# Math 414-501 — Spring 2014

## Assignments

Assignment 1

• Problems.
1. Chapter 0, exercises: 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 := wp is orthogonal to v.

3. Use the L2 inner product given in Definition 0.5, pg. 6, with a = 0 and b = 1. Let f(x) = x2 and g(x) = x+1. Find the projection of g onto f. Verify that if p is the required projection, then q:= g − p is orthogonal to f.

4. Use the L2 inner product given in Definition 0.5, pg. 6, with a = −1 and b = 1. Find the straight line y = A+Bx that best fits the function f(x) = ex/2 in the L2[−1,1] sense.
Due Friday, 1/24/2014

Assignment 2

• Problems.
1. Chapter 1, exercises: 1, 4, 7, 8

2. In this problem, use the L2 inner product given in Definition 0.5, pg. 6, with a = −1 and b = 1,
1. Verify that (normalized) Legendre polynomials below form an orthonormal basis for $P_2$. $p_0(x)=\frac{1}{\sqrt{2}}, \quad p_1(x) = \sqrt{\frac {3}{2}}\, x, \quad p_2(x)=\sqrt{\frac{5}{8}}(3x^2-1)$
2. Expand the polynomials $p(x) =x^2 -x$ and $q(x) = 3x+1$ in the basis $\{p_0,p_1,p_2\}$ — i.e., find $a$'s and $b$'s such that $p(x) =a_0p_0(x) + a_1p_1(x) + a_2p_2(x)$ and $q(x) =b_0p_0(x) + b_1p_1(x) + b_2p_2(x)$.

3. For the $a$'s and $b$'s that you found above, let $a = (a_0\ a_1 a_2)^T$ and $b = (b_0\ b_1 b_2)^T$. Verify that $\langle p,q\rangle_{L^2[-1,1]} = a_0b_0+a_1b_1+a_2b_2=b^Ta$.

3. Let V be a vector space with a complex inner product < ·,· >. Suppose that the set B = {u1, u2, ..., un} is an orthonormal basis for V. Show that if
v = a1u1 + a2u2 + ... + anun and w = b1u1 + b2u2 + ... + bnun,
then <v, w > = ∑j aj bj = bT a. (Hint: Put the expression for v in the inner product and then use activity and homogeneity. Finally, identify the coefficients that multiply the a's in the resulting sum.)
Due Friday, 1/31/2014

Assignment 3

• Problems.
1. Chapter 1, exercises: 10, 11, 18, 21, 22, 23(a,b,c,d), 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), we derived the Fourier series for $g(x) = x$ on $-\pi \le x <\pi$. Use the series for $g$ and equations (1.35) and (1.36) in the text to show that the Fourier series for $f(x) = \frac{1}{12}(x^3 - \pi^2x)$ is given by $f(x) = \sum_{n=1}^\infty\frac{(-1)^n}{n^3} \sin(nx)$
2. Sketch the $2\pi$-periodic function to which this series converges pointwise. Determine whether or not the series is unformly convergent.
3. Use the series for $f(x)$ to show that $\sum_{k=1}^\infty \frac{(-1)^{k+1}}{(2k-1)^3}=\frac{\pi^3}{32}$. (Hint: use $x=\pi/2$.)
4. Use Parceval's equation to find $\sum_{n=1}^\infty \frac{1}{n^6}$.
Due Wednesday, 2/12/2014

Assignment 4

• Read sections 2.1 and 2.2.
• Problems.
1. Chapter 2 exercises: 1, 2, 4.
2. Find the Fourier transform of $f(t) = e^{-|t|}$. In addition, 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 are Fourier transforms and inverse Fourier transforms related? Use the answer to this and #6)

3. Find the Fouirer 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. $h(t) = \left\{\begin{array}{cl} -1 & \text{if }-3 \le t \le 0 \\ 1 & \text{if }\ 0 < t \le 3 \\ 0 & \text{otherwise}. \end{array} \right.$
Due Friday, 2/21/2014

Assignment 5

• Problems.
1. Chapter 2 exercises: 8, 10, 11, 12 (skip the plots for $\hat h(\lambda)$), 13.
2. 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] = A e− α t 0min(1,t) eατ f(τ)dτ, if t ≥ 0, and that L[f] = 0 if t < 0.

Due Friday, 3/7/2014

Assignment 6

2. Suppose that x is an n-periodic sequence (i.e., xSn). 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.)
3. Consider the Gaussian function $f(t)=e^{-t^2}$. Its Fourier transform is $\hat f(\lambda) =2^{-1/2}e^{-\lambda^2/4}$. Numerically approximate $\hat f$ using the FFT with interval $[-3, 3]$ and $n$ = 128, 256, and 1024. Graph both $\hat f$ and its approximation $\hat f_{ap}$ for these three values of $n$. (See section 3.1.4 for details.)