Math 658 - Fall 2008
Homework
Assignment 1 - Due Wednesday, September 3, 2008
- Read sections 1.3, 2.1-2.3.
- Problems
- Verify the Fourier series for the functions in entries 1, 5, 6,
13, 18, 19 of Table 1, p. 26-28.
- Consider the trig polynomial T(x) = ∑|n|≤N
dn einx.
- Show that the Fourier coefficents of T are just the
dn's for |n|≤ N and 0 for |n|> N.
- Show that if f has the Fourier series ∑n
cn einx, then
(2π)-1
∫02π f(x)T(x)dx
=
∑|n|≤N
cndn.
- Show that (2π)-1 ∫02π
|T(x)|2dx = ∑|n|≤N
|dn |2, so Bessel's inequality becomes an
equation for trig polynomials.
- Show that if f has the Fourier series ∑n
cn einx, then
(2π)-1
∫02π|f(x) − T(x)|2dx
= (2π)-1
∫02π|f(x)|2dx −
∑|n|≤N |cn|2
+ ∑|n|≤N |dn
− cn|2
Assignment 2 - Due Friday, September 12, 2008
- Read sections 2.4-2.6.
- Problems
- Section 2.2: 2, 7
- Section 2.3: 2(a), 5, 7(b,c)
- Consider the Fourier series in #17 in Table 1 on p. 28. Estimate
the number of terms your would need to take to get the (unifrom)
error below 10−5.
- Let f and g be piecewise continuous, 2π periodic
functions. Define the convolution
f∗g(θ) :=
(2π)-1 ∫02πf(θ
-φ)g(φ)dφ.
Let f and g have the Fourier series ∑n
cn einθ and ∑n
dn einθ, respectively.
- Show that the Fourier series for f∗g is ∑n
cn dn einθ.
- Show that this series is uniformly convergent. (Hint. You will
need the M-test, Schwarz's inequality and Bessel's inequality.)
- Verify that the Fourier series
for f(−θ) is ∑n
cn einθ. Using
this series and assuming that f∗g(θ) = ∑n
cn dn einθ, show that
Parseval's (Plancheral's) equation holds for piecewise continuous f:
(2π)-1
∫02π|f(φ)|2dφ =
∑n |cn|2
- Use the same notation as in the previous problem. Define the
function H(θ) := f(θ)g(θ). Let ∑n
Cn einθ be the Fourier series for
H. Show that Cn =
∑k cn−k dk . In addition,
show that for n fixed the series
∑k cn−k dk
is absolutely convergent. (If we consider the sequences c =
{cn} and d = {dn}, the sequence C
={Cn} is called the convolution of c and d. We
write C=c∗d. Notice the similarity to f∗g.)
Assignment 3 - Due Wednesday, October 1, 2008
- Read sections 3.1-3.3, 7.1.
- Problems
- Section 2.6: 1
- Let gN(θ) be the function defined in the problem
above on p. 61; also, let a, b be fixed and satisfy 0 < a < b
< 2π. Show that gN converges to 0 uniformly
on the interval [a,b].
- Let Σn(u) be the Cesàro mean of the
Dirichlet kernels Dk(u). (See pgs. 60 and 34 for
definitions.) Show that
Σn(u) = [2π(n+1)
sin2(½u)]− 1
sin2(½(n+1)u).
-
Let σn(θ) be the Cesàro mean of the
partial sums sn(θ) of a Fourier series for a PC function f.
- Show that σn(θ) = ∑|k|≤
n(1-|k| ∕(n+1)) ckeikθ.
- Show that σn(θ) is also given by the
convolution (see Assignment 2)
σn(θ) = 2πΣn
∗f(θ).
- Bonus: show that if f is continuous, then σn
converges uniformly to f.
- Let Ka(θ) be the 2π periodic extension of the
function defined in entry #12 in Table 1 on p. 28, and let f be a
2π periodic PS function.
- For θ fixed, show that Ka∗f(θ) is,
in the sense of integrals, the average of f over the interval
[θ - a, θ + a].
- Given the complex form of the Fourier series for f, use the
results from Assignment 2 to find the complex form of the Fourier
series for Ka∗f.
- Let f be the 2π-periodic extension of the function f(θ)
= 1, 0 < θ < π, and f(θ) = − 1, −
π < θ < 0. At the discontinuities, just take f to be
0. Let the parameter a satisfy 0 < a < π/8. Find the
Fourier series of Ka∗f. By doing the integrals
involved, directly calculate the function
Ka∗f. Make a sketch of several periods of
Ka∗f for different values of a.
- Show that ℓ2, the set of bi-infinite sequences
having finite energy, is a Hilbert space. The inner product for this
space is
< x, y > = ∑k
xk yk .
Assignment 4 - Due Friday, October 24, 2008
- Read sections 7.5, 7.6 (DFT), and 9.1.
- Problems
- Section 7.1: 2, 3(a), 5 (You may use the FT and convolution
theorem.)
- Section 7.2: 4, 6, 13(a,b)
- Section 7.3: 3, 8, 10
- Show that if a linear filter L is causal (cf.
Linear Filters) and if the signal
f(t) = 0 for t < 0, then
L[f](t) = ∫0t f(t - τ)h(τ)dτ.
Assignment 5 - Due Friday, November 7, 2008
- Read sections 9.1 and 9.4.
- Problems
- Let f(t) be band-limited, with its Fourier transform f^(ω)
(f "hat") having support in [−2π, 2π]. The
Nyquist rate for f is 2π/π = 2, and the
Nyquist frequency is ½·2 = 1.
-
Show that if f is sampled at the Nyquist frequency, instead of the
Nyquist rate, then the sampling theorem doesn't recover f. Instead, it
gives another band-limited function, falias. Show that on
[−π, π],
f^alias(ω) = f^(ω − 2π) + f^(ω)
+ f^(ω + 2π),
and that f^alias = 0 otherwise.
- Referring to the previous part, consider the (single-frequency)
band-limited function
f(t) = sin(5πt/3+π/6),
whose angular frequency 5π/3 is in the band [−2π,
2π]. What is falias(t) in this case? Use your favorite
software to make a plot showing f(t), falias(t) on the
interval − 6 ≤ t ≤ 6, and also use circles or x's to
indicate samples of these two functions at integer values of t.
- Section 7.5: 2, 3, 6(a,b).
- Find the n-dimensional Fourier transform of the Gaussian, g(r)
=(2π)− n/2 exp(− r2/2), where r =
|x|.
- Prove the three formulas listed at the end of section 2 on p. 3
of the
Notes on the Discrete Fourier Transform.
- Section 9.4: 3.
Updated 10/29/08 (fjn)