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.
Use the L^{2} inner product given in Definition 0.5, pg. 6,
with a = 0 and b = 1. Let f(x) = x^{2} 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.
Use the L^{2} 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) = e^{x/2} in the L^{2}[−1,1] sense.
Due Friday, 1/24/2014
Assignment 2
Read sections 1.1.1-1.1.3, 1.2.1-1.2.5
Problems.
Chapter 1, exercises: 1, 4, 7, 8
In this problem, use the L^{2} inner product given in
Definition 0.5, pg. 6, with a = −1 and b = 1,
Verify that (normalized) Legendre polynomials below form an
orthonormal basis for $P_2$.
\[
p_0(x)=\frac{1}{\sqrt{2}}, \quad p_1(x) = \sqrt{\frac {3}{2}}\, x,
\quad p_2(x)=\sqrt{\frac{5}{8}}(3x^2-1)
\]
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 $q(x) =b_0p_0(x)
+ b_1p_1(x) + b_2p_2(x)$.
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 $\langle
p,q\rangle_{L^2[-1,1]} = a_0b_0+a_1b_1+a_2b_2=b^Ta$.
Let V be a vector space with a complex inner product <
·,· >. Suppose that the set B =
{u_{1}, u_{2},
..., u_{n}} is an orthonormal basis for V. Show that
if
v = a_{1}u_{1} +
a_{2}u_{2} + ... +
a_{n}u_{n} and w =
b_{1}u_{1} +
b_{2}u_{2} + ... +
b_{n}u_{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 activity and
homogeneity. Finally, identify the coefficients that multiply the a's in the
resulting sum.)
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 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
\[
f(x) = \sum_{n=1}^\infty\frac{(-1)^n}{n^3} \sin(nx)
\]
Sketch the $2\pi$-periodic function to which this series
converges pointwise. Determine whether or not the series is unformly
convergent.
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 $x=\pi/2$.)
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.
Chapter 2 exercises: 1, 2, 4.
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:
$te^{-|t|}$ (Use #2.)
$e^{-2|t-3|}$ (#6 and #7)
${\rm sign}(t)e^{-|t|}$ (Hint: differentiate $e^{-|t|}$;
use #4.)
$(1+(t-2)^2)^{-1}$ (Hint: How are Fourier transforms and
inverse Fourier transforms related? Use the answer to this and #6)
Chapter 2 exercises: 8, 10, 11, 12 (skip the plots for $\hat
h(\lambda)$), 13.
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} ∫_{ 0}^{min(1,t)}
e^{ατ} f(τ)dτ,
if t ≥ 0, and that L[f] = 0 if t < 0.
Due Friday, 3/7/2014
Assignment 6
Read section 3.1.
Problems.
Chapter 3 exercises: 2 (Hint: use the additional problem below.),
4, 5, 10, 12, 13.
Suppose that x is an n-periodic sequence (i.e., x
∈ 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.)
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.)