Quals UCLA

7/30/2019 Quals UCLA

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ANALYSIS QUALIFYING EXAM, SPRING 2012

Instructions: Work any 10 problems and therefore at least 4 from Problems 1-6 andat least 4 from Problems 7-12. All problems are worth ten points. Full credit on oneproblem will be better than part credit on two problems. If you attempt more than 10problems, indicate which 10 are to be graded.

1: Some of the following statements about sequences functions fn in L3([0, 1]) are false.

Indicate these and provide an appropriate counterexample.

(a) If fn converges to f almost everywhere then a subsequence converges to f in L3

.(b) If fn converges to f in L

3 then a subsequence converges almost everywhere.(c) If fn converges to f in measure (= in probability) then the sequence converges to fin L3.(d) If fn converges to f in L

3 then the sequence converges to f in measure.

2: Let X and Y be topological spaces and XY the Cartesian product endowed with theproduct topology. B(X) denotes the Borel sets in X and similarly, B(Y) and B(X Y).(a) Suppose f : X Y is continuous. Prove that E B(Y) implies f1(E) B(X).(b) Suppose A B(X) and E B(Y). Show that A E B(X Y).

3: Given f : [0, 1] R belonging to L1(dx) and n {1, 2, 3, . . .} define

fn(x) = n

(k+1)/nk/n

f(y) dy for x [k/n, (k + 1)/n) and k = 0, . . . , n 1.

Prove fn f in L1.

4: Let S = {f L1(R3) : f dx = 0}.(a) Show that S is closed in the L1 topology.(b) Show that S

L2(R3) is a dense subset of L2(R3).

5: State and prove the Riesz Representation Theorem for linear functionals on a (sepa-rable) Hilbert space.

6: Suppose f L2(R) and that the Fourier transform obeys f() > 0 for almost every .Show that the set of finite linear combinations of translates of f is dense in the Hilbertspace L2(R).

Problems 7-12 on next page.

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2 ANALYSIS QUALIFYING EXAM, SPRING 2012

7: Let {un(z)} be a sequence of real-valued harmonic functions on D := {z C : |z| < 1}that obey

u1(z) u2(z) u3(z) 0 for all z D.Prove that z infn un(z) is a harmonic function on D.8: Let be the following subset of the complex plane:

:= {x + iy : x > 0, y > 0, and xy < 1}.Give an example of an unbounded harmonic function on that extends continuously to and vanishes there.

9: Prove Jordans lemma: If f(z) : C C is meromorphic, R > 0, and k > 0, then

f(z)eikz dz

100k supz |f(z)|where is the quarter-circle z = Rei with 0 /2. (It is possible to replace 100here by /2, but you are not required to prove that.)

10: Let us define the Gamma function via

(z) =

0

tz1et dt,

at least when the integral is absolutely convergent. Show that this function extends toa meromorphic function in the whole complex plane. You cannot use any particularproperties of the Gamma function unless you derive them from this definition.

11: Let P(z) be a polynomial. Show that there is an integer n and a second polynomialQ(z) so that

P(z)Q(z) = zn|P(z)|2 whenever |z| = 1.

12: Show that the only entire function f(z) obeying both

|f(z)| e|z| and f n1+|n|

= 0 for all n Z

is the zero function. Here denotes differentiation.


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