Math 504 Spring Term 2009, Solutions 1
1. 1.1. Exercise 2. Prove that if f1 , f2 . . . are real continuous functions of R and if for each x R we
have f (x) limn fn (x) exists, then cfw_x : 0 f (x) < 1 is measurable. Hint: Prove that cfw_x : f (x) < 1 =
k1
MATH 504 ANALYSIS IN EUCLIDEAN
SPACES, SPRING TERM 2009, SOLUTIONS 4
1. 1.5 Exercise 1. Show that there are , in L1 (S 1 ) such that +
2 = 2 2 + 2 2.
1
1
1
Let (x) =
11

02
1
2 , (x) = x.
1 /2 1
( 2 x)dx
0
xdx =
and 1 = 1 , so RHS= 1.
2
Then + 1  =
MATH 504 SPRING TERM 2009, SOLUTIONS 5
1. Evaluate the series
(1)n1
.
(2n 1)3
n=1
e(nx)
n=0 (2in)3 .
By Problems 3, when 0 x 1, B3 (x) =
in in
n=1
n3
Moreover
n=0
e(n/4)
n3
=
and in in = 0 when n is even, is 2i when n 1 (mod 4) and
is 2i when n 3 (mod 4
Math 504 Analysis in Euclidean Spaces, Spring Term 2009, Solutions 6
1. Suppose that cfw_un is uniformly distributed (mod 1), and let c be a real number. Put
vn = un + c. Show that cfw_vn is uniformly distributed.
n
1
By Weyl I, for h N we have n m=1 e(
Math 504 Spring Term 2009, Solutions 1
1. 1.1. Exercise 2. Prove that if f1 , f2 . . . are real continuous functions of R and if for each x R we
have f (x) limn fn (x) exists, then cfw_x : 0 f (x) < 1 is measurable. Hint: Prove that cfw_x : f (x) < 1 =
k1
MATH 504 ANALYSIS IN EUCLIDEAN
SPACES, SPRING TERM 2009, PROBLEMS 6
Return by Monday 23rd February
1. Suppose that cfw_un is uniformly distributed (mod 1), and let c be a real number.
Put vn = un + c. Show that cfw_vn is uniformly distributed.
2. Let n
MATH 504 ANALYSIS IN EUCLIDEAN
SPACES, SPRING TERM 2009, PROBLEMS 2
Return by Monday 26th January
These exercises are essentially the same as in the text, so I have included a cross
reference.
1. 1.4. Exercise 5. Prove that if cfw_zn is a sequence of com
MATH 504 ANALYSIS IN EUCLIDEAN
SPACES, SPRING TERM 2009, PROBLEMS 3
Return by Monday 2nd Febraury
1. Let the polynomials Bp (x) for k N be dened on R by B1 (x) = x 1 , Bp+1 (x) =
2
x
1
p (k ) denote the Fourier coecient (relative
Bp (y )dy 0 (1 y )Bp (y )
MATH 504 ANALYSIS IN EUCLIDEAN
SPACES, SPRING TERM 2009, PROBLEMS 4
Return by Monday 9th Febraury
1. 1.5 Exercise 1. Show that there are , in L1 (S 1 ) such that +
2 = 2 2 + 2 2.
1
1
1
2
1
+
2. 1.5 Exercise 8. Prove that if f L1 (S 1 ) and if for each
MATH 504 ANALYSIS IN EUCLIDEAN
SPACES, SPRING TERM 2009, PROBLEMS 5
Return by Monday 16th Febraury
1. Evaluate the series
(1)n1
.
(2n 1)3
n=1
This is L(3, ) where is the nontrivial Dirichlet character modulo 4. Hint:
Problems 3 can be useful.
2. Find a Fo
MATH 504 ANALYSIS IN EUCLIDEAN
SPACES, SPRING TERM 2009, SOLUTIONS 2
1. 1.4. Exercise 5. Prove that if cfw_zn is a sequence of complex numbers such that
1
limn zn = z , then limn n (z1 + + zn ) = z . Give an example in which the
second limit exists but t
Math 504 Spring Term 2009, Solutions 3
x
1
1. Let the polynomials Bp (x) for k N be dened on R by B1 (x) = x 2 , Bp+1 (x) = 0 Bp (y )dy
1
(1 y )Bp (y )dy and let Bp (k ) denote the Fourier coecient (relative to the family e(kx) on L2 (S1 ).
0
(i) Prove t
MATH 504 ANALYSIS IN EUCLIDEAN
SPACES, SPRING TERM 2009, SOLUTIONS 11
1. In class we showed that if f L1 (R) and f satises
b
f (t)e(xt)dt
f (x) =
(0)
b
and
f (x) = 0
(1)
holds for x > a, then f is identically 0. Prove that the conclusion follows provide
MATH 504 ANALYSIS IN EUCLIDEAN
SPACES, SPRING TERM 2009, SOLUTIONS 2
1. 1.4. Exercise 5. Prove that if cfw_zn is a sequence of complex numbers such that
1
limn zn = z , then limn n (z1 + + zn ) = z . Give an example in which the
second limit exists but t
Math 504 Spring Term 2009, Solutions 3
x
1
1. Let the polynomials Bp (x) for k N be dened on R by B1 (x) = x 2 , Bp+1 (x) = 0 Bp (y )dy
1
(1 y )Bp (y )dy and let Bp (k ) denote the Fourier coecient (relative to the family e(kx) on L2 (S1 ).
0
(i) Prove t
MATH 504 ANALYSIS IN EUCLIDEAN
SPACES, SPRING TERM 2009, SOLUTIONS 4
1. 1.5 Exercise 1. Show that there are , in L1 (S 1 ) such that +
2 = 2 2 + 2 2.
1
1
1
Let (x) =
11

02
1
2 , (x) = x.
1 /2 1
( 2 x)dx
0
xdx =
and 1 = 1 , so RHS= 1.
2
Then + 1  =
MATH 504 SPRING TERM 2009, SOLUTIONS 5
1. Evaluate the series
(1)n1
.
(2n 1)3
n=1
e(nx)
n=0 (2in)3 .
By Problems 3, when 0 x 1, B3 (x) =
in in
n=1
n3
Moreover
n=0
e(n/4)
n3
=
and in in = 0 when n is even, is 2i when n 1 (mod 4) and
is 2i when n 3 (mod 4
Math 504 Analysis in Euclidean Spaces, Spring Term 2009, Solutions 6
1. Suppose that cfw_un is uniformly distributed (mod 1), and let c be a real number. Put
vn = un + c. Show that cfw_vn is uniformly distributed.
n
1
By Weyl I, for h N we have n m=1 e(
Math 504 Analysis in Euclidean Spaces, Spring Term 2009, Solutions 7
1. Prove that if f , g L1 (R), then f g 1 f 1 g 1 .
By denition (f g )(x) = R f (y )g (x y )dy . Hence by the triangle inequality, (f g )(x)
f (y )g (x y )dy and f g 1 = R (f g )(x
MATH 504 ANALYSIS IN EUCLIDEAN
SPACES, SPRING TERM 2009, SOLUTIONS 8
1. Let L1 (R+ , x1 dx) denote the set of functions f dened on the positive real
numbers R+ such that 0 f (x)x1 dx < . (i) Show that the convolution product
(f g )(x) = 0 f (x/y )g (y )
MATH 504 ANALYSIS IN EUCLIDEAN
SPACES, SPRING TERM 2009, SOLUTIONS 9
1. (Exercise 2.6.5 [corrected]) Prove that if f, f L1 (R) and f is continuous on R, then
limu0+ exp(2 2 t2 u)f (t) = f (x) pointwise.
By denition exp(2 2 t2 u)f (t) = R R exp(2 2 t2 u)f
MATH 504 ANALYSIS IN EUCLIDEAN
SPACES, SPRING TERM 2009, SOLUTIONS 10
1. This is based on the material in 2.7.1. Suppose that f S L1 (R) is given and
u C 2 (R) L1 (R) satises u u = f .
(i) Show that limx u(x) = limx u (x) = 0. Hint; the summability of
MATH 504 ANALYSIS IN EUCLIDEAN
SPACES, SPRING TERM 2009, PROBLEMS 1
Return by Wednesday 21st January
These exercises are essentially the same as in the text, so I have included a cross
reference.
1. 1.1. Exercise 2. Prove that if f1 , f2 . . . are real co