Math 118 - Winter 2009 - Midterm Exam
Name:
Student ID:
Signature:
Instructions: Print your name and student ID number, write your signature to indicate
that you accept the honor code. During the test, you may not use computers, phones, or
any other electronic device. Read each question carefully, and show all your work. Justify
all your answers. You have 75 minutes (3:15p-4:30p) to answer all the questions.
Question
Score
Maximum
1
35
2
15
3
25
4
25
Total
100
Problem 1. Consider the function f (x) = x on the interval [−π, π].
(a) (10 pts.) Calculate the Fourier series expansion fˆk of f (x). [Hint: use integration by
parts.]
(b) (5 pts.) Show that the series
to an integral.]
P
k≥1
k −1 diverges to infinity. [Hint: compare the series
1
(c) (5 pts.) Is the Fourier series fˆk that you calculated earlier absolutely summable, i.e.,
do we have
X
|fˆk | < ∞?
k∈Z
(d) (10 pts.) The function f can be mapped onto the interval [−1, 1] by means of the
change of variables g(x) = f (πx). Calculate the expansion of g(x) into Chebyshev
polynomials.
2
(e) (5 pts.) Does the Chebyshev expansion of g converge faster than the Fourier series
expansion of f ? Explain very concisely the intuitive reason for this behavior. [Hint:
this is a question that you have a shot at answering even if you did not do parts
(a)–(d).]
3
Problem 2. (15 pts.) Make judicious use of Fourier analysis to compute the value of
Z
∞
−∞
sin x
x
4
2
dx.
Problem 3. (25 pts.)
Consider a vector fj of length N , and assume that N is even.
Consider the vector gj defined by
gj = (−1)j fj .
How does the discrete Fourier transform of gj relate to that of fj ? [Hint: eiπ = −1.]
5
Problem 4.
(a) (10 pts.) Show that, for x ∈ R,
|eix − 1| ≤ |x|.
[Hint: write eix − 1 as a definite integral.]
(b) (15 pts.) A function is said to be Lipschitz continuous over R if there exists M < ∞
such that, for all x, y ∈ R such that x 6= y,
|
f (x) − f (y)
| ≤ M.
x−y
It is a notion of regularity stronger than continuity, but weaker than differentiability.
Show that if
Z ∞
|k||fˆ(k)| dk < ∞,
−∞
then f is Lipschitz continuous over R. [Hint: use part (a).]
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