 0.3, 0.5.1, 0.5.2 (The sections from chapter 0 are to give background for convergence in the mean and wavelets.)
• Problems:
  1. Chapter 1: 15, 18, 27, 33

  2. Let $f(x) = \frac{1}{12}(x^3 - \pi^2x)$, $-\pi \le x \le \pi$.
     1. In the text (cf. Example 1.9, p. 51), we derived the Fourier series for $f''(x) = x$ on $-\pi \le x <\pi$. Use this series and equations (1.35) and (1.36) on p. 74 to show that \[ f(x) = \sum_{n=1}^\infty\frac{(-1)^n}{n^3} \sin(nx) \]
     2. Explain why the series for $f$ converges pointwise. Use this fact to find the value of the series $\sum_{k=1}^\infty \frac{(-1)^k}{(2k-1)^3}$.
     3. Use the Fourier series for $f$ and Parceval's theorem to find the value of the series $\sum_{n=1}^\infty \frac{1}{n^6}$.

Assignment 5 — Due Wednesday, 3/3/2021.

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

  2. Find the Fourier transform of each of the following functions:
     1. $f(t) = 1 - t^2, \ -1\le t \le 1$
     2. $g(t)= e^{-|t|}, \ -\infty < t <\infty$

Assignment 6 - Due Wednesday, 3/10/2021.

• Read sections 2.1 and 2.2.
• Problems.
  1. Chapter 2 exercise: 4.
  2. Recall that 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. $te^{-|t|}$ (Use #2.)
     2. $e^{-2|t-3|}$ (#6 and #7)
     3. ${\rm sign}(t)e^{-|t|}$ (Hint: differentiate $e^{-|t|}$; use #4.)
     4. $(1+(t-2)^2)^{-1}$  (Hint: How is this function related to $\hat f(\lambda)$, where $f(t)=e^{-|t|}$? Once you've gotten this, 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 7 - Due Monday, 3/22/2021.

• 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 Wednesday, 4/7/2021.

• Read sections 0.5.2 (orthogonal projections), 4.2, 4.3 and 5.1
• Problems.
  1. Find the orthogonal projection of $\begin{pmatrix} 2& 3 & -1 & 2\end{pmatrix}^T$ onto the space spanned by $\begin{pmatrix} 1 &0 &-1 & 1\end{pmatrix}^T/\sqrt{3}$ and $ \begin{pmatrix} 0&2 &-1 & -1\end{pmatrix}^T/\sqrt{6}$.

  2. Chapter 4: 2, 5, 6, 7

  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. Start putting together groups for projects.

Assignment 9 - Due Friday, 4/16/2021.

• Read sections 5.1 and 5.2.
• Problems.
  1. Chapter 5: 3, 4, 8, 11, 13
  2. Recall that the projection of a function $f$ onto $V_j$, which has the orthonormal basis $\{ \phi_{j,k} \}_{k=-\infty}^\infty$, where $\phi_{j,k}(x)=2^{j/2}\phi(2^jx-k)$, is given by \[ P_{V_j}f(x)=\sum_{k=-\infty}^\infty \langle f, \phi_{j,k}\rangle \phi_{j,k}(x). \] Using the orthogonal basis $\{\phi(2^jx-k)\}_{k=-\infty}^\infty$, the projection is given by \[ P_{V_j}f(x)=\sum_{k=-\infty}^\infty a_k^j \phi(2^jx-k). \] Show that for the Haar MRA $a_k^j$ is the average of $f$ over the interval $[2^{-j}k,2^{-j}(k+1)]$. (The average of a function $f$ over an interval $[a,b]$ is $\text{avg}(f)=\frac{1}{b-a}\int_a^b f(x)dx$.)

  3. Finish putting together groups for projects.

Updated 3/28/2021 (fjn)