For the inner products and vectors below, verify Schwarz's inequality
Use the inner product in Example 0.3, pg. 4, and the column
vectors v and w given below:
Use the L2 inner product given in Definition 0.5, pg. 6, with a = 0 and b = 1, and the ``vectors'' (functions)
f(x) = x2 and g(x) = x+1.
Due Friday, 1/27/2012
Assignment 2
Read sections 1.2.1-1.2.5
Problems.
Section 0.8: 15
Section 1.4: 1, 4, 7, 8, 9, 10, 11
Let V be a vector space with a complex inner product <
·,· >. Suppose that the set B =
{u1, u2,
..., un} is an orthonomal basis for V. Show that
if
v = a1u1 +
a2u2 + ... +
anun and w =
b1u1 +
b2u2 + ... +
bnun,
then <v, w > = ∑j ajbj
= bT 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.)
Due Wednesday, 2/8/2012
Assignment 3
Read section 1.3
Problems.
Chapter 1, exercises: 20, 22, 23(a,b,c,d), 32(c,d,e,f), 33. (In
32(f), numerically evaluate the integral using a computer.)
Suppose that f is a 2π-periodic function that has a continuous
derivative f′. If the Fourier series for f and f′ are
f(x) = a0 + ∑n ancos(nx) +
bnsin(nx) and f′(x) = a′0 +
∑n a′ncos(nx) +
b′nsin(nx),
then show that the coefficients of the two series are related this way
for n ≥ 1:
a′n = n bn, b′n = −
n an and a′0 = 0.
(Integrate by parts. 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).
In the text (cf. Example 1.9), we derived the Fourier series for
g(x) = x on − π < x < π. Use that series and the
previous problem to show that the Fourier series for f(x) =
(x3 − π2x)/12 is given by
f(x) = ∑n=1∞ (−1)n
sin(nx)/n3.
Due Friday, 2/17/2012
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:
t e-|t| (Use #2.)
e-2|t-3| (#6 and #7)
sign(t)e-|t| (Hint: differentiate e-|t|;
use #4.)
1/(1+(t-2)2) (Hint: How are Fourier transforms and
inverse Fourier transforms related? Use the answer to this and #6)
Find the Fouirer transforms of these functions.
g(t) = 1 if a < t < b and g(t) = 0 otherwise.
h(t) = -1 if -a < t < 0, h(t) = 1 if 0 < t < a, and
h(t) = 0 otherwise.
Due Friday, 2/24/2012
Assignment 5
Read sections 2.3-2.5.
Problems.
Chapter 2 exercises: 8, 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 ∫ 0min(1,t)
eατ f(τ)dτ,
if t ≥ 0, and that L[f] = 0 if t < 0.
Let h(t) be the impulse response function in exercise 12 of Chapter
2. For any signal f, show that
L[f] = d−1∫ t−dt
f(τ)dτ
Due Friday, 3/9/2012
Assignment 6
Read sections 3.1-3.2
Problems.
Suppose that x is an n-periodic sequence (i.e., x
∈ Sn). Show that
∑j=mm+n-1xj =
∑j=0n-1xj.
Chapter 3 exercises: 2 (Hint: use the previous problem.), 4, 5,
10, 12, 13, 16.
Consider the Gaussian function f(t) = e−
t2. The Fourier transform of this function is
F(ω) = 2− 1/2 e−
ω2/4. Numerically approximate F using the FFT
with interval [−3, 3] and n = 128, 256, and 1024. Graph both F
and its approximation Fap for these three values of n.
Due Wednesday, 3/28/2012
Assignment 7
Read sections 4.1-4.3
Problems.
Chapter 4 exercises: 2, 5, 6, 7, 11.
Consider a function f that is 0 for |x| ≥ 2 and is
cos(πx/4) for |x| ≤ 2. Find the projection of f onto
V0, V1 and W0 from the projection
formula given Theorem 0.21, with the bases being
{φ1,k}, {φ0,k}, and
{ψ0,k}, respectively. Do the coefficients satisfy the
formulas invoving a's and b's in Theorem 4.12?
Due Wednesday, 4/4/2012
Assignment 8
Read sections 5.1 and 5.2
Problems.
Chapter 5 exercises: 2, 5, 8(c,d,e,f)
Let the pk'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).
Show these pk's satisfy the four properties in Theorem
5.9, p. 196.
Find all four filters corresponding to these coefficients: low
pass and high pass decomposition and reconstruction filters.
Use Theorem 5.9 and the wavelet ψ(x) above to show that ∫
ψ(x)dx = 0.