MATH 172 HOMEWORK 5 SOLUTIONS
Exercise 3.4: Note that t = B |f | > 0 for some ball B = B (0, r) (centered at the origin
with radius r). For any x Rd with |x| 1,
f (x)
1
B (x, r + |x|)
|f (y )|dy
B (x,r +|x|)
1
B (x, r + |x|)
|f (y )|dy
B (0,r )
t
B (x,
Math 172
Problem Set 5 Solutions
2.4 Let E = cfw_(t, x) : 0 < x b, x t b. To prove integrability of g, rst
observe that
b
b
b
|g(x)|dx =
0
x
0
f (t)
dt dx
t
b
b
x
0
f (t)
dtdx.
t
Next note that |f (t)/t|E is a measurable function on R2 since the pull-bac
MATH 172 HOMEWORK 4 SOLUTIONS
Exercise 2.13: Let N R be a non-measurable subset. Consider the sets A, B R2
dened by
A = cfw_0 [0, 1], B = N cfw_0.
Both A and B are measurable because they have measure 0 in R2 . However, the sum-set
A + B = N [0, 1] is not
MATH 172: PROBLEM SET 5
DUE THURSDAY, FEBRUARY 20, 2014
Problem 1. Do Stein-Shakarchi, vol. 3, Ch. 2, Exercise 21.
Problem 2. This is the compact version of the previous problem.
Suppose f, g are measurable functions on [, ]d , and extend them to be 2 per
MATH 172: PROBLEM SET 8
DUE FRIDAY, MARCH 13, 2015, 2:15PM
(BEGINNING OF LECTURE)
Problem 1. Find the Fourier transform on R of the following functions:
(i) f (x) = [a,a] , a > 0.
(ii) f (x) = [0,) eax , where a > 0.
(iii) f (x) = |x|n ea|x| , where a > 0
MATH 172: THE FOURIER TRANSFORM BASIC
PROPERTIES AND THE INVERSION FORMULA
ANDRAS VASY
The Fourier transform is the basic and most powerful tool for studying translation invariant analytic problems, such as constant coecient PDE on Rn . It is
based on the
Math 172
Problem Set 7 Solutions
Problem 1.
(i) We rst note that KP is a good kernel, in the sense that:
KP 0,
KP (r, )d = 1 for any r,
and for any
> 0, there exists > 0 such that for any r [1 , 1),
KP (r, )d > 1 .
We also know that continuous functions a
MATH 172: TEMPERED DISTRIBUTIONS AND THE FOURIER
TRANSFORM
ANDRAS VASY
We have seen that the Fourier transform is well-behaved in the framework of
Schwartz functions as well as L2 , while L1 is much more awkward. Tempered
distributions, which include L1 ,
MATH 172: INNER PRODUCT SPACES, SYMMETRIC
OPERATORS, ORTHOGONALITY
ANDRAS VASY
When discussing separation of variables, we noted that at the last step we need to
express the inhomogeneous initial or boundary data as a superposition of functions
arising in
Math 172
Problem Set 8 Solutions
Problem 1.
(i) We have
a
[a,a] eix dx =
(F[a,a] )() =
eix dx
a
R
1
(eiax eiax )
i
2
= sin a.
=
(ii) We have
(F[0,) eax )() =
eax eix dx =
0
ex(a+i) dx =
0
1
a + i
where we have used the fact that limx |ex(a+i) | = 0 since
Math 172
Problem Set 3 Solutions
1.4a Denote by Cn the set obtained after the nth step in the construction of C.
n ) = 1 n 2k1 k . Each Cn is obviously measurable,
By construction, m(C
k=1
and hence so is C = n C. It thus follows directly from monotonicit
Math 172
Problem Set 2 Solutions
8a. Linear functions are clearly continuous, and continuous functions preserve
compact sets (this is easy to prove from the Heine-Borel property and the fact
that the inverse image of an open set under a continuous functio
MATH 172: CONVERGENCE OF THE FOURIER SERIES
ANDRAS VASY
We now discuss convergence of the Fourier series on compact intervals I. Convergence depends on the notion of convergence we use, such as
(i) L2 : uj u in L2 if uj u L2 0 as j .
(ii) uniform, or C 0
Math 172
Problem Set 6 Solutions
Problem 1 4.5a As suggested in the hint, we rst consider a function f (x) =
|x| when |x| 1, and zero everywhere else. Using polar coordinates, and
letting d denote the area of the unit sphere in Rd , we compute
1
f
2
L2
|f
MATH 172: PROBLEM SET 6 (EXPANDED, EXTENDED)
DUE WEDNESDAY, FEBRUARY 25, 2015
Problem 1. Do Stein-Shakarchi, vol. 3, Ch. 4, Exercise 5.
Problem 2. Do Stein-Shakarchi, vol. 3, Ch. 4, Exercise 6.
Problem 3. Do Stein-Shakarchi, vol. 3, Ch. 4, Exercise 7.
Pro
MATH 172: PROBLEM SET 7
DUE WEDNESDAY, MARCH 4, 2015
Problem 1.
(i) Let KP be the one-dimensional Poisson kernel:
1
1 r2
.
2 1 2r cos + r2
Show that for f L1 ([, ]), f KP f in L1 as r
1.
(ii) Show that 2-periodic C functions are dense in L1 ([, ]d ).
KP (
Math 172
Problem Set 1 Solutions
1. As stated in the hint, if x, y C and |x y| > 1/3k , then x and y lie in
dierent intervals of Ck . There is thus no path in Ck connecting them and hence
none in C = Ck . Therefore C is totally disconnected. To show that
MATH 172: MOTIVATION FOR FOURIER SERIES:
SEPARATION OF VARIABLES
ANDRAS VASY
Separation of variables is a method to solve certain PDEs which have a warped
product structure. First, on Rn , a linear PDE of order m is of the form
a (x) u = f (x),
|m
where a
Math 172
Problem Set 4 Solutions
2.2 Let R be a rectangle in Rd . We will rst show that R R L1 0 as
1. First observe that if x Rd and |i 1| < for all i, we have
|x x| =
(x1 1 x1 )2 + + (xn n xn )2 < |x|.
Therefore the symmetric dierence R R cfw_y : distR
MATH 172: PRACTICE MIDTERM
FRIDAY, FEBRUARY 7, 2014
This is a closed book, closed notes, no calculators/computers exam. As usual, m
is the (Lebesgue) exterior measure on Rd , and m is the corresponding measure.
You may quote any theorem from the textbook
MATH 172 PRACTICE MIDTERM SOLUTIONS
Problem 1. (25 points)
(i) State the denition of a set in Rd being (Lebesgue) measurable.
Solution. A subset E of Rd is (Lebesgue) measurable if for any
there exists an open set O with E O and m (O E ) .
>0
(ii) Suppose
MATH 172 HOMEWORK 6 SOLUTIONS
Exercise 3.11: Note that
f (x) = axa1 sin(xb ) + xa cos(xb )(b)xb1 = xa1 (a sin xb bxb cos xb ).
If a > b, then
1
1
|f (x)|dx a
0
1
xa1 dx + b
0
xab1 dx < .
0
This implies that f is of bounded variation. To see why, let 0 = x
MATH 172 HOMEWORK 3 SOLUTIONS
Exercise 2.4: Without loss of generality, we may assume that f 0 (otherwise, write
f = f+ f with f+ , f 0, and work with f+ , f separately). We have
b
b
b
g (x)dx =
0
0
x
f (t)
dtdx.
t
By Fubinis theorem (Theorem 3.2), this i
MATH 172 HOMEWORK 1 SOLUTIONS
Exercise 4 (Cantor-like sets) Let Jk, (1 2k1 ) be the 2k1 intervals removed at
the k th step of the construction. By countable additivity,
2k1
m(C ) = 1
2k1
m(Jk, ) = 1
k=1
=1
k
> 0.
k
This proves part (a).
To prove part (
MATH 172 HOMEWORK 7 SOLUTIONS
Exercise 3.14 (a): For any r > 0, dene a function Gr on [a, b] by the formula
F (x + h) F (x)
.
h
0<h<r
Gr (x) = sup
We claim that Gr is a measurable function. In fact, for any y R, the set
cfw_x : Gr (x) > y =
x:
0<h<r
F (x
MATH 172 HOMEWORK 8 SOLUTIONS
Exercise 6.1: We need to impose the further condition that M is closed under nite
intersections.
To prove that M is a -algebra, it suces to show that M is closed under countable
unions. Let A1 , A2 , be arbitrary sets in M. L
MATH 172: PROBLEM SET 6
DUE FRIDAY, FEBRUARY 28, 2014, 2:15PM
(BEGINNING OF LECTURE)
Problem 1. For the purposes of this problem, dene L1 ([0, 1]) and L2 ([0, 1]) as
the completions of C ([0, 1]) in the L1 , resp. L2 -norm (using the Riemann integral).
Wi
MATH 172: PROBLEM SET 6
DUE FRIDAY, MARCH 7, 2014, 2:15PM
(BEGINNING OF LECTURE)
Problem 1. Do Stein-Shakarchi, vol. 3, Ch. 5, Exercise 1.
Problem 2. Do Stein-Shakarchi, vol. 3, Ch. 5, Exercise 2.
Problem 3. Do Stein-Shakarchi, vol. 3, Ch. 2, Exercise 22.
MATH 172: PROBLEM SET 8
DUE FRIDAY, MARCH 14, 2014, 2:15PM
(BEGINNING OF LECTURE)
Problem 1.
(i) On R3 , nd the Fourier transform of the function g (x) = |x|1 . (Hint: to
do this eciently, consider g (x) as the limit of ga (x) = ea|x| |x|1 , and use
your
Math 172
Problem Set 1 Solutions
1. As stated in the hint, if x, y C and |x y | > 1/3k , then x and y lie in
dierent intervals of Ck . There is thus no path in Ck connecting them and hence
none in C = Ck . Therefore C is totally disconnected. To show that