Math 5052 - Homework 1
Due 1/22/09
1. (Problem 66, page 142) Let 1
n
1 cn t
be the Maclaurin series for (1 t)1/2 .
(a) The series converges absolutely and uniformly on compact subsets of (1, 1), as does
the termwise dierentiated series ncn tn1 . Thus, if
gm
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uw
7
29m
ff
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Problem 2. Suppose T is a bounded linear map from real-valued U to itself. Define, T on complexvalued
U7 by setting
TO + 139} == (W) + Tm
far a. f, g 6 LP. Show that T maps complex-valued LP to i
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;;E_?£? Mfr? 2*} UK S?{} Skid xmfé'fii
._fv§{k§~m§:{y)} 5,; Mi Ma} s? W 113 5?)
L j a _ Ni 86! . .-
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5
Homework 7
Math 5052
Amos Dettonville
Exercise 1 (8.15 in Folland). Let sinc x := (sin x)/x and sinc 0 = 1.
(a) If a > 0, then 1
[a,a] (x) = 1[a,a] (x) = 2a sinc 2ax.
Solution. For x = 0,
a
1
[a,a] (x) =
e2ix d
a
=a
1 2ix
e
2ix
=a
1
=
e2iax e2iax
2ix
sin
Math 5052 - Homework 3
Due 2/05/09
1. (Problem 27, page 164) There exist meager subsets of R whose complements have Lebesgue
measure zero.
2. (Problem 30, page 164) Let Y = C([0, 1]) and X = C 1 ([0, 1]), both equipped with the uniform
norm.
(a) X is not
Math 5052 - Homework 4
Due 2/12/09
(Its not necessary to write up solutions to the many exercises referenced in the ones assigned below.)
1. (Problem 49, page 170) Suppose that X is an innite-dimensional Banach space.
(a) Every nonempty weakly open set in
Math 5052 - Homework 5
Due 2/19/09
1. (Problem 67, page 178, The Mean Ergodic Theorem). Let U be a unitary operator on the
Hilbert space H, M = cfw_x : U x = x, P the orthogonal projection onto M (Exercise 58), and
Sn = n1 n1 U j . Then Sn P in the strong
Math 5052 - Homework 6
Due 2/26/09
1. (Problem 12, page 187) If p = 2, the Lp norm does not arise from an inner product on Lp , except
in trivial cases when dim(Lp ) 1. (Show that the parallelogram law fails.)
2. (Problem 13, page 187) Lp (Rn , m) is sepa
Math 5052 - Homework 2
Due 1/29/09
1. (Problem 7, page 155) Let X be a Banach space.
(a) If T L(X, X) and I T < 1 where I is the identity operator, then T is invertible; in
fact, the series (I T )n converges in L(X, X) to T 1 .
0
(b) If T L(X, X) is inver
Math 5052 - Homework 8
Due 03/26/09
1. (Problem 1, page 215) Let X be a LCH space, Y a closed subset of X (which is an LCH space in
the relative topology), and a Radon measure on Y . Then I(f ) = (f |Y ) d is a positive linear
functional on Cc (X), and th
Math 5052 - Homework 7
Due 03/05/09
1. (Problem 27, page 196. Hilberts Inequality) The operator T f (x) = 0 (x + y)1 f (y) dy
satises T f p Cp f p for 1 < p < , where Cp = 0 x1/p (x + y)1 dx. (For those who
know about contour integrals: Show that Cp = csc
Math 5052 - Homework 9
Due 04/02/09
1. (Problem 16, page 224) Suppose that I C0 (X, R) and I + , I are the functionals constructed
in the proof of Lemma 7.15. If is the signed Radon measure associated to I , then the positive
and negative variations of ar
Math 5052 - Homework 10
Due 04/09/09
1. (Problem 2, page 239) Observe that the binomial theorem can be written as follows:
(x1 + x2 )k =
| |= k
k!
x
!
(x = (x1 , x2 ), = (1 , 2 ).
Prove the following generalizations:
(a) The multinomial theorem: If x Rn
Math 5052 - Homework 11
Due 04/16/09
1. (Problem 12, page 254) Work out the analogue of Theorem 8.22 for the Fourier transform on Tn .
2. (Problem 13, page 254) Let f (x) =
1
2
x on the interval [0, 1), and extend f to be periodic on R.
(a) f (0) = 0, an
Math 5052 - Homework 12
Due 04/23/09
1. (Problem 23, page 256) In this exercise we develop the theory of Hermite functions.
(a) Dene operators T, T on S(R) by T f (x) = 21/2 [xf (x) f (x)] and T f (x) =
21/2 [xf (x) + f (x)]. Then (T f )g = f (T g) and T