MAT3400/4400 H13 Solution outline
Problem 1
1a) Obviously, since we are considering the -algebra P(X), every function from X into K is
measurable, sowe dont have to bother about measurability.
Let f (
MAT3000/4000 Spring 2013 Solution of Extra exercises 3 and 4
Extra exercise 3
Let (X, A, ) be a measure space, M = cfw_f : X R | f is A-measurable and E A.
We consider the measure space (E, AE , E ),
Solution to exam in MAT3400/4400, Linear analysis with applications.
Exam date Monday, December 5, 2011.
Problem 1. The n-th Fourier coecient of f for n = 0 is
0
1
(einx )dx
2
1 einx
1
inx
e dx =
=
MAT3400/4400 - Fall 2013 - Solution of Extra Exercise 9
Let a, b R, a < b and let m denote the Lebesgue measure on [a, b]. Let k : R K be a
continuous function on R = [a, b] [a, b]. For each x [a, b],
MAT3400/4400 - Fall 2013 - Solutions of Extra Exercises 6 and 7
Extra-Exercise 6
Let p [1, ), (X, A, ) be a measure space and K denote R or C.
Let M = f : X K | f is A-measurable and Lp = cfw_f M | f
Solutions to the exam problems in MAT3400/4400, Fall 2012
Problem 1. Using Fourier series show that if f : [0, 2] C is a C 1 -function such that
2
f (0) = f (2) and 0 f (t)dt = 0, then
2
2
|f (t)|2 dt
MAT3400/4400 - Fall 2013 - Solution Extra exercise 2
We recall that J = cfw_ cfw_(a, b] | a, b R, a < b cfw_(a, ) | a R cfw_(, b] | b R,
while S denotes the collection of all subsets of R consisting o
Solution to exam in MAT3400/4400, Linear analysis with applications.
Exam date Monday, December 6, 2010.
Problem 1. The n-th Fourier coecient of f (x) = x2 on [, ] is (for
n = 0)
1
x2 einx dx
2
x2 ei
MAT3400/4400 - Fall 2013 - Solution of Extra Exercise 8
Let (X, A, ) be a measure space, K denote R or C,
M = f : X K | f is A-measurable ,
and L denote the set of all functions in M that are essentia