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Kings College LondonUniversity Of London
This paper is part of an examination of the College counting
towards the award of a degree.
Examinations are governed by the College Regulations under the
authority of the Academic
Board.
ATTACHthis paper to your script USING THE STRING PROVIDED
Candidate No: . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . Desk No: . . . . . . . . . . . . . . . . . . . . . .
.
MSc and MSci Examination
7CCMMS11T Fourier Analysis (MSc Programme)
7CCMMS11U Fourier Analysis (MSci Programme)
Summer 2013
Time Allowed: Two Hours
All questions carry equal marks. Full marks will be awarded for
complete
answers to FOUR questions. Only the best FOUR questions will
count towards
grades A or B, but credit will be given for all work done for
lower grades.
You are permitted to use a Calculator.
ONLY CALCULATORS APPROVED BY THE COLLEGE MAY BE USED.
DO NOT REMOVE THIS PAPERFROM THE EXAMINATION ROOM
TURN OVER WHEN INSTRUCTED
2013 cKings College London
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- 2 - 7CCMMS11
1. A Define the concepts of linear independence and
orthonormality in a Hilbert
space. LetL2[0, 1] be the Hilbert space L2[0, 1] of
square-integrable functions
f : [0, 1] Cwith the scalar product
(f, g) =
10
f(x) g(x) dx .
Prove that if the family of functions {ek}kZ is orthonormal in
L2[0, 1], then
it must be linearly independent. Give an example to show that
the converse
is not necessarily true. [15 marks]
B Give an example of an orthonormal basis for the n-dimensional
space Cn.
Show thatL2[0, 1] is infinite-dimensional. [5 marks]
C Show that the function f(x) = ex2012
belongs to the Schwartz class S(R).Is the function f(x) =ex
2013
in the Schwartz class S(R)? [5 marks]
2. A Prove that the family {e4ikx}kZ is orthonormal in L2[0, 1].
[5 marks]
B Find a function f L2[0, 1] with fL2[0,1] = 1 that is
orthogonal to the
entire family{e4ikx}kZ. [5 marks]
C LetMbe the set of all functions in L2[0, 1] that are
orthogonal to the entire
family {e4ikx}kZ. Prove that M is a linear subspace of the
vector space
L2[0, 1]. Show that it is infinite-dimensional. [15 marks]
See Next Page
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- 3 - 7CCMMS11
3. A Prove that the family {e2ikx}kZ is orthonormal in L2[0, 1].
[5 marks]
B Prove that jZ
1
2(2j+ 1)2
e2i(2j+1)x converges inL2[0, 1]. [5marks]
[You may use Bessels inequality without proving it.]
C Compute the Fourier coefficients F(k) =
10
F(x) e2ikx dx, k Z, of the
functionF : [0, 1] C defined byF(x) =x forx [0, 1/2] andF(x) =
1x
forx [1/2, 1]. [5 marks]
D FindjZ
1
(2j+ 1)22e2i(2j+1)x in L2[0, 1]. [5 marks]
E Show that
j=0
1(2j+ 1)4
= 4
96. [5 marks]
4. A Letf : R C be a continuously differentiable function that
is periodic with
period 1. If
f(k) =
10
f(x) e2ikx dx, k Z,
are its Fourier coefficients, prove that lim|k|
{|k f(k)|}= 0. [8 marks]
B Show that for the function
f0(x) =
1, 0 x
