Wk 13
MPL–186
RMS-03
M.Phil. DEGREE EXAMINATION –
JANUARY 2009.
Mathematics
(AY 2005-06 and CY 2006 batches only)
FOURIER TRANSFORMS
Time : 3 hours
Maximum marks : 75
Answer any FIVE questions.
Each question carries 15 marks.
1.
(a)
If m * E  0 prove that E is measurable.
(3)
(b) Prove that the collection of measurable sets
is a  - algebra.
(12)
2.
(a) Construct a measurable set which is not a
Borel set.
(11)
(b)
If f is a measurable function and if f  g a.e.
then prove that g is measurable.
(4)
Wk 13
3.
(a) Let f be a measurable function on R and
finite a.e.. . then prove that there exists a sequence fn
of continuous functions such that fn  f a.e.
(10)
(b) Let E be a set of finite measure. Then for any
given  0 there exists a step function s such that
m({ x :  E ( x )  s( x )}  .
(5)
4.
State and prove Lebesgue’s monotone convergence
theorem.
5.
(a)
If f  L (  ) , then prove that
 fd   f d .
X
X
(8)
(b)
Let f  g a.e.   on X. Then prove that
 fd   g d , where
E
 is a measure on a  - algebra
E
m and E  m .
6.
(7)
State and prove Fubini’s theorem.
2
MPL–186
Wk 13
7.
If g  L and g is continuous at a point x
(a)
then
prove
h ( x ) 


lim
  0 ( g  h ) ( x )  g( x )
that
where
e ( t ) e itx dm(t ) and   0.
(7)

(b)
g( x ) 


If
f  L1
and
fˆ  L1
and
if
fˆ(t )e ixt dm(t ) , ( x  R ) then prove that g  C 0

and f  g a.e.
(8)
Let E be measurable set in R 1 . Let M E  { fˆ  L2 :
8.
f  0 . a.e. on E}. Then prove the following :
(a)
ME
is
a
closed
translation
invariant
subspace of L2 .
(b) Every closed translation invariant subspace
of L is M E for some measurable set E in R 1
MA  MB
(c)
if
and
only
if
M (( A  B )  ( B  A ))  0 .
2
9.
If  is a non-zero complex homomorphism on L1 ,
prove that there exists a unique t  R such that
 ( f )  fˆ(t ) .
3
MPL–186
Wk 13
10.
Suppose A and C are positive constants and f is an
entire



function
such
f (z)  c e A
that
z
and
2
f ( x ) dx   . Prove that there exists for all z
F  L2 (  A, A ) such that f ( z ) 

A
F (t )e itz dt .
A
———————
4
MPL–186