Tamil Nadu Open University (TNOU) 2006 M.Phil Mathematics Fourier transforms - Question Paper

(b) If / iz) = jF(t) e^ltz dt, where 0 <A<

-A

and Fe 1} (-A, A) prove that |/()| <Ce^Z where C=)\ Fit) | dt.

M.Phil. DEGREE EXAMINATION - JUNE 2006. Mathematics Paper III - FOURIER TRANSFORMS Time : 3 hours    Maximum marks : 75

Answer any FIVE questions.

Each question carries 15 marks.

1.    (a) Prove that the interval (a, ) is measurable.

(b) Prove that the outer measure of a set is translation invariant.

2.    (a) Prove that every Borel set is measurable.

(b) Let < E_i > be a sequence of disjoint measurable sets and A any set. Then prove

that m*(AnUE_j) = 'Ym*(AnE ).

3.    (a) Prove that Lebesque measure is invariant under translation modulo 1.

(b) State and prove one version of Littlewoods third principle.

4.    (a) Let    {/} be a sequence of measurable functions on R, and suppose that

(i)    0 < (x) < f₂ (x) <....<> for every

xe R,

(ii)    f_n (x) f (x) as n > for every xe R. Then    prove that f is measurable, and f/ d_M -> jfd_M as n -> oo.

/if    R

(b) Prove that if f is a real function on R such that {x ' fix) > r} is measurable for every rational number r, then f is measurable.

5.    (a) Suppose {/} is a sequence of complex measurable functions defined a.e. on R such that

Z J¹ fn 1 < Then prove that the series

n=1 R

f {x) = Yj^fn M converges for almost all x,

71\

e I) (//) and Z jf_n di- jf dfi.

=1 R    R

(b) Let {E_R} be a sequence of measurable sets

oo

in R, such that ju( E_R ) < .

R=1

Then prove that almost all x G R 1 ie in at most finitely many of the sets E_R.

6.    (a) Let (JT, *,//) and (1,3,A) be a -finite measure spaces. Suppose Q x 3. If

<p (x) = A (Q_x), 'F (y) = n (Q^y) for every x e X and y e Y, then prove that 0 is -measurable, T is

3 - measurable , and jfid/i = jdA.

x    y

(b) Prove that in Fubinis theorem the requirement that o -finiteness of measure spaces cannot be dispensed with.

7.    If f and g are in L'(R') prove that the convolution h of/and# is also in L'{R') and that 1 < J|/||_i .

8.    (a) If / G L' then prove that f E C₀.

(b) State and prove the Inversion theorem.

9.    State and prove the Plancherel Theorem.

10.    (a) Let Fe 1} (0, ) and let

f (z) = Ji'⁷ (t) e^ltz dt, zeti* o

where n* = {z = x + iy y > 0}. Then prove that f is holomorphic in n⁺ and its restriction to horizontal lines in 7t* form a bounded set in L²(-.

3    MPL-636

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